Quantum Computing Environments 3030897451, 9783030897451

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Sitharama S. Iyengar Mario Mastriani K. J. Latesh Kumar   Editors

Quantum Computing Environments

Quantum Computing Environments

Sitharama S. Iyengar • Mario Mastriani K. J. Latesh Kumar Editors

Quantum Computing Environments

Editors Sitharama S. Iyengar Florida International University Miami, FL, USA

Mario Mastriani Quantum Information Group Florida International University Aventura, FL, USA

K. J. Latesh Kumar Quantum Information Group Florida International University Miami, FL, USA

ISBN 978-3-030-89745-1 ISBN 978-3-030-89746-8 https://doi.org/10.1007/978-3-030-89746-8

(eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to my dear wife Manorama Iyengar who passed away several weeks back

Preface

The amalgamation of computer science, information theory with quantum physics, and current trending technologies is a striking concept. The mission of this book is to provide various true computer science concepts on quantum computing to facilitate the smooth transition from conventional computing to quantum. The goal is also to build an exciting research and teaching area reachable to broad multidisciplinary audiences. In particular, we focus to help and close the gap between the physicsbased quantum computing with computer science technologies and conventional computing. The book also provides a comprehensive insight into the quantum mechanics and quantum computer science technique tools that have evolved and impacted in flourishing business through the quantum computing era. It also includes chapters that discuss the most primitive quantum schemes such quantum internet, quantum radar, and quantum compiling. Additionally, the quantum entanglement is described in the most simple way, and a powerful approach is applied to narrate the quantum information processing. This book is designed for undergraduate, graduate, and postgraduate students but not limited to computer science. It greatly offers an extensive knowledge on various quantum techniques alongside computer science methods and highly beneficial to academicians and researchers. The intent of mastering the book is presented below: • A strong grounding in quantum physics with computer science (conventional computing) • Quantum basics, construction, and analysis of quantum algorithm

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Preface

• Detailed approach of quantum computing and computing environment fundamentals of the computer science techniques that gives the reader a wider coverage of how multidisciplinary concepts are encapsulated with quantum physics along with computer science and information theory. Miami, FL, USA Aventura, FL, USA Miami, FL, India

Sitharama S. Iyengar Mario Mastriani K. J. Latesh Kumar

Acknowledgments

The authors would like to thank their colleagues and friends for their continuous support. Also, To all the frontline workers during COVID19 (Health, Civic, and Law and Order)

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Contents

1

Quantum Information Processing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ritajit Majumdar

1

2

Quantum Compiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marco Maronese, Lorenzo Moro, Lorenzo Rocutto, and Enrico Prati

39

3

The Future Quantum Internet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabio Cavaliere, Rana Pratap Sircar, and Tommaso Catuogno

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4

Quantum Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Kadir Durak, Zeki Seskir, and Bulat Rami

5

Quantum Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Do Ngoc Diep

6

Future Perspectives of Quantum Applications Using AI . . . . . . . . . . . . . . . . 193 H. U. Leena and R. Lawrance

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

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About the Editors

Sitharama S. Iyengar is currently the Distinguished University Professor, Founding Director of the Discovery Lab, and Director of the US Army-funded Center of Excellence in Digital Forensics at Florida International University, Miami. He has been involved in research and education in high-performance intelligent systems, data science and machine learning algorithms, sensor fusion, data mining, and intelligent systems. Since receiving his Ph.D. degree in 1974 from MSU, USA, he has directed over 65 Ph.D. students, 100 Master’s students, and many undergraduate students who are now faculty at major universities worldwide or scientists or engineers at national labs/industries around the world. He has published more than 600 research papers, has authored/co-authored and edited 26 books. His h-index is 63 with over 19,000 citations and is among the list of top 2% cited scholars of Stanford study this year. His books are published by MIT Press, John Wiley and Sons, CRC Press, Prentice Hall, Springer Verlag, IEEE Computer Society Press, etc. One of his books titled Introduction to Parallel Algorithms has been translated into Chinese. During the last 30 years, Dr. Iyengar has brought in over 65 million dollars for research and education. Recently, he has received $2.25 million (USD) funding from the United States Army Research Office to develop a Center of Excellence in Digital Forensics in collaboration with HBCU institutions. This is considered to be one of the biggest achievements in his whole career in FIU, many such grants are in the past. He has been providing the students and faculty with a vision for active learning and collaboration at Jackson State University, Louisiana State University, Florida International University, and across the world. Dr. Iyengar’s career is a distinguished one, marked by his incredible record of success in groundbreaking research, inspirational teaching, and excellence in community service. It is his consistent drive to fight for and promote the minority and underrepresented groups which is his passion. Through his national and international contributions, he has consistently provided opportunities for minority students and underrepresented groups to participate in his research endeavors and to develop local, state, and national programs to promote minority and underrepresented groups in computer science and STEM xiii

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About the Editors

education programs. He has developed enormously successful models and programs that have been replicated in universities around the world. Through NSF, he developed a comprehensive network of computer education and coordinated computer science workshops and short courses which introduced computer science to over 5000 minority students and assisted minority faculty in advancing educational concepts and research. In his most current initiatives in providing computer science advising and student tutors, he has been able to significantly increase retention rates at his university in STEM areas. Dr. Iyengar has also provided outreach to industry and to a variety of groups in the local high school community. Industry affiliations have resulted in internships with multiple Fortune 500 companies for his students. Informally as well as formally through his NSF-sponsored Research Experience for Teachers, he has worked with local science teachers in high schools and middle schools to open up many of his labs for weekend work and interaction with the students to participate with undergraduate students in areas such as computer hardware, cyber security, and robotics. He has invited and sponsored the Girls Who Code organization to provide summer seminars for local high school girls—a tremendous success in preparing and recruiting high school girls for STEM careers. His research has been funded by National Science Foundation (NSF), Defense Advanced Research Projects Agency (DARPA), Multi-University Research Initiative (MURI Program), Office of Naval Research (ONR), Department of Energy/Oak Ridge National Laboratory (DOE/ORNL), Naval Research Laboratory (NRL), National Aeronautics and Space Administration (NASA), US Army Research Office (URO), and various state agencies and companies. He has served in the US National Science Foundation and the National Institute of Health Panels to review proposals in various aspects of computational science and has been involved as an external evaluator (ABET-accreditation) for several Computer Science and Engineering Departments across the country and the world. Dr. Iyengar has also served as a research proposal evaluator for the National Academy. Dr. Iyengar, a computer scientist of international repute, is a pioneer in the field and has made fundamental contributions in the areas of information processing for sensor fusion networks, robotics, and high-performance algorithms, all relevant to critical event detection systems as seen in the following: 1. Co-inventor of the Brooks–Iyengar algorithm for noise tolerant distributed control which bridges the gap between sensor fusion and Byzantine fault tolerance, providing an optimal solution to the fault-event disambiguation problem in sensor networks (1996) 2. Co-inventor of a novel, paradigm shifting method for grid coverage of surveillance and target location in distributed sensor networks (2002) 3. Provided seminal work for automated analyses and interpretation of satellite imagery of the ocean and other unknown terrain (1994) 4. Co-invented the Cognitive Information Processing Shell, a complex event processing architecture and engine, which recognizes and responds to complex patterns in mission critical, real-time applications (2010)

About the Editors

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5. Solved an open problem in graph recognition, laying foundation for fast parallel computing for large-scale data sets (1988) The impact of his research contributions can be seen in places like Raytheon, Telecordia, Motorola, the United States Navy, DARPA agencies, etc. Dr. Iyengar is a Member of the European Academy of Sciences, a Life Fellow of the Institute of Electrical and Electronics Engineers (IEEE), a Fellow of the Association of Computing Machinery (ACM), a Fellow of the American Association for the Advancement of Science (AAAS), a Fellow of the Society for Design and Process Science (SDPS), and a Fellow of the American Institute for Medical and Biological Engineering (AIMBE). He has received various national and international awards including the outstanding Test of Time Research (for his seminal work which has impacted billions of computer and internet users worldwide) and Scholarly Contribution Award from 2019 IEEE Congress on Cybermatics, the Times Network NRI (Non-Resident Indian) of the Year Award for 2017, most distinguished Ramamoorthy Award at the Society for Design and Process Science (SDPS 2017), the National Academy of Inventors Fellow Award in 2013, and the NRI Mahatma Gandhi Pradvasi Medal at the House of Lords in London in 2013 among others. He was awarded Satish Dhawan Chaired Professorship at IISc, then Roy Paul Daniel Professorship at LSU. He has received the Distinguished Alumnus Award of the Indian Institute of Science. In 1998, he was awarded the IEEE Computer Society’s Technical Achievement Award and is an IEEE Golden Core Member. Professor Iyengar is an IEEE Distinguished Visitor, SIAM Distinguished Lecturer, and ACM National Lecturer. In 2006, his paper entitled “A Fast-Parallel Thinning Algorithm for the Binary Image Skeletonization” was the most frequently read article in the month of January in the International Journal of High-Performance Computing Applications. His innovative work called the Brooks–Iyengar algorithm along with Professor Richard Brooks from Clemson University is applied in industries to solve the real-world applications. Dr. Iyengar’s work had a big impact; in 1988, he and his colleagues discovered “NC algorithms for Recognizing Chordal Graphs and K-trees” [IEEE Trans. on Computers 1988]. This breakthrough result led to the extension of designing fast parallel algorithms by researchers like J. Naor (Stanford), M. Naor (Berkeley), and A. A. Schaffer (AT&T Bell Labs). Professor Iyengar earned his undergraduate and graduate degrees at UVCE-Bangalore, and the Indian Institute of Science, Bangalore, respectively, and a doctoral degree from Mississippi State University. Dr. Iyengar has been a Visiting Professor or Scientist at Oak Ridge National Laboratory, Jet Propulsion Laboratory, Naval Research Laboratory, and has been awarded the Satish Dhawan Visiting Chaired Professorship at the Indian Institute of Science, the Homi Bhaba Visiting Chaired Professor (IGCAR), and a professorship at the University of Paris-Sorbonne. Mario Mastriani received the degree of Electronics Engineering: Automatic Control in 1989, the Ph.D. degree of Electronics Engineering: Despeckling of Satellite SAR images in 2006, the Ph.D. degree of Computer Sciences: Image Compression in 2009, the Ph.D. degree of Sciences and Technology: Video Compression in

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About the Editors

2011, the Ph.D. degree of Informatics Sciences: Quantum Computing in 2014, and the Post Doc of Computer Sciences: Quantum Computing in 2016. He was/is a reviewer of the following IEEE’s journals: T. on Neural Networks, Signal Processing Letters, T. on Image Processing, T. on Signal Processing, T. on Medical Imaging, T. on Geoscience and Remote Sensing, Geoscience and Remote Sensing Letters, T. on Biomedical Engineering, Communications Letters, T. on Fuzzy Systems, T. on Multimedia, T. on Nanobioscience, T. on Instrumentation and Measurement, T. on Systems, Man, and Cybernetics: Systems, and T. on Circuits and Systems for Video Technology. Besides, he is a reviewer of Springer-Nature: Journal of Digital Imaging, and Springer-Nature: Quantum Information Processing, Elsevier: Information Fusion; SPIE: Journal of Optical Engineering, SPIE: Journal of Electronic Imaging; Taylor & Francis: International Journal of Remote Sensing, Taylor & Francis: Remote Sensing Letters, Taylor & Francis: International Journal of Computers and Applications; IET: Electronics Letters, IET: Software; Latin American Applied Research Journal; Wiley: International Journal of Circuits Theory and Applications; Editorial Science (South Africa): Journal of Mathematics and Computer Science Research; ACM Computing Surveys; Editorial Old City Publishing (USA): International Journal of Unconventional Computing; MDPI (Switzerland): Entropy, MDPI (Switzerland): Sensors, MDPI (Switzerland): Mathematical and Computational Applications; and World Scientific (Singapore): Modern Physics Letters B. He was a reviewer of three books for zbMATH of Cambridge University Press. He have received scholarships for all his postgraduate studies, even for the Postgraduate Course of Nuclear Engineering awarded by National Commission of Atomic Energy. He is an author of four books and a co-author of other two books. He has three patents. He is an author of 70 scientific articles. He was a professor of 12 universities in three countries. He was a research advisor in several Ph.D. thesis. He has worked for several government agencies in the area of SAR imaging, as well as multi- and hyperspectral imagery. He also worked in medical images in numerous institutions in the country and abroad. He was a Senior Research Associate in Superresolution of Multi-spectral Imagery in Satellogic. He was co-founder of five startups in Quantum Technology, all in the USA. His current interest is Quantum Communications (Entanglement, Teleportation, QKD, Quantum Internet), and Quantum Signal and Image Processing. K. J. Latesh Kumar received his Ph.D. and Master’s from the University of Malaysia, and he is currently in the Department of Computer Science, Siddaganga Institute of Technology, India. Prior to this, he served in IT industries for 10 years. Dr. Latesh is a pioneer in the field of cloud-based storage computing, cyber security, storage protocol engineering, and high-performance network computing. He started his carrier as a Software Developer at Lionbridge Technologies, later associated to Hewlett Packard, India and US as Technical Solution Consultant. His clients included Data Protection, Automobile, Airports, and Banking. Few years later, he moved to NetApp and served as Product Manager of Network Attached Storage Business Unit and also in the Engineering of NFS and pNFS protocol at Bangalore and California. Dr. Latesh has published numerous papers in various journals,

About the Editors

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conferences, book chapters, and technical articles in reputed medias like Springer, IEEE, ACM, Elsevier, IT next, HP, and NetApp Technical Library worldwide. He has presented 100+ technical talks and presentations across globe in the field of cloud-based storage computing, data protection, best cyber practices, Linux server management, and agile method of Software Developments. Dr Latesh is an active Research Consultant, he has investigated several projects in the area of UNIX kernel building, storage, data security, and cloud computing with companies ABB, Dataknots, IdeaInfinity, Startups, and currently, he is researching in cyber security and cloud-based storage computing. Dr. Latesh has received several awards including distinguished “Excellency for Engineering and Technical Commitment on UNIX Server Management” at Hewlett Packard and Best International Journal award at PSRC, Indonesia. He is a state professional badminton player at India.

Chapter 1

Quantum Information Processing Ritajit Majumdar

Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy. —Richard P. Feynman

Consider a molecule with M atoms and n electrons. Simulation of such a system is essential to study its energy spectrum, which, in its turn, would lead to the reaction rate and other properties of the system. However, the number of parameters of such a system scales ∼ O(2n ) and hence becomes intractable by modern supercomputers even for a moderate value of n. Many such problems of interest are found in physics, chemistry, computer science, economics, etc., that scale exponentially with the number of parameters and hence cannot be efficiently simulated in a classical computer. However, nature solves such problems regularly, and nature is not classical. This led Feynman to hypothesize that it may be possible to tackle these problems by changing our mode of computation to make use of quantum mechanical properties [1]. However, the components of modern computers also consist of electrons and other quantum mechanical particles. What then makes a quantum computer quantum? A computer is said to be a quantum computer if it actively makes use of quantum mechanical phenomena such as superposition, and entanglement for its computation. Such phenomena are not observable in the macroscopic world. Current computers, although are, in essence, made of quantum mechanical particles, do not make use of these phenomena. Hence the current computers cannot be called quantum. This chapter provides a brief introduction to the exciting field of quantum information processing. We cover basic quantum mechanics and introduce the language of quantum information processing. We review necessary linear algebra

R. Majumdar () Indian Statistical Institute, Kolkata, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Iyengar et al. (eds.), Quantum Computing Environments, https://doi.org/10.1007/978-3-030-89746-8_1

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R. Majumdar

required for this study. We then introduce the concept of entanglement, and Bell Inequality. A review of quantum gates is provided next, followed by an introduction to some basic quantum algorithms. This is not a complete review of this field, but we hope that it is sufficient to get an interested reader to get started in this field. We have provided ample references for the interested readers to follow through so that they can learn the various subtopics in further detail.

1.1 Schrødinger Equation The evolution of a wavefunction ψ(x, t) is governed by the Schrødinger equation: i h¯

∂ψ(x, t) = H ψ(x, t), ∂t

(1.1)

where H , called the Hamiltonian of the system, encapsulates the information of the kinetic and potential energy of the system, and h¯ is Plank’s constant. Any state ψ(x, t), which satisfies the Schrødinger equation, is a valid quantum state. We shall briefly look into some properties of a quantum state that shall be of importance in the later parts: 1. Schrødinger equation is linear. In other words, if ψ1 (x, t) and ψ2 (x, t) are two valid quantum states, then so is c1 ψ1 (x, t) + c2 ψ2 (x, t), where c1 , c2 are scalar. Such a state c1 ψ1 (x, t) + c2 ψ2 (x, t) is termed as the superposition of ψ1 (x, t) and ψ2 (x, t). 2. Solving Eq. 1.1 gives ψ(t) = exp(

−i H.t)ψ(0), h¯

(1.2)

i.e., the wavefunction at some time t is obtained by acting the operator exp( −i h¯ H.t) on the wavefunction at time t = 0. This operator is unitary since the complex conjugate of this operator, exp( hi¯ H.t), is its inverse. 3. The probability of finding the quantum state ψ(x, t) within some position window [a, b] at time t is given by 

b

ψ(x, t)dx.

(1.3)

a

Since the particle must exist at some point in the space, the probability of finding the particle in all space should be 1. This is called the Normalization Condition and is mathematically represented as 

∞ −∞

ψ(x, t)dx = 1.

(1.4)

1 Quantum Information Processing

3

The Normalization Condition, in turn, implies that the norm of any quantum state ψ must be unity. For more details on quantum mechanics, we refer the readers to [2, 3]. We shall, henceforth, dive into the abstraction and terminologies used in quantum information processing.

1.2 Linear Algebra for Quantum Information Processing and Quantum Computing A quantum state can be a single, or a system of electrons. It can also be the polarization of light, or spin particles. However, for our study, we shall not bother ourselves with the physical realization of a quantum state. Rather, we shall represent a quantum state as an abstract mathematical system. However, before going into those abstractions, we shall briefly review some necessary linear algebra.

1.2.1 Linear Vector Space A linear vector space V over a field Fq consists of a set of vectors {a, b, c . . . } that are closed under vector addition and scalar multiplication [4]. The properties of a linear vector space are: Closure under vector addition: If a, b ∈ V, then a + b ∈ V. Addition is associative: If a, b, c ∈ V, then (a + b) + c = a + (b + c). Addition is commutative: For a, b ∈ V, a + b = b + a. Existence of identity element under vector addition: ∃ a zero vector 0 such that ∀ v ∈ V, 0 + v = v. 5. Existence of inverse element under vector addition: ∀v ∈ V, ∃ a unique −v such that v + (−v) = 0. 6. Closure under scalar multiplication: ∀ a ∈ Fq , and ∀ v ∈ V, av ∈ V. 7. Distributive property: For some scalar c, d ∈ Fq , and a, b ∈ V, 1. 2. 3. 4.

c(a + b) = ca + cb, (c + d)a = ca + da. 8. Associativity w.r.t scalar multiplication: For scalar c, d ∈ Fq , and a ∈ V, c(da) = (cd)a. A set of vectors {g0 , g1 , . . . , gp } is said to span a vector space V, if ∀v ∈ V, ∃a0 , . . . , ap ∈ Fq such that

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v = a0 g0 + a1 g1 + . . . + ap gp . Moreover, if {g0 , g1 , . . . , gp } are linearly independent, then they form a basis for the vector space V. A set of vectors {g0 , g1 , . . . , gp } are said to be linearly independent if p i=0

αi gi = 0, αi ∈ Fq ⇒ αi = 0∀i.

The number of vectors in the basis is called the dimension of the vector space. In our above example, the vector space V is p + 1-dimensional. Therefore, a vector space is completely defined by its basis vectors since any element of the vector space can be expressed as a linear combination of them. A set of vectors U ⊂ V is a subspace of the vector space V if U satisfies all the properties of a vector space for the vector addition and scalar multiplication operators defined over V, but every element in U also belongs to V. Therefore, U is a vector space in its own right, but its elements belong to a (usually) bigger vector space V. If U is a subspace of V over a field Fq , then it is often represented as U ≤q V.

1.2.2 Hilbert Space Any member of a linear vector space is usually represented as a column matrix: ⎞ α0 ⎜ ⎟ α = ⎝ α1 ⎠ , .. .α ⎛

p

where αi ∈ Fq , 0 ≤ i ≤ p. Henceforth, we shall consider the complex field, i.e., C. In quantum information processing, the bra-ket notation, introduced by the Nobel laureate Paul Dirac, is used to represent vectors. In this notation, the vector α is represented as |α . | is called the ket notation. This is just a way of representation without any change in the mathematical implications. The complex conjugate transpose vector (α † ), represented by the bra notation, is 

α| = α0∗ α1∗ . . . αp∗ . Any element of a vector space is a vector. However, in quantum information, we shall like to have information such as the norm of a vector, the overlap of one vector with another, etc., which are, in general, scalar. Therefore, it is necessary to extend

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our underlying space to allow such operations and accommodate those information. First we shall define the inner product between two vectors. Definition 1 The inner product of two vectors |α , |β is a function that satisfies the following properties: (i) α|β = β|α ∗ . (ii) For some scalar a, α|aβ = a α|β . (iii) α|α ≥ 0, with equality if and only if |α = 0. There can be multiple functions that satisfy the properties of inner product. In quantum information processing, the inner product between two vectors ⎛

⎞ α0 ⎜ ⎟ |α = ⎝ α1 ⎠ .. .α p

⎛

and

⎞ β0 ⎜ ⎟ |β = ⎝ β1 ⎠ .. .β p

is defined as ⎛ ⎞ p  β0

⎜ β1 ⎟  ∗ ∗ ∗ ∗ α|β = α0 α1 . . . αp ⎝ ⎠ = αi βi . .. i=0 .β p

The norm of a vector |α , || |α ||√2 , is, therefore, the square root of the inner product of the vector with itself, i.e., α|α . Two vectors |α and |β are said to be orthogonal if α|β = 0. A Hilbert space can be defined as a complex vector space (i.e., Fq = C), where inner product is defined. From the point of view of linear algebra, Hilbert space has some other properties as well. However, for the purpose of studying quantum information processing, it suffices to stick to this definition of Hilbert space. For more detail on this topic, we refer the readers to [5, 6].

1.3 Postulates of Quantum Mechanics In this section we discuss the postulates of quantum mechanics in detail. Some of these postulates have been hinted at while discussing the Schrødinger equation, and we encourage the readers to try to find correspondence between the two approaches. Postulate 1 Every quantum state is associated with a Hilbert space, and the quantum state is a unit vector in that Hilbert space. If |ψ is a quantum state, then ∃ Hilbert space Hψ such that |ψ ∈ Hψ and ψ|ψ = 1. Postulate 2 Any operator U acting on a quantum state |ψ must be unitary. The evolution of |ψ from time t1 to t2 is governed by |ψ(t2 ) = U (t2 , t1 ) |ψ(t1 ) .

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Since a quantum state is a vector, U must be a matrix. Every matrix is not invertible [4]. However, for unitary matrices, U † = U −1 , where U † implies the complex conjugate transpose of U . Therefore, any quantum operation is invertible, and hence quantum computation is reversible. Moreover, the inner product of U |ψ

with itself gives ψ| U † U |ψ = ψ|ψ . In other words, unitary operation preserves the norm of a quantum state. Postulate 3 Given a quantum state |ψ and a set of measurement operators {Mm }, the probability of obtaining outcome m is † Mm |ψ

p(m) = ψ| Mm

and the post-measurement state is

M √ m |ψ . p(m)

The measurement operators follow that



Mi = I . We shall be particularly

i

interested in projection operators. A measurement operator P is a projector if P 2 = P . Henceforth, by measurement, we shall imply projection. A Hilbert space can have multiple, often infinite, number of bases. If {|b1 , |b2 . . . |bd } is a basis of a Hilbert space of dimension d, then so is {|c1 , |c2 . . . |cd }, where |ci = U |bi , 1 ≤ i ≤ d, for some unitary operator U . A quantum state |ψ can be represented as |ψ = λb1 |b1 + λb2 |b2 + . . . + λbd |bd

= γc1 |c1 + γc2 |c2 + . . . + γcd |cd

= ..., where λ and γ are scalars. Measurement is always associated with some basis. A basis {|b1 , |b2 , . . . , |bd } is said to be orthonormal if ∀i, j , bi |bj = δij . Born Rule If a quantum state |ψ is measured in an orthonormal basis {bi }, then the measurement outcome is one of the basis states |bi with probability | bi |ψ |2 . In other words, the state |ψ collapses to the state |bi with probability |λbi |2 . If this state is measured again in the same basis, then the outcome is |bi with certainty, which satisfies the property of a projector. λbi is termed as the probability amplitude. Postulate 4 The Hilbert space associated with multiple quantum states is the tensor product of the Hilbert space of the individual quantum states. If |ψ1 , |ψ2 , . . ., |ψn

are n quantum states, then their composite system is |ψ1 ⊗ |ψ2 ⊗ . . . ⊗ |ψn .

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A quantum bit (or qubit) is a 2-dimensional quantum system. In accordance to classical computation, the computational basis is denoted as |0 =

1 0

|1 =

0 . 1

A general qubit is a superposition of those basis states |ψ = α |0 + β |1 ,

α, β ∈ C,

|α|2 + |β|2 = 1.

 T  T If |ψ = a b and |φ = c d , then a c |ψ ⊗ |φ = ⊗ b d ⎛ ⎞ ⎛ ⎞ c ac ⎜a d ⎟ ⎜ad ⎟ ⎟ ⎜ ⎟ =⎜ ⎝ c ⎠ = ⎝ bc ⎠ . b d bd Therefore the dimension of the 2-qubit composite system is 22 . In general, an n-qubit composite system resides in a 2n -dimensional Hilbert space. Another important basis is the Hadamard basis, which is denoted as 1 1 |+ = √ 2 1

1 1 |− = √ . 2 −1

We urge the readers to verify that |+ = √1 (|0 + |1 ) and |− = √1 (|0 − |1 ). 2 2 A quantum state can be represented in both these bases. For example, |ψ = a |+ + b |−

a−b a+b = √ |0 + √ |1 . 2 2 We shall look into an example for further clarity. Consider a quantum state  1 2 |ψ = √ |+ + |−

3 3 √ √ 1+ 2 1− 2 = √ |0 + √ |1 . 6 6

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On the one hand, if we measure the state |ψ in the Hadamard basis,  then the 1 2 √ outcome is |+ with probability | | = 0.33 or |− with probability | 23 |2 = 0.66. 3 On the other hand, if we measure the state in the computational basis, then the outcome is |0 with probability 0.97 or |1 with probability 0.03. However, this probability is not obtainable from a single measurement of a qubit. In order to obtain the probability, one must prepare multiple identical copies and measure them. The fraction of the outcomes will give an estimate of the probability amplitudes. Lemma 1 If two known states |ψ and |φ are orthogonal, then there exists a measurement that can distinguish them. Proof Since |ψ and |φ are orthogonal, we can measure the states in the basis {|ψ , |φ }. From Born Rule, if the state is |ψ , then the measurement outcome is always |ψ , and vice versa.  Lemma 2 It is not possible to distinguish two non-orthogonal states |ψ and |φ

in a single measurement. Proof Since |ψ and |φ are non-orthogonal, |ψ ψ| + |φ φ| = I does not form a basis. If we choose some other basis {|b1 , |b2 }, then the quantum states can be represented in this basis as |ψ = a |b1 + b |b2 , |φ = c |b1 + d |b2 , for some a, b, c, d = 0 ∈ C. Upon measurement, the resultant state is either |b1 or |b2 . However, both |ψ and |φ have non-zero probability to give the outcome |b1

or |b2 and hence it is not possible to determine which state produced the outcome. Therefore, a single measurement cannot distinguish two non-orthogonal states.  Ideally, it requires infinite measurements to uniquely distinguish two nonorthogonal states. We refer the readers to [7, 8] for further information on quantum state discrimination. However, it is possible to probabilistically distinguish among non-orthogonal quantum states by positive operator valued measure (POVM).

1.3.1 Positive Operator Valued Measure (POVM) We touch upon POVM very briefly in this section. We shall motivate this type of measurement using an example. Consider two non-orthogonal quantum states |0 and |+ that cannot be distinguished uniquely using a single projective measurement. However, let us consider the following measurement operators: M1 = |1 1|

M2 = |− −|

M 3 = I − M1 − M2 .

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The third measurement operator is necessary since



Mi = I . If after

i

measurement, the triggered outcome is M1 , then one can infer that the state was |+ (since 0|1 = 0). Similarly, if the triggered outcome of the measurement is M2 , then one can infer that the state was |0 . However, if the triggered outcome is M3 , then nothing can be inferred about the original quantum state. Therefore, a generalized measurement is where the measurement operators  M1 , M2 , . . . MK follow the necessary condition Mi = I , but Mi Mj = 0 ∀i = j . i

The post measurement state for such a measurement cannot be uniquely determined. This type of measurement is useful where the objective is to distinguish among nonorthogonal states with high probability without caring for the post-measurement state. The readers can look into different types of quantum measurements in [9, 10].

1.4 Entanglement The composite system of two quantum states |ψ and |φ is denoted as |ψ ⊗ |φ . However, in literature, the tensor notation is often omitted, and the composite state is represented as |ψφ . We shall, henceforth, omit the tensor notation unless necessary. Consider the quantum states |ψ = a |0 + b |1 and |φ = c |0 + d |1 . Their composite state will be |ψφ = (a |0 + b |1 ) ⊗ (c |0 + d |1 ) = ac |00 + ad |01 + bc |10 + bd |11 . Such a quantum state, which can be represented as the tensor product of two component qubits, is called separable state. On the contrary, recall that the superposition of two quantum states is also a valid quantum state. Therefore, since |00 and |11 are valid quantum states, so is α |00 + β |11 , |α|2 + |β|2 = 1. Lemma 3 It is not possible to write the state |ψ = α |00 + β |11 as a tensor product of two component states. Proof Let us assume, on the contrary, that it is possible to write the state |ψ as the tensor product of two component qubits a |0 + b |1 and c |0 + d |1 . Then, α = ac

β = bd

ad = bc = 0.

However, if ad = 0, then either a = 0 or d = 0. But if a = 0, then α = 0, and if d = 0, then β = 0. A similar argument can be made from the observation that bc = 0. Therefore, α |00 + β |11 = (a |0 + b |1 ) ⊗ (c |0 + d |1 ), for any a, b, c, d ∈ C. 

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Such a quantum state, which is not separable, is said to be entangled. Consider an entangled state with α = β = √1 , where the first qubit is with Alice and the 2 second qubit is with Bob. If Alice measures her qubit in the computational basis, then the state of her qubit collapses to |0 or |1 with probability 0.5. However, if her state collapses to |0 (|1 ), then the state of Bob will collapse to |0 (|1 ) as well with certainty. In other words, even without measuring Bob’s qubit, Alice has changed its state and she knows precisely what the state of Bob’s qubit is from her measurement outcome. We encourage the readers to convince themselves that this is not possible for a separable state. Therefore, two entangled states share some correlation between them. Correlations exist in our day to day life as well. For example, (i) if we toss a coin and find head, we can state with certainty that tail did not appear; (ii) if the sum of two variables is 10, and the value of one variable is provided, then the other variable can be determined with certainty. However, the correlation in an entangled state seems to be a bit different from classical correlation. Although each state has an equal probability to be measured |0 or |1 , measurement of one qubit disturbs the state of the other irrespective of the spatial distance between them. This leads to an argument that perhaps quantum theory is incomplete, and there is some hidden variable λ, unknown to us, which, when known, can explain the waveform collapse [11]. In the next subsection, we shall discuss Bell’s theorem [12], which shows that quantum correlation is indeed different from classical correlation.

1.4.1 Bell’s Theorem Consider the following four entangled states: 1 |φ ± = √ (|00 ± |11 ), 2 1 |ψ ± = √ (|01 ± |10 ). 2 The readers are encouraged to show that these four states, called the Bell states [12], are orthogonal to each other. (Hint: Take the inner product of two states and show that it is equal to 0.) For a 2-qubit system, which resides in a 4-dimensional Hilbert space, the four Bell states form a basis. In other words, any 2-qubit system can be represented as a linear combination of the Bell states. The four Bell states are termed as maximally entangled states.

1 Quantum Information Processing

1.4.1.1

11

A Game of Communication Between Two Parties

Let us consider two parties Alice and Bob who are being charged of a crime. They are being interrogated in two separate rooms, and they are not allowed to communicate with each other once the interrogation starts. Alice is provided a question A1 or A2 randomly. Similarly Bob is also provided a question B1 or B2 randomly. Both of them can only answer in yes (+1) or no (−1). Upon receiving their answers, the police will cross-check them, and they will be sentenced or released accordingly. Let V (Ai ) and V (Bj ) be the answers of Alice and Bob, respectively, for questions Ai and Bj . The questions are devised such that Alice and Bob will be released only if  V (Ai ).V (Bj ) =

−1 when i = j = 2, +1 otherwise.

Is it possible for Alice and Bob to come up with a strategy prior to the interrogation so that they always satisfy the above condition? Lemma 4 There exists no classical strategy such that Alice and Bob can satisfy the above-mentioned conditions for any pair of Ai and Bj . Proof Let us assume that it is possible to devise such a strategy to satisfy the conditions always. The conditions, written explicitly, are V (A1 ).V (B1 ) = 1 V (A1 ).V (B2 ) = 1, V (A2 ).V (B1 ) = 1 V (A2 ).V (B2 ) = −1. Multiplying the four equations, we get [V (A1 ).V (A2 ).V (B1 ).V (B2 )]2 = −1. Since the square of any number is always ≥0, it is not possible to satisfy the above equation. Hence, there is no classical strategy for Alice and Bob to satisfy the conditions in every case.  In fact, it can be shown that there exists no strategy such that Alice and Bob can win this game with probability >0.75.

1.4.1.2

A Quantum Mechanical Strategy

In a quantum world, Alice and Bob can share the Bell state |φ − prior to the interrogation. Let us suppose that Alice measures her particle in some spin direction n and Bob measures his particle in some spin direction m.  Spin direction essentially

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implies the basis of measurement. The outcome (eigenvalue) of their measurement is either +1 or −1. Alice and Bob would answer the outcome of their measurement. In such a scenario, the probability of their outcome [9]:  P rob(V (Ai ).V (Bj )) =

1  n) 4 (1 − m. 1  n) 4 (1 + m.

when i = j, otherwise.

Let us explicitly walk through the P rob(+1, +1) scenario. The other scenarios will be similar. Let Alice decide a basis {|0 , |1 }, and Bob decide a basis {|a1 , |a2 }, or vice versa. The choice of basis is such that the inner product between |0 and |a1 gives an angle of cos(135). Therefore, P rob(+1, +1) =

1 1 −1 1 (1 − m.  n) = (1 − cos(135)) = (1 − ( √ )). 4 4 4 2

However, this probability can occur in two different ways (the choice of basis can be interchanged). Therefore, P rob(+1, +1) = 12 (1 + √1 )  85%. 2 It can be shown similarly that there exists a choice of basis for each V (Ai ) and each V (Bj ) such that the required winning probability is ∼ 85% [12]. If Alice and Bob decide upon this choice of basis prior to the interrogation, then during interrogation they shall measure their qubit in a basis according to the question provided to them. Their answer (+1 or −1) will be the outcome of their measurement. Then the two parties can win this game with probability 0.85. This thought experiment by Bell conclusively proves that quantum entanglement correlation is different from classical correlation, thus disproving the hidden variable theory.

1.5 Quantum Circuit 1.5.1 Classical Circuit In a classical circuit, bits are manipulated by gates. Some of the well-known gates are AND gate, OR gate, NOT gate, XOR gate, NAND gate, and NOR gate. Out of these NOT gate is a single-input-single-output gate, while the others are two-inputone-output gates. AND gate or OR gate taken together with NOT gate is universal, i.e., any Boolean logic can be represented using these gates only. Equivalently, NAND and NOR gates are universal gates by themselves [13]. We show the truth table of NAND gate in Table 1.1, which will be referred to in the later sections. Note that, from the truth table of NAND gate (and also applicable to all the above-mentioned gates except for NOT gate), if the output is 0, it is not possible to uniquely determine the inputs. Such a gate is called irreversible. In other words, if Sin is the set of inputs that contains two bits, the output is a set Sout that contains a single bit. In these gates, the information of one bit is lost due to the gate operation.

1 Quantum Information Processing Table 1.1 Truth table of NAND gate

13 Inputs A B 0 0 0 1 1 0 1 1

Output A NAND B 1 1 1 0

Fig. 1.1 A NOT gate

Table 1.2 Truth table of NOT gate

Input 0 1

Output 1 0

It was shown by Landauer [14] that the heat generated during the computation is due to the loss of information; the loss of each bit of information dissipates heat equal to kT ln2, where k = 1.3805 × 10−23 J/K is the Boltzmann constant, and T is the temperature in Kelvin. Although this amount of heat is negligible for a single bit loss, when millions of bits are operated on per second in a real computer, the heat dissipation becomes significant. It was later shown by Bennett [15] that if the heat dissipation is to be avoided, then the computation must be reversible.

1.5.2 Reversible Computation Consider the NOT gate (Fig. 1.1) that has a single input and a single output. The truth table of NOT gate is shown in Table 1.2. Note that it is possible to determine the input of a NOT gate uniquely from its output. Such a gate, for which the input(s) can be uniquely determined from their output(s), is called a reversible gate. The conditions for a gate G to be reversible are: 1. Necessary Condition: The number of inputs should be equal to the number of outputs. 2. Sufficient Condition: Given an output o, there should be a unique input i such that G acting on i produces o. In other words, the function represented by the gate G must be bijective. It is possible to convert an irreversible gate into a reversible gate by introducing some ancillary bits to the inputs and some garbage outputs. Ancillary inputs and

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Fig. 1.2 A reversible NAND gate Table 1.3 Truth table of reversible NAND gate Inputs a0 0 0 0 0 1 1 1 1

a1 0 0 1 1 0 0 1 1

ancilla 0 0 0 0 0 0 0 0

Outputs out0 0 0 0 0 1 1 1 1

out1 0 0 1 1 0 0 1 1

ancilla XOR (a0 N AN D a1 ) 1 1 1 1 1 1 0 0

garbage outputs are extra bits, which are introduced to ensure that the number of inputs is equal to the number of outputs and that the function is bijective. For example, in Fig. 1.2 we show a possible design of a reversible NAND gate and illustrate its truth table in Table 1.3. Multiple designs may be possible for a reversible NAND gate. In this representation, the ancilla bit is set to 0 always. The first two outputs carry along the input bits, and the third output is a0 NAND a1 XOR-ed with the ancilla bit. Since ancilla bit is always 0, this third output simply contains a0 NAND a1 . In this design, there are two garbage outputs and one ancilla input. Using such a technique of introducing multiple ancilla inputs and garbage outputs, any irreversible gate can be converted into a reversible one. However, the general aim of such a circuit synthesis is to reduce the number of extra bits incorporated.

1.5.3 Quantum Gates A qubit |ψ(0) undergoes one or more evolutions to the state |ψ(t) . |ψ(t) = U (t, t1 ).U (t1 , t2 ) . . . U (tk , 0) |ψ(0) .

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Table 1.4 Action of Pauli operators on basis states

Basis State |0

|1

Action of σx σz |1 |0

|0 − |1

σy −i |1

i|0

Each unitary operator U (tj , ti ) evolves the qubit from |ψ(ti ) to |ψ(tj ) . These unitary operators, which evolve the qubits, are called quantum gates. A qubit is a 2-dimensional vector. Therefore a single qubit quantum gate should be a (2 × 2) unitary matrix. Furthermore, since the dimension of an n-qubit composite system is 2n , an n-qubit quantum gate should be a (2n × 2n ) unitary matrix. Since a quantum gate is a unitary matrix U , any quantum gate is reversible (since each unitary gate U has an inverse U † ). Therefore, quantum circuits are by nature reversible. If R is a reversible gate, then the corresponding quantum gate will be UR , which is a unitary matrix. It may be necessary to add extra ancilla qubits and garbage outputs in order to convert from R to UR . Here, we introduce some of the important quantum gates (Table 1.4): 1. Pauli Gates: There are four Pauli gates I , σx , σy , and σz . These four operators span the space of all 2 × 2 unitary operators [16]. Out of them, the identity gate I keeps the input state unchanged. The actions of the Pauli gates on the three basis states is shown in Table 1.4, and their matrix representations are depicted below: σx =

01 10

σz =

1 0 0 −1

σy = iσz σx .

Note that the Pauli gates are Hermitian, i.e., σi σi = I , ∀i ∈ {1, 2, 3, 4}. We invite the readers to work out how the Pauli operators would act on any arbitrary superposition state. 2. Hadamard Gate: This gate can create superposition. The matrix representation and the action of this gate on the basis states are as follows:

1 1 1 H =√ 2 1 −1

1 H |0 = √ (|0 + |1 ) 2

1 H |1 = √ (|0 − |1 ). 2

The states H |0 and H |1 are important in numerous quantum computing protocols and algorithms and are given their own notation: H |0 = |+ and H |1 = |− . Note that +|− = 0, i.e., these two states, called the Hadamard basis, form another basis of the two qubit Hilbert space. The Hadamard gate is a Hermitian gate as well. In Fig. 1.3 we illustrate the action of Hadamard gate on the two bases states  = |0 and |1 . Recall that inner product of two vectors a and b is defined as  a |b

 || a ||.||b||cosθ , where θ is the angle between the two vectors. Furthermore, note that +|0 = −|1 = √1 . Every quantum state has unit norm. Therefore, the 2

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Fig. 1.3 Hadamard basis

Fig. 1.4 Controlled NOT gate

Table 1.5 Action of CNOT gate

Input Control 0 0 1 1

Target 0 1 0 1

Action of CNOT Control Target 0 0 0 1 1 1 1 0

angle between |0 and |+ (and also between |1 and |− ) is 45◦ (since cos(45◦ ) = √1 ). 2 Furthermore, the action of σx and σz are equivalent to each other up to a change of basis. This property has a significant implication in quantum error correction. σx

|1 −→ |0

σz

|− − → |+

|0 −→ |1

|+ − → |−

σx

σz

3. CNOT Gate: The controlled-NOT or CNOT is a two-qubit gate. Here, one of the qubits is the control, while the other one is the target (refer to Fig. 1.4). In the CNOT gate, the control qubit always remains unchanged. The target qubits is flipped only when the control qubit is 1 (refer to Table 1.5). CNOT gate is a Hermitian gate. Lemma 5 There exists no unitary operators U1 and U2 such that Ui , i ∈ {1, 2}, acts on qubit i, and CNOT = U1 ⊗ U2 .

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Proof Let us assume that there exist U1 and U2 such that CN OT = U1 ⊗U2 and Ui , i ∈ {1, 2}, acts on qubit i. We focus on the action of U2 alone. From Table 1.5, we note that there exist combinations such that U2 |0 = |0 and U2 |0 = |1

(rows 1 & 3). Therefore, U2 does not preserve the inner-product and hence is not unitary [4].  Therefore, CNOT is a global gate, which cannot be represented as the tensor product of single qubit gates. In fact CNOT has the ability to create entanglement. The circuit of Fig. 1.5a creates the Bell State |φ + , which is verified by the measurement outcome in Fig. 1.5a. 4. SWAP gate: An ideal quantum circuit assumes that every qubit is able to interact with every other qubit. However, in modern quantum computers, it is not so. The underlying connectivity is restricted, and, therefore, there exist qubits qi , qj , i = j , such that it is not possible to perform a two-qubit operation on them. However, it may be so that both qi and qj are connected to some other qubit qk . In such a scenario, it is necessary to SWAP one of the qubits qi or qj to the position qk and perform the intended two-qubit operation. The action of SWAP gate is depicted as SWAP(|ψ ⊗ |φ ) = |φ ⊗ |ψ . The SWAP gate can be implemented using three cascading CNOT gates as shown in Fig. 1.6. We invite the readers to verify that the SWAP gate is a global gate as well.

(a)

(b)

Fig. 1.5 Bell state creation and measurement outcome. (a) Circuit for Bell State creation. (b) Measurement outcome of the state |φ +

Fig. 1.6 Realization of a SWAP gate using three CNOT gates

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Table 1.6 Truth table of Toffoli gate

Inputs Control1 0 0 0 0 1 1 1 1

Control2 0 0 1 1 0 0 1 1

Target 0 1 0 1 0 1 0 1

Outputs Control1 0 0 0 0 1 1 1 1

Control2 0 0 1 1 0 0 1 1

Target 0 1 0 1 0 1 1 0

5. Toffoli Gate: The controlled-controlled-NOT, or Toffoli gate, is a three-qubit gate. It has two controls and a single target. Similar to the CNOT gate, the control qubits remain unchanged forever, whereas the target qubit is flipped only if both the control qubits are 1. The truth table of the Toffoli gate is shown in Table 1.6. Note that if the inputs and outputs of the Toffoli gates are a, b, and c and a  ,  b , and c respectively, then a = a

b = b

c = c ⊕ ab.

If c is set to 1, then c = 1 ⊕ ab = ¬(ab). Thus, Toffoli gate is equivalent to a NAND gate. Hence, any computation that is possible in a classical computer is possible in a quantum computer as well. However, unlike reversible NAND gate, a Toffoli gate can accept superposition states as inputs. 6. S and T Gates: All the above-mentioned gates are Hermitian. S and T gates are two single qubit gates, which are unitary but not Hermitian. The matrices corresponding to these two gates are depicted below: 1 S= 0

1 0 , T = 0 exp(i π4 )



1 0 S = , 0 −i

1 0 † T = . 0 exp(−i π4 )

0 , i

†

We invite the readers to prove that S = T 2 and σz = S 2 . 7. Parameterized Gates: In order to create arbitrary unitary operations, it is necessary to have some parameterized (or rotation) gates. Various unitary actions are attainable by changing the parameters of these gates. There are three primary rotation gates, named after their axes of rotations.

cos( θ2 ) −i.sin( θ2 ) , Rx (θ ) = −i.sin( θ2 ) cos( θ2 )

cos( θ2 ) sin( θ2 ) Ry (θ ) = , sin( θ2 ) cos( θ2 )

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0 exp(−i θ2 ) . Rz (θ ) = 0 exp(i θ2 ) For further properties of quantum gates, we refer the readers to [9, 17]. The set of all quantum gates can be broadly classified into three sets, namely Pauli Group, Clifford Group, and non-Clifford C3 group. Here we briefly introduce each of these groups.

1.5.4 Pauli Group The Pauli matrices have been defined earlier. They span the set of 2 × 2 unitary operators. The Pauli group (also called the C1 group) is defined as [9] C1 = {eP : P ∈ {I, σx , σz , σy }, e ∈ {±1, ±i}}. N -qubit tensor product of Pauli operators also forms a group for any N : −1 GN = {⊗N i=0 Pi | Pi ∈ C1 ∀i}.

Pauli matrices have the property that for any P1 , P2 ∈ GN , they always either commute or anti-commute. This property plays a pivotal role in quantum error correction [18].

1.5.5 Clifford Group An operator U is said to belong to the Clifford group (also called the C2 group) if it, when conjugated with the Pauli group, maps it back to the Pauli group [19]: C2 = {U | U C1 U † ∈ C1 }. For example, CNOT is a Clifford gate. One can easily verify that CNOT (I ⊗ σx ) CNOT = I ⊗ σx

CNOT (σx ⊗ I ) CN OT = σx ⊗ σx

CNOT (σx ⊗ σx ) CNOT = σx ⊗ I CNOT (I ⊗ σz ) CNOT = σz ⊗ σz

CNOT (σz ⊗ I ) CN OT = σz ⊗ I

CN OT (σz ⊗ σz ) CN OT = I ⊗ σz . Other combinations can be formulated similarly. Thus, CNOT maps any combination of two qubit Pauli gates to two qubit Pauli gates and is, therefore, a

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Clifford gate. This particular formulation has applications in error correction and fault tolerance [20].

1.5.6 C3 Group An operator U is said to belong to the C3 group if it, when conjugated with the Pauli group, maps it back to the Clifford group [9]: C3 = {U | U C1 U † ∈ C2 }. The T gate, in particular, belongs to this set of gates. In general, the Pauli and Clifford gates can be easily simulated classically [21]. However, they cannot create universal quantum computation. In other words, any arbitrary unitary cannot be efficiently approximated using only Pauli and Clifford groups. Generation of nonClifford gates are usually costly but are essential for universal quantum computation [22].

1.5.6.1

No Cloning Theorem

A major difference between the classical and quantum circuits is that fan-out is not allowed in quantum circuits. The reason for this is the no-cloning theorem. Lemma 6 (No-cloning Theorem [23]) There exists no universal cloner U that can copy any arbitrary quantum state. We first discuss the innate meaning of cloning. Given a quantum state |ψ and a blank state |b , the action of a cloner U should be U |ψ |b = |ψ |ψ . We now prove this theorem. Proof Let us assume that there exists such a cloner U . If |ψ and |φ are two arbitrary quantum states and |b is a blank state, then the action of U can be represented as U |ψ |b = |ψ |ψ ,

(1.5)

U |φ |b = |φ |φ .

(1.6)

Since U must be a unitary operator, it should preserve inner product of the states. Therefore, the inner products of the LHS and RHS of Eqs. 1.5 and 1.6 must be equal: (ψ| b|)(|φ |b ) = (ψ| ψ|)(|φ |φ ) ⇒ ψ|φ = ψ|φ ψ|φ

= (ψ|φ ) . 2

(since b|b = 1 f or any |b )

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The above equation is of the form x = x 2 , whose solutions are either x = 0 or x = 1. x = 0 ⇒ ψ|φ = 0, i.e., |ψ and |φ are orthogonal. x = 1 ⇒ ψ|φ = 1, i.e., |ψ = |φ . Therefore, the cloner U can clone only orthogonal qubits (which can be associated with classical states as well). In other words, a generalized cloner for arbitrary quantum states does not exist.  No-Cloning Theorem restricts certain operations such as fan-out, feedback loop, etc., in quantum circuits. Nevertheless, we have already discussed that since it is possible to implement a NAND gate in quantum computers, any operation that a classical computer can perform can also be done in a quantum computer. However, the restriction of fan-out, feedback loop, etc., may lead to increased requirement of qubits.

1.6 Quantum Algorithms In this section we shall peek into some of the problems in which quantum computers can provide some speedup. The model of computation that we shall focus on is called the Oracle model. In such a model, there is a n-qubit Oracle, which is a blackbox, i.e., the working principle of the Oracle is unknown to the user. However, the user can query this Oracle. In other words, if the Oracle executes some function Uf , the user can provide some input x, and the Oracle will provide its corresponding output Uf (x). The goal of the quantum algorithms is not always to learn this function Uf . Rather, we shall be interested in determining some properties of this function only.

1.6.1 Deutsch Algorithm The first quantum algorithm, which showed definitive quantum speedup, is the Deutsch algorithm [24]. Although this algorithm is relatively uninteresting and does not seem to have any real-world application, it does provide a thorough insight on the design of quantum algorithms. And this algorithm also introduces the extremely important notion of phase kickback, which shall be used henceforth in many other algorithms.

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Problem Setting Given a function f : {0, 1} → {0, 1}, determine whether the function is constant or balanced:  constant if f (0) = f (1), f = balanced otherwise. A classical computer will require two queries to solve this problem. The first query is with x = 0, and then again with x = 1. Deutsch showed that a quantum computer can solve this problem with a single query.

1.6.1.1

Phase Kickback

First let us walk through the idea of phase kickback. We consider a circuit, where the first qubit is the input and the second qubit is the ancilla. The Oracle is operated on both of these (Fig. 1.7). We shall move through this circuit step by step along each barrier as shown in the figure. Let the input qubit be in some state |x , whereas the ancilla is prepared in state |1 . Each step in the following shows the change in the quantum states along each barrier: |0 − |1

√ 2 |0 − |1

Oracle ⊕ |f (x) ) −−−→ |x ( √ 2 1 = √ (|x |0 ⊕ f (x) − |x |1 ⊕ f (x) ). 2 I ⊗H

|x |1 −−−→ |x

Now let us look deeper into the ancilla qubit. When x = 0, f (x) =

⎧ ⎨0 → state is ⎩1 → state is

Fig. 1.7 The circuit for phase kickback

√1 2 √1 2

|0 (|0 − |1 ) |0 (|1 − |0 ) = − √1 |0 (|0 − |1 ). 2

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Similarly, when x = 1, f (x) =

⎧ ⎨0 → state is ⎩1 → state is

√1 2 √1 2

|1 (|0 − |1 ) |1 (|1 − |0 ) = − √1 |1 (|0 − |1 ). 2

In other words, Uf |x

|0 − |1

|0 − |1

= (−1)f (x) |x √ . √ 2 2

(1.7)

Therefore, under the action of the Oracle Uf , the information of f (x) appears as a phase along with the original state. This technique is termed as phase kickback.

1.6.1.2

Back to Deutsch Algorithm

A classical machine will have to query the Oracle twice. However, a quantum device can provide a superposition of the inputs. In Fig. 1.8 we show the circuit diagram for Deutsch algorithm and once again move step-by-step along the barriers. In any quantum algorithm, every qubit is initialized to |0 . Necessary quantum gates are operated to change the initialization of the qubits: I ⊗X

|0 |0 −−→ |0 |1

1 1 1 1 H ⊗H −−−→ √ (|0 + |1 ) √ (|0 − |1 ) = √ |0 √ (|0 − |1 ) 2 2 2 2 1 1 + √ |1 √ (|0 − |1 ) 2 2 1 1 (|0 − |1 ) Oracle + √ (−1)f (1) |1

−−−−→ √ (−1)f (0) |0 ⊗ √ 2 2 2 (|0 − |1 ) (phase kickback) ⊗ √ 2

Fig. 1.8 Circuit diagram for Deutsch algorithm

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=

1 √ [(−1)f (0) |0 + (−1)f (1) |1 ] 2 (Dropping the second qubit henceforth)

=

1 (−1)f (0) √ (|0 + (−1)f (0)⊕f (1) |1 ) 2

H

− →

|f (0) ⊕ f (1)

((−1)f (0) is a global phase, hence ignored).

We encourage the readers to convince themselves about the action of the last Hadamard gate (Hint: What is the state when f (0) ⊕ f (1) = 0 (or 1)?). After measurement, the outcome is f (0) ⊕ f (1) =

 0 if f(0) = f(1) 1 otherwise.

Therefore, in a quantum computer, a single query is sufficient to determine whether the function is constant or balanced.

1.6.2 Deutsch Jozsa Algorithm Although Deutsch algorithm shows that quantum computers can provide speedup over classical computers, the problem itself is not very interesting. This problem is somewhat generalized to an n-qubit system in the Deutsch–Jozsa algorithm [24]. Problem Setting Given an n-bit function f : {0, 1}n → {0, 1}, determine whether the function is constant or balanced. In this setting, a function f : {0, 1}n → {0, 1} is: (i) constant if f (x) = 0 or f (x) = 1 for all inputs x, (ii) balanced if f (x) = 0 for half of the inputs x, and f (x) = 1 otherwise. Once again, we are only interested in the global property of the function (constant or balanced) and not the function itself. A constant function can either have f (x) = 0 or f (x) = 1. This algorithm does not aim to discriminate between them. Neither does this algorithm deal with any other types of functions.

1.6.2.1

Classical Approach

Lemma 7 A classical algorithm requires O(2n ) queries to determine whether the function is constant or balanced.

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Proof If the function is balanced, then there are exactly 2n−1 inputs for which f (x) = 0 (or f(x) = 1). In the worst case, for a balanced function, the first half can have the same value, while the second half has the other value. Therefore 2n−1 + 1 = O(2n ) queries are required to determine the nature of the function. 

1.6.2.2

Quantum Approach

For the working principle of the quantum algorithm, we shall heavily rely on the property of Hadamard gate. If x is an n-qubit state, then

H

⊗n

|x =

n −1 2

z=0

(−1)

x.z

|z ,

H

⊗n

n −1 2

(−1)x.z |z = |x .

z=0

In Fig. 1.9 we show the circuit diagram of the Deutsch–Jozsa algorithm for n = 4. The first four qubits are the inputs, whereas the last qubit is the ancilla qubit. We shall once more move through the steps of this algorithm. However, our calculation will be for any value of n. I ⊗X

|0 |0 −−→ |0 |1

⎞ ⎛n 2 −1 ⊗(n+1) 1 1 H −−−−−→ √ ⎝ |x ⎠ √ (|0 − |1 ) n 2 2 x=0 ⎞ ⎛n 2 −1 1 1 Oracle −−−−→ √ ⎝ (−1)f (x) |x ⎠ √ (|0 − |1 ) 2 2n x=0

Fig. 1.9 Circuit diagram for Deutsch–Jozsa algorithm for n = 4

(Phase kickback)

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= H ⊗n

−−→

⎞ ⎛n 2 −1 1 ⎝ (−1)f (x) |x ⎠ (Dropping the ancilla qubit) √ 2n x=0 ⎞ ⎛n 2 −1 2n −1 1 ⎝  (−1)f (x) (−1)x.z |z ⎠ 2n

=

x=0 z=0

⎞ 2n −1 2n −1 1 ⎝  (−1)f (x)+x.z |z ⎠ 2n ⎛

x=0 z=0

Now, from the above expression, the coefficient of the state |0 ⊗n is

1 2n

n −1 2

(−1)f (x) :

x=0

 2n −1 1  1 if f(x) = 0 for all x, (−1)f (x) = n 2 −1 if f(x) = 1 for all x. x=0 But those are the criteria of a constant function. Therefore, if the function is n −1 2 constant, then the probability of obtaining outcome |0 ⊗n is | 21n (−1)f (x) |2 = 1. x=0

In other words, when the function is constant, the only possible outcome of the measurement is |0 ⊗n . Furthermore, when f is balanced, half of the terms of the coefficient of |0 ⊗n is +1, and the remaining are −1. Therefore, the probability of obtaining |0 ⊗n when the function is balanced is 0. Therefore, we run the circuit, as shown in Fig. 1.9, and measure the input qubits. If the outcome is |0 ⊗n , then the function is constant. If we obtain any of the other 2n − 1 outcomes, then the function is balanced. Therefore, the quantum algorithm could solve this problem in a single query as opposed to 2n−1 + 1 queries required for a classical computer. The Deutsch–Jozsa algorithm, thus, shows an exponential speedup over its classical counterpart.

1.6.3 Simon Algorithm The earlier two algorithms were designed specially to show the power of quantum computers over classical ones. They have little or no real-world application. However, Simon’s algorithm [25] aims to find the periodicity of a function. Periodicity is used in multiple branches of science such as signal processing, cryptography, etc.

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Problem Settings Given a function f : {0, 1}n → {0, 1}n , there exists a = a0 a1 . . . an−1 such that f (x ⊕ a) = f (x) for all x. Find a. In other words, for a function f that is periodic with periodicity a, the Simon algorithm aims to find the periodicity of the function.

1.6.3.1

Classical Approach

Birthday Paradox [26] In a population of n randomly chosen people, birthday paradox seeks to find the probability that two persons share the same birthday. Obviously, if n ≥ 367, then this probability is 1 (considering leap year). However, √ the probability exceeds 0.5 if n = 23, which is slightly greater than 365. √ Lemma 8 In order to obtain the period a in a classical computer, O( 2n ) queries to the Oracle are required. Proof We require to find two inputs x1 = x2 such that f (x1 ) = f (x2 ). Since the function is periodic, two different inputs can produce the same output only if x1 = x2 ⊕√a. Then a = x1 ⊕ x2 . The probability of finding two such √ inputs exceeds 0.5 in O( 2n ) queries (from birthday paradox). Therefore, O( 2n ) queries are sufficient to determine a with high probability. 

1.6.3.2

Quantum Approach

Simon’s algorithm makes use of entanglement and superposition to solve this problem in polynomial time. In Fig. 1.10, we show the circuit diagram of Simon’s algorithm for n = 3. In Simon’s algorithm, the number of ancilla qubits is the same as that of the data qubits. This is because the action of the Oracle brings the value of f (x) to the ancilla qubits. Here f (x) is an n-bit value, and the global property to retrieve, the period, is also an n-bit value. Therefore, n ancilla qubits are required to hold the value. In Fig. 1.10, q0 − q2 are the data qubits, and a0 − a2 are the ancilla qubits. As before, we shall walk through systematically along each barrier. However, the calculation will be for any arbitrary n. |0 ⊗n |0 ⊗n

⎞ ⎛n 2 −1 1 ⎝ −−→ √ |x ⎠ |0 ⊗n 2n x=0 H ⊗n

2 −1 2 −1 1  1  −−−−→ √ (|x |0⊗n ⊕ f (x) ) = √ (|x |f (x) ) 2n x=0 2n x=0 n

n

Oracle

At this point, the data and ancilla qubits are entangled. Now, we measure the ancilla qubits. Upon measurement, it will collapse to some value f (xr ). However, due to the structure of the problem, there exist xr and xr ⊕ a such that f (xr ) =

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Fig. 1.10 Circuit diagram for Simon’s algorithm for n = 3

f (xr ⊕ a). Therefore, when the ancilla qubits collapse to some f (xr ), the state of the data qubits is 1 √ (|xr + |xr ⊕ a ). 2 Continuing our analysis along the circuit: 2n −1

 1 1 H ⊗n ((−1)xr .z |z + (−1)(xr ⊕a).z |z ) √ (|xr + |xr ⊕ a ) −−→ √ n+1 2 2 z=0  = (−1)x.z |z . z

The final step needs further explanation. Since x, z, a ∈ {0, 1}n ,  a.z =

0 mod 2 1 mod 2.

¯ = 0. In other If a.z = 1 (mod 2), then (−1)x.z ⊕ (−1)x.z⊕1 = (−1)x.z ⊕ (−1)x.z words, the probability of obtaining a z for which a.z = 1 is zero. Therefore, upon measuring the data qubits, the state collapses to some |zi such that

zi .a = 0. Note that a = a0 a1 . . . an is an n-bit string. It is not possible to determine ai ∀i from a single equation zi .a = 0. Therefore, the above exercise is repeated O(n) times in order to obtain n linearly independent equations of the form zi .a = 0. Since each

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time the Oracle is queried only once, the entire algorithm requires O(n) queries to the Oracle. Once n linearly independent equations of the form zi .a = 0 are obtained, classical techniques, such as Gaussian elimination method [4], can solve for a in O(n) time. Therefore, the time complexity of Simon’s algorithm is O(n), which is √ still a significant speedup over the O( 2n ) time complexity of classical algorithm.

1.6.4 Bernstein–Vazirani Algorithm Bernstein-Vazirani algorithm [27] can be considered as a further generalization of the Deutsch–Jozsa algorithm. Problem Settings Given a function f : {0, 1}n → {0, 1}, there exists an n-bit string a = a0 a1 . . . an−1 such that for every input x = x0 x1 . . . xn−1 , f (x) = x.a mod 2. Find a.

1.6.4.1

Classical Approach

Classically, the user can provide strings of the form z0 z1 . . . zn−1 to the black-box function f such that only the i-th location of the input string zi = 1, and zj = 0 ∀j = i. There are n such unique strings for 0 ≤ i ≤ n − 1. For example, if the input is z = 100 . . . 0, then f (z) = (100 . . . 0).(a0 a1 a2 . . . an−1 mod 2 = a0 mod 2. Therefore, by providing an input z0 z1 . . . zn−1 to the function, where zi = 1 and zj = 0 ∀j = i, one obtains the value of ai mod 2. By querying the black-box function with n such inputs, one can determine a.

1.6.4.2

Quantum Approach

The quantum mechanical approach to this problem is similar to the Deutsch–Jozsa problem. The circuit for n = 3 bit Bernstein–Vazirani algorithm is shown in Fig. 1.11. However, we show a generalized calculation next. For an n-bit string, the quantum circuit requires n + 1 qubits, the last one being the ancilla qubit prepared in the |− = √1 (|0 − |1 ) state. The first n qubits are 2

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Fig. 1.11 Circuit diagram for Bernstein–Vazirani algorithm for n = 3

initialized to |0 . The first step is to create an equal superposition of the first n qubits by applying a Hadamard gate on each of them. 2 −1 1 1  1 H ⊗n |0 ⊗n √ (|0 − |1 ) −−−→ √ |x √ (|0 − |1 ) n 2 2 x=0 2 n

2 −1 1  1 Oracle −−−−→ √ (−1)x.a |x √ (|0 − |1 ) 2n x=0 2 n

2 −1 1  (−1)x.a |x

√ 2n x=0

(P hase Kickback)

n

=

(Dropping the ancilla qubit)

H ⊗n

−−−→ |a

Therefore, a single call to the Oracle to sufficient in a quantum setting to determine the unknown string a. Thus, the Bernstein–Vazirani algorithm provides a polynomial speedup over its classical counterpart.

1.6.5 Grover Algorithm A milestone in the research of Quantum Algorithm is the algorithm by Grover for unstructured database search [28]. This algorithm has multiple real-world applications and is used as a module in various other algorithms. Problem Settings Given an unstructured database of N = 2n elements and a marked state w, find w.

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31

Classical Approach

Since the database is unstructured, the classical algorithm cannot exploit any structures to speed up its search. In the worst case, the marked state may be at the last position, and therefore the search will require a time proportional to N. To further ensure that no approach can give any speedup to this problem, let us assume that the marked state can be in any of the positions with equal probability. Therefore, if the marked state is in location 1, a single search suffices. However, the average search time is N 1  1 1 N(N − 1) N −1 i = (1 + 2 + . . . + N) = · = = O(N ). N N N 2 2 i=1

No classical strategy can provide a speedup to the linear time requirement for this problem.

1.6.5.2

Quantum Approach

In the quantum mechanical algorithm, we shall use the Oracle approach. The previous algorithms have also used this approach without explicit mention. In this approach, we consider a black-box Oracle. The user can provide a location x to the Oracle, which will return whether it is the marked state or not. This can be thought of as similar to the usual classical approach. However, in the classical approach, the user himself or herself checks whether a location contains the marked state. Here, there is a separate Oracle that provides this information. The Oracle is a blackbox since the user is not provided access to the working principle of this Oracle. Therefore, the complexity of the algorithm becomes proportional to the number of calls to the Oracle. Design of the Oracle Initially, all the N = 2n times are provided a binary encoding, i.e., every element x ∈ {0, 1}n . For example, if we want to search from {a, b, c, d}, then we can encode these elements as a = 00, b = 01, c = 10, d = 11. Corresponding to each element x ∈ {0, 1}n , the qubit is prepared as |x . Let UOracle be the Oracle. The action of the Oracle is  − |x if |x = |w , UOracle |x = |x otherwise. In other words, the Oracle applies a phase of −1 to the marked state |w and leaves all other states unchanged. Note that, henceforth in the Grover algorithm, |0

would imply the state |0 ⊗n . Operators on |0 will also be, therefore, of dimension 2n × 2n . The steps of the algorithm are elaborated as below:

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Fig. 1.12 Creating an equal superposition of all the states

Step 0 Prepare an equal superposition state |ψ of n qubits, i.e., 2 −1 1  |ψ = H |0 = √ |x = N x=0 n



N −1  1 |x + √ |w . N N |x =|w

At this situation, the probability of measuring the marked state w is w|ψ = N1 , which is exponentially small. Fig. 1.12 shows the diagrammatic representation of step 0. In this figure N = 16. Note that the probability of measuring all the states is the same, and the average value is denoted by the dotted red line. Step 1 The Oracle applies a phase of −1 on the marked state. Therefore, the application of UOracle on |ψ results in a state: 

N −1  1 |x − √ |w . N N |x =|w

Note that the probability of obtaining the state |w is still N1 . However, the average value of all the states has decreased slightly due to this phase flip (refer Fig. 1.13 where state |15 is the marked state). Step 2 Now we define a reflection operator Uref = 2 |ψ ψ| − I . This operator reflects each of the state along the mean value. The action of this operator on the state UOracle |ψ is depicted in Fig. 1.14.

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Fig. 1.13 Action of the Oracle on the superposition state

Fig. 1.14 Action of the Oracle on the superposition state

We immediately note that now after applying the Grover operator G = Uref .UOracle , the probability of obtaining the marked state after measurement is higher than that of the other states. Applying the Grover operator multiple times will further increase the probability of measuring the marked state. We now resort to a geometrical description of the Grover algorithm, which is easier to analyze.

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1.6.5.3

Geometric Approach to Grover Algorithm

As before, the action of the Oracle is defined as  UOracle |x =

− |x if |x = |w , |x otherwise.

In addition to the Oracle, we also define a Unitary U , which, similar to the Oracle, keeps the state |0 unchanged and applies a phase of −1 to all other states.  U |x =

|x if |x = |0 , − |x otherwise.

We now want to find a vector orthogonal to the superposition state |ψ . One such vector is the state H |x , |x = |0 . x|H |ψ = x|H.H |0 = x|0 = 0. Therefore, |ψ and |ψ ⊥ = H |x , |x = |0 , span the vectorspace consisting  of the marked state |w and the superposition of all other states NN−1 |x . The actions of the operator Uψ⊥ = H.U.H on the two orthogonal states are as follows: H.U.H |ψ = H.U.H (H |0 ) = H.U |0 = H |0 = |ψ , H.U.H |ψ ⊥ = H.U.H (H |x ) = H.U |x = −H |x = − |ψ ⊥ . Define the Grover operator: G = Uψ⊥ .UOracle .

(1.8)

Recall that the original state was expressed as the superposition of the marked state and all other states.  N −1  1 |i + √ |w

|ψ = N N  =

|i =|w

N −1 1 |ψ other + √ |w

N N

= cosθ |ψ other + sinθ |w , where sin2 θ =

1 N.

We invite the readers to convince themselves that ψother |w = 0.

(1.9)

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Fig. 1.15 Geometrical representation of the states

Fig. 1.16 Action of the Oracle on the superposition state

(Hint: |ψ other is essentially the state |w removed from the state |ψ ). Therefore, |ψ other and |w are also orthogonal vectors spanning the vector space (Fig. 1.15). Applying the operator UOracle on the state |ψ (Eq. 1.9) creates the state (refer to Fig. 1.16): UOracle |ψ = cosθ |ψ other − sinθ |w .

(1.10)

¯ = H |x = −sinθ |ψ other + cosθ |w , we have Now, note that for |ψ

¯ ψ|UOracle |ψ = 0. Therefore, we can express |w and |ψ other in terms of |ψ

¯ as follows: and |ψ

¯ , |w = sinθ |ψ + cosθ |ψ

¯ . |ψ other = cosθ |ψ − sinθ |ψ

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Fig. 1.17 A single Grover iteration

Replacing the values of |w and |ψ other in Eq. 1.10 yields ¯ . UOracle |ψ = cos2θ |ψ − sin2θ |ψ

(1.11)

Now applying the operator Uψ⊥ on the state of Eq. 1.11 yields Uψ⊥ UOracle |ψ = H.U.H (cos2θ H |0 − sin2θ H |x ) ¯ = cos2θ |ψ + sin2θ |ψ

= cos2θ (cosθ |ψ other − sinθ |w ) + sin2θ(sinθ |ψ other + cosθ |w ) = cos3θ |ψ other + sin3θ |w .

Two consecutive reflections (Uψ⊥ and UOracle ) constitute a rotation. And we find from the above calculations that a single operation of the Grover operator is equivalent to a rotation by an angle 3θ (refer Fig. 1.17). At this stage, measuring the state in {|ψ other , |w } basis will yield a result |w with a higher probability (can you find the similarity with Fig. 1.14?) After a single iteration of the Grover operator, the final state is cos3θ |ψ other + sin3θ |w . We invite the readers to prove that after k such iterations of the Grover operator, the resultant state will be cos((2k + 1)θ ) |ψ other + sin((2k + 1)θ ) |w . We require that sin((2k + 1)θ )  π2 , which will make the probability of obtaining |ψ other after measurement  0. For small θ , sinθ = θ . Therefore, π 2 π 1 (2k + 1) √  2 N (2k + 1)θ 

1 [Since sinθ = √ ] N √ ⇒ k = O( N).

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Therefore, Grover algorithm provides a quadratic speedup over the classical algorithm for unstructured database search. It is straightforward to use Grover’s algorithm for multiple marked states. For a problem √ scenario with k marked states, the algorithm can find a marked state within O( N) steps as before. However, the algorithm finds any one of the marked states, and each of the marked states is equally probable. This algorithm has far-fetched results. For example, consider any NP-complete problem. These are some of the toughest problems in the class NP, and it requires exponential time to solve any of these problems (this is a loose definition; for a proper definition of NP-completeness, we refer the readers to [29]). However, for NP-complete problems we always have a polynomial time verifier, i.e., if a candidate solution to the problem is provided, then the solution can be verified in polynomial time. All the NP-complete problems are equivalent to each other up to a polynomial time reduction. In a quantum scenario, we can consider the Oracle UOracle to be the polynomial time verifier. Then the application of the Grover’s search algorithm provides a quadratic speedup to any NP-complete problem. In this section we have discussed primarily Oracle-based quantum algorithms. However, we have left out three well-studied and important quantum algorithms— namely Quantum Phase Estimation, Quantum Fourier Transform, and Shor’s Factorization Algorithm. We refer the readers to [17, 30] for these algorithms. Moreover, recent researches study hybrid quantum–classical algorithms for near-term devices. We refer the readers to [31–33] for the recent developments in quantum–classical hybrid algorithms. Acknowledgments The author would like to acknowledge Prof. Guruprasad Kar, Physics and Applied Mathematics Unit, Indian Statistical Institute, whose class he attended in the University. His teaching style has been followed in multiple portions of this chapter. All the circuit figures were created using the IBM Quantum Simulator Qiskit [34].

References 1. Feynman, R. P. (1982). Simulating physics with computers. International Journal of Theoretical Physics, 21(6/7), 467–488. 2. Griffiths, D. J., & Schroeter, D. F. (2018). Introduction to quantum mechanics (3rd ed.). Cambridge University Press. 3. Sakurai, J. J., & Napolitano, J. (2017). Modern quantum mechanics (2nd ed.). Cambridge University Press. 4. Strang, G. (1993). Introduction to linear algebra (Vol. 3). Wellesley, MA: WellesleyCambridge Press. 5. Hoffman, K., & Kunze, R. A. (2004). Linear algebra (2nd ed.). PHI Learning. 6. Kreyszig, E. (1978). Introductory functional analysis with applications (Vol. 1). New York: Wiley. 7. Chefles, A. (2000). Quantum state discrimination. Contemporary Physics, 41(6), 401–424. 8. Barnett, S. M., & Croke, S. (2009). Quantum state discrimination. Advances in Optics and Photonics, 1(2), 238–278. 9. Nielsen, M. A., & Chuang, I. (2002). Quantum computation and quantum information.

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10. Braginsky, V. B., Braginski, V. B., & Khalili, F. Y. (1995). Quantum measurement. Cambridge University Press. 11. Einstein, A., Podolsky, B., & Rosen, N. (1935). Can quantum-mechanical description of physical reality be considered complete? Physical Review, 47(10), 777. 12. Bell, J. S. (1964). On the Einstein Podolsky Rosen paradox. Physics Physique Fizika, 1(3), 195. 13. Salivahanan, S., & Kumar, S. P. (2003). Digital circuits and design. Vikas Publishing House. 14. Landauer, R. (1961). Irreversibility and heat generation in the computing process. IBM Journal of Research and Development, 5(3), 183–191. 15. Bennett, C. H. (1973). Logical reversibility of computation. IBM Journal of Research and Development, 17(6), 525–532. 16. Shor, P. W. (1995). Scheme for reducing decoherence in quantum computer memory. Physical Review A, 52(4), R2493. 17. Benenti, G., Casati, G., & Strini, G. (2004). Principles of quantum computation and information-volume I: Basic concepts. World scientific. 18. Gottesman, D. (1997). Stabilizer codes and quantum error correction. arXiv preprint quantph/9705052. 19. Gottesman, D. (1998). Theory of fault-tolerant quantum computation. Physical Review A, 57(1), 127. 20. Fowler, A. G., Mariantoni, M., Martinis, J. M., & Cleland, A. N. (2012). Surface codes: Towards practical large-scale quantum computation. Physical Review A, 86(3), 032324. 21. Gottesman, D. (1998). The heisenberg representation of quantum computers. arXiv preprint quant-ph/9807006. 22. Litinski, D. (2019). A game of surface codes: Large-scale quantum computing with lattice surgery. Quantum, 3, 128. 23. Wootters, W. K., & Zurek, W. H. (1982). A single quantum cannot be cloned. Nature, 299(5886), 802–803. 24. Deutsch, D., & Jozsa, R. (1992). Rapid solution of problems by quantum computation. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 439(1907), 553–558. 25. Simon, D. R. (1997). On the power of quantum computation. SIAM Journal on Computing, 26(5), 1474–1483. 26. Ross, S. (2009). A first course in probability (Vol. 6). Upper Saddle River. 27. Bernstein, E., & Vazirani, U. (1997). Quantum complexity theory. SIAM Journal on Computing, 26(5), 1411–1473. 28. Grover, L. K. (1996). A fast quantum mechanical algorithm for database search. In Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing (pp. 212–219). 29. Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). Introduction to algorithms. MIT Press. 30. Kaye, P., Laflamme, R., Mosca, M., et al. (2007). An introduction to quantum computing. Oxford University Press. 31. McClean, J. R., Romero, J., Babbush, R., & Aspuru-Guzik, A. (2016). The theory of variational hybrid quantum-classical algorithms. New Journal of Physics, 18(2), 023023. 32. Farhi, E., Goldstone, J., & Gutmann, S. (2014). A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028. 33. Hadfield, S. (2018). Quantum algorithms for scientific computing and approximate optimization. arXiv preprint arXiv:1805.03265. 34. Aleksandrowicz, G., Alexander, T., Barkoutsos, P., Bello, L., Ben-Haim, Y., Bucher, D., Cabrera-Hernández, F., Carballo-Franquis, J., Chen, A., Chen, C., et al. (2019). Qiskit: An open-source framework for quantum computing. Accessed on: Mar, 16.

Chapter 2

Quantum Compiling Marco Maronese, Lorenzo Moro, Lorenzo Rocutto, and Enrico Prati

2.1 Introduction This chapter is entirely dedicated to the compiling stack of quantum computers. Although the topic has been developed at several layers and by multiple approaches during the history of quantum computation, it has never been reviewed systematically.

MM, LM, and LR equally contributed to this work as first author. M. Maronese · L. Rocutto Quantum Team - Istituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche, Milano, Italy Dipartimento di Informatica - Scienza e Ingegneria - DISI, Alma Mater Studiorum - Università di Bologna, Bologna, Italy Istituto Italiano di Tecnologia, Genova, Italy L. Moro Quantum Team - Istituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche, Milano, Italy Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milano, Italy Consorzio Interuniversitario delle Telecomunicazioni, Parma, Italy E. Prati () Quantum Team - Istituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche, Milano, Italy Consorzio Interuniversitario delle Telecomunicazioni, Parma, Italy Università degli Studi di Milano, Milano, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Iyengar et al. (eds.), Quantum Computing Environments, https://doi.org/10.1007/978-3-030-89746-8_2

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In computer science, the firmware is a class of computer software aimed at providing the low-level control of a specific hardware, in order to enable hardware independence. A compiler is a computer program aimed at translating computer code between two languages, called the source and the target [1]. Usually it translates from a high-level programming language to a lower-level language (like assembly language) so that the latter can be executed. For the purpose of this chapter, we may simplify by saying that the compiler applies an algorithm to generate a firmware. High-level quantum algorithms require error-free qubits and logic gates, so the main purpose of a quantum compiler is twofold: translating ideal quantum gate operations used in quantum algorithms into machine-level operations, and, because of the special nature of quantum computers, to fight against the loss of quantum information during time because of decoherence [2]. Because of the complexity of managing quantum systems in practice, the compiling process of the quantum firmware for quantum computer requires a number of stacked operations. In general we may refer to quantum compiling as classical software algorithms needed to connect the physical operations on a quantum hardware [3], which may range from semiconductor [4] or superconductor chip [5] to system based on trapped ions [6] or neutral atoms [7], with the source code of a high-level quantum algorithm written in terms of error-free quantum logic gates. A quantum computer should be seen as the quantum version of a co-processor. Similar to a graphical processing unit (GPU), it requires a classical chip with classical software to be exploited. The quantum processing unit (QPU) is called by those parts of the code involving a quantum algorithm. The task of the full quantum compiler stack is made complex by its twofold role. Addressing the quantum firmware to enable hardware independence requires both to map constrained physical operation into high-level gates and to organize groups of physical qubits so to behave collectively as error-free logical qubits. In this chapter, we discuss in detail how the two aspects are managed by following the layered architecture of quantum computers [8]. Such layered architecture and the field of quantum compiling have been developed originally by targeting the gate-model quantum computer (like those of IBM, Rigetti, IonQ). Here, we extend the approach to adiabatic quantum computers [9]. Instead, one-way qumodes-based [10] and topological quantum computers [11] are in a too early stage to be included in the analysis. Similar to the ISO/OSI stack, which conceptualizes the different layers of a network, both gate-model and adiabatic quantum computers can be abstracted by stacking distinct layers with different roles connecting the hardware with the quantum algorithm level. While the compiling of gate-model quantum computers involves synthesis of quantum gates at both the physical and the logical layers, adiabatic quantum computers require embedding methods to increase the limited connectivity of physical qubits. In the following, the layered architecture of quantum computers is introduced for the gate-model quantum computer as well as its extension to adiabatic quantum computer. Next, a section is dedicated to the quantum compiling techniques for gate-model quantum computers, another section to those for embedding in adiabatic quantum computers, and finally a section is dedicated to discuss and compare

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commercial products developed to support the improvement of performances of quantum computers.

2.2 Layered Architecture and Quantum Compiling Stack In this section, we introduce the concept of the layered architecture of quantum computers. We outline how each layer is separately addressed, and we explain how this architecture, originally conceived for gate-model quantum computers, can be extended to adiabatic quantum computers.

2.2.1 The Five-Layer Architecture of Quantum Computers To clarify the kind and the encapsulation layer at which the quantum compiling operates, the most straightforward starting point is by introducing the layered architecture of quantum computers. As highlighted, a quantum processing unit works as a co-processor operated by a classical computer. Therefore, the layered architecture only defines the stack of such a QPU. The main advantage of a modular architecture consists of its hierarchical organization, thanks at least nominally to a conceptual separation among different kinds of operations, which are supposed to intervene with some order. Like the TCP/IP OSI/ISO layered architecture [12], which is adequate for textbooks but is systematically violated in practice, the layered architecture of quantum computers may be seen as a conceptualization of different kinds of operations that can be more relaxed than the rigid layers would suggest. Therefore, the layers should be considered as a tool to abstract one’s architecture rather than some constraint to adapt the design. The layered architecture of quantum computers was introduced in 2012 as a conceptual framework for the specific implementation [8] of optically controlled semiconductor quantum dots. Independent of such specific implementation, the architecture has been developed as a general tool suitable for any hardware technology. The layered architecture for quantum computer’s physical design consists of five layers, where each one has a prescribed set of tasks to accomplish. An interface separates all adjacent layers to provide services from the lower layer to the one above it. Ideally, to execute an operation, a layer must issue commands to the layer below and process the results. Some procedures can still be in principle operated between non-adjacent layers. The most substantial constraint of the hierarchical layered architecture of Ref. [8] is the synchronization of time operation, keeping in mind that the lowest layer will inevitably be asynchronous, being the time evolution of qubits driven by a timedependent Hamiltonian grounded in a physical hardware. As each layer should output instructions to layers below in a specific sequence and handling errors is

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Layer 5: Application Quantum algorithms and interface to classical user Application measurement

Application qubit

Application gates

Layer 4: Logical Construct a substrate supporting universal quantum computation Logical measurement

Logical qubit

Logical CNOT

Injected ancilla state

Layer 3: Quantum error correction QEC corrects arbitrary system errors if rate is below threshold Measure Z-basis

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Virtual 1-qubit gate

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Layer 2: Virtual Open-loop error-cancellation such as dynamical decoupling OND readout

Physical qubit

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2-Qubit gate

Layer 1: Physical Hardware apparatus including physical qubits and control operations Fig. 2.1 The five layers of the architecture of quantum computers. The architecture can be naturally extended to adiabatic quantum computers. The architecture is grounded in Layer 1 Physical. Next, from the bottom to the top, there are: the Layer 2 Virtual, the Layer 3 Quantum Error Correction, the Layer 4 Logical, and finally the Layer 5 Application. The latter is not discussed in this chapter as it has to do with high-level algorithms only while compiling is not involved. Reproduced with permission under the License Creative Commons 3.0 from JONES, N. Cody, et al. Layered architecture for quantum computing. Physical Review X, 2012, 2.3: 031007. https://doi.org/10.1103/PhysRevX.2.031007

inescapable, a control loop must manage the overall system time evolution. In parallel, syndrome measurements are processed to correct errors. The advantage of the layered architecture for quantum engineers is to focus on individual challenges within an overall design (Fig. 2.1).

2.2.2 Description of the Five Layers of Quantum Computers In the following the five layers are described. Layer 1—Physical There are several different physical implementations on various host systems of a two-level system suitable to be operated as qubits. The most successful are currently superconductive qubits such as flux qubits used by DWave quantum annealer [13] and transmon qubits by the IBM [14], semiconductor qubits [4, 15], trapped ion qubits [16], and neutral atom qubits [17]. Here, we do

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not discuss the approach of one-way quantum computers based on photons [18] involving qumodes instead of qubits. The physical layer is devoted to control and measure physical qubits. The methods employed at this layer are hardware dependent, intending to hide the physical properties and inaccuracies from higher levels. The physical layer provides unitary control of a qubit by at least two adjustable degrees of freedom, such as rotation around two axes on the Bloch sphere, by three freely adjustable parameters. In some cases, the natural precession of the qubit in the reference frame of the laboratory can be exploited to limit the control to one degree of freedom only. For instance, short pulses of either electric, magnetic, or electromagnetic fields (such as microwave pulses for superconductor qubits and lasers for trapped ion qubits [19]) can represent the control of the qubit. As dephasing naturally occurs, such control pulses characterize the error source model and provide some adjustments. The physical layer involves 1-qubit gates, 2-qubit gates, and a readout mechanism. Layer 2—Virtual The virtual layer collects the quantum dynamics of qubits and shapes them into virtual qubits and quantum gates. In computer science, a virtual object behaves according to a predetermined set of rules, without a specified object structure. The typical example is the ready-to-use three-spin qubit [20, 21] forming a robust all-electrically controlled two-level system that acts as the effective qubit, involved in creating logical qubits in the gate-model quantum computer by the layers above. In some cases, the virtual qubit is built by a single physical qubit. One of the aims of the virtual layer is to eliminate systematic errors. Compensation sequences at Layer 2 can correct correlated errors due to imperfections in the control operations of the gates in Layer 1 [22], such as electronic noise, fluctuations in laser intensity, or the strength of the coupling of a quantum-dot spin to spins or a microwave or a laser. The compensation sequence may work if the correlated errors happen on time scales longer than operations of the chosen architecture. Hence, a compensation sequence is effective, and the virtual gate has a lower net error than each of the constituent gates in the sequence. In the original Ref. [8] introducing the 5 layers, the authors point out that many compensation sequences are quite general, so error reduction works without knowledge of the type or magnitude of the error. On the contrary, if one tunes the time-dependent Hamiltonian of the control operations [23], the method falls under the name of dynamically corrected gates. One good reason to use artificial intelligence is connected to the need to characterize the accuracy of operations in the virtual layer. Layer 3—Quantum Error Correction The quantum error correction (QEC) layer at its maximum degree of effectiveness is supposed to support fault-tolerant quantum computing. QEC is needed because of the insufficient ability to correct errors of Layer 2. Quantum error correction is unfeasible on NISQ hardware for the simple reason that tens of physical qubits need to be allocated to ensure that the information of one qubit survives long enough. Typical quantum error correction methods are concatenated Steane code and surface code [24]. While in Layer 2, the fundamental principle is to make correlated errors to cancel each other, here isolated general errors are removed. Ideally, Layer 3 would complete the hardware-

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aware section of the stack and present to Layer 4 fault-tolerant qubits and logic gates only. In practice, in the NISQ era, the qubits presented to Layer 4 are not sufficient for fault tolerance, but they can be used for small simulations. Notice that QEC methods such as the mentioned Steane code correct errors in real time, while topological codes, such as the surface code, track the faulty qubits and recover the answer after being monitored with the platform. Layer 4—Logical The logical layer is the first hardware-independent layer from the bottom to the top. Such independence is supported by the quantum error correction layer, which guarantees fault tolerance by providing error-free gates. Usually, the reasoning is based on gate-model quantum circuits. The need of a logical layer arises from the limited number of gates offered by Layer 3, such as Clifford gates [25]. An easy way to see the reason for such a limit is by thinking that a finite number of syndrome qubits imposes a lower limit to the resolution to distinguish a rotation gate error. Layer 4 compiles all the possible unitary gates, given the limited set of error-free quantum gates provided by Layer 3. For a gate-model quantum computer implementing surface codes, the logical layer will provide logical Pauli frames (i.e., the stored Pauli gates to be applied to correct the corresponding error at the end of the computation), distillation of ancilla states, the full Clifford group (if this is provided partially from Layer 3), and the approximation of arbitrary quantum gates. Layer 5—Application The application layer is hardware independent and relies on either the logical (for the gate model) or the virtual (for the adiabatic quantum computer) qubits as arranged by the management of the layers below. This is a high-level programming layer as the classical computer code delivers the algorithm as a sequence of high-level operations, consisting of a quantum circuit in the gate-model quantum computer or the embedding with the annealing schedule on the adiabatic quantum computer. The algorithms consist of the mathematical flow involving the qubits to achieve either a deterministic or probabilistic result. The quantum firmware is applied to the high-level algorithm elements to be translated into physical operations of the quantum hardware. Still, it is not directly involved in the quantum compiling itself. Layer 5 can be used to determine the error correction strategy and the number of hardware resources to be collected from the low levels.

2.2.3 Transposing the Five-Layer Architecture to Adiabatic Quantum Computers Optimization problems require to find the global minimum of a certain cost function that can be seen as an energy function. A well-known classical approach to solve optimization problems is simulated annealing (SA). In SA, we make use of thermal fluctuations to let the system overcome energy barriers standing between the actual state and the ground state for the energy function. At higher temperatures,

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exploration is fast. Temperature is then slowly lowered so that the system is forced into a minimum, which hopefully corresponds to the lowest energy achievable. It is interesting to wonder if such a paradigm can be extended to a quantum mechanical system. Consider a mechanical system with an associated energy function that we want to minimize using SA. One can introduce artificial degrees of freedom of quantum nature, which in turn introduce quantum fluctuations. In principle, quantum fluctuations can be considered as the tendency of the quantum system to explore the phase space of the classical configurations. We can initially set a high amplitude for such quantum fluctuations, so to make the system eager to explore. Then the strength has to gradually decrease to finally vanish, as the system hopefully moves to the ground state. In this way, we are using quantum fluctuations to mimic the effects that temperature has in SA. An algorithm that controls a quantum system by implementing a schedule where quantum fluctuations’ strength is gradually reduced is called a quantum annealing (QA) algorithm. The physical idea underlying such a procedure is to keep the system close to the instantaneous ground state of the quantum system, analogously to the quasi-equilibrium state to be kept during the time evolution of SA. In QA, quantum tunneling between different states replaces thermal hopping in SA. QA is a generic algorithm applicable, in principle, to any combinatorial optimization problem and is used as a method to reach an approximate solution within a given finite amount of time. Although SA is usually considered a useful and effective method for solving such problems, evidence exists that QA can outperform SA in certain cases. In reference [26], the authors compare the required running time for SA and QA in some optimization problems. The problem consists of searching for the optimal configuration among a finite set of configurations each represented by n bits. The authors show that there exist problems for which QA running time is polynomial in n and SA running time grows more than polynomially in n. Despite this capability, QA has yet to find useful practical applications. Indeed, the major drawback of QA is that a full practical implementation should rely on a quantum computer since time-dependent Schrödinger equations with a very large scale have to be solved. Unfortunately, quantum computers are still at an early stage of development, and only small size problems can be effectively solved. Nonetheless, QA theory suggests that future quantum devices could tackle problems considered difficult for SA methods. In recent years, adiabatic quantum computers (AQCs) have undergone a fast development. Such devices are a physical realization of the QA concept and are currently investigated as an alternative paradigm of quantum computation. Even if the layered architecture of quantum computers has been developed for the gate-model architecture, adapting to the specific features of adiabatic quantum computers is a natural generalization that provides insights of the two methods. In the remaining of this subsection, the adaptation of the layered architecture to adiabatic quantum computers is outlined. Layer 1—Physical As the aim of an adiabatic quantum computer is to maintain a many-body quantum system in its ground state while transforming its Hamiltonian

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from a generally non-interacting to a connected form, the main point of the control of the physical layer consists of preserving the gap between the ground state and the excited state. The main methods fall into the category of energy gap protection (EGP) [27] and dynamical decoupling (DD) [28], respectively. Layer 2—Virtual In a planar chip of superconductive qubits the number of connections from each qubit is necessarily limited, currently of the order o 23−4 . In order to increase the number of connections, the strategy consists of virtualization of qubits by strongly connecting pairs or groups of qubits so they behave as a single qubit. This method is called embedding and can be considered the heart of quantum compiling for an adiabatic quantum computer [29, 30]. Layer 3—Quantum Error Suppression and Correction Adiabatic quantum computers have been considered incompatible with a naive implementation of stabilizer codes [31]. Layer 3 consists mainly of providing error-suppressed virtual qubits. Quantum error correction is possible by considering non-equilibrium dynamics in encoded AQC by cooling local degrees of freedom, i.e., qubits [32]. Layer 4—Logical Independently from the work operated by the Layer 3, consisting of either error suppression or error correction or both, the logical layer can be exploited to cast more general problems than QUBo problems so to solve higherorder unconstrained binary optimization (HUBO) [33]. Hence, the logical qubits are replaced by HUBO embeddings, based on the QUBO embedding addressed at the bottom layers [33]. Layer 5—Application Adiabatic quantum computers are programmed at high level by algorithms of three kind: minimization of functionals, graph partitioning, and sampling statistical distributions [29].

2.3 Quantum Compiling of Gate-Model Quantum Computers This section addresses the quantum compilation problem for gate-model quantum computers as a crucial step to translate high-level quantum algorithms in terms of elementary operations implementable on real-world quantum hardware.

2.3.1 Gate-Model Quantum Computers To understand why quantum compilers are fundamental to run quantum algorithms on quantum hardware, we first need to recall how gate-model quantum computers process the information [34].

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Gate-model quantum computers work, in some aspects, similarly to their classical counterpart. Although classical computers process information via logical and arithmetical operations and gate-model quantum computers by exploiting quantum phenomena such as superposition and entanglement, they both rely on applying transformations on few bits at once to perform computations. Unitary transformations mathematically describe quantum computation. Therefore, unitary matrices acting on the state of two-level physical systems called qubits represent quantum gates. To achieve general-purpose quantum computation, we need to build devices that can implement any quantum computation, i.e., any unitary transformation acting on the qubits. However, physical constraints limit quantum computers to apply few set of gates, and quantum errors make every instruction count. Therefore, there is the need for specialized software, i.e., quantum compilers, to provide robust control of the computation and map the quantum computation into ordered sequences of gates implementable on real quantum hardware. In gate-model quantum computers, quantum compiling synthesizes quantum logic gates at two different layers of the stack. At Layer 2 it consists of mapping arbitrary quantum gates into the constrained unitary operations of the physical qubits of a specific hardware technology. At Layer 4 it consists of mapping the set of quantum gates resulting from the quantum error correction layer, which is usually limited, to arbitrary quantum gate to be made available to Layer 5.

2.3.2 The Standard Circuit Model The standard circuit model is one of the first well-known theoretical results in the quantum computation framework to achieve gate-model quantum computers [35]. According to the model, it is always possible to achieve quantum computation as an ordered sequence of transformations acting on single and two-qubit subsystems, i.e., as a circuit of quantum gates. Although the resulting circuit requires a finite number of gates that manipulate the information locally and entangle the qubits in pairs, it is completely equivalent to the n-qubit computation. Here, we outline the steps of the demonstration, without the proof: First step. It consist of showing that any unitary transformation U on a ddimensional Hilbert space can be decomposed as a sequence of CNOT gates and multi-controlled two-level unitary matrices Cn -U. Second step. Next the multi-controlled two-level unitary matrices Cn -U are decomposed as a sequence of CNOT gates and controlled single-qubit unitary matrices C-U. Third step. Finally the controlled single-qubit unitary matrices C-U are written as a sequence of single-qubit unitary matrices and CNOT gates. The standard circuit model seems to simplify building an n-qubit gate-model quantum computer considerably. Instead of fabricating a device that controls all

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the n-qubits at once, it is possible to achieve the computation by manipulating them individually or in pairs by employing CNOT gates. However, such a result has a little practical advantage. On the one hand, it requires a device that can implement every possible single-qubit unitary transformation. On the other hand, real quantum hardware cannot implement all single-qubit gates due to quantum noise and fabrication constraints in their architecture. More importantly, its main drawback is the length of the quantum circuit resulting from such a procedure, which scales exponentially with the number of qubits. Such a massive number of gates would lead to an unmanageable noisy computation since real-world quantum computers cannot implement quantum gates exactly due to noise and inevitable non-idealities resulting from the manufacturing. Additionally, each gate requires a finite time to be executed. Therefore, any quantum algorithm with an exponential number of logical operations to be performed would hardly offer any computational advantage over the classical counterpart. Such problems are highly relevant nowadays, where gate-model quantum computers can rely on a few qubits and little error correction techniques are applicable. The standard circuit model leaves unsatisfied due to its limited practical value. It would be more convenient to build general-purpose quantum computers that can implement few quantum gates rather than all of them. However, a finite set of quantum gates cannot generate any unitary transformations perfectly (unless using infinite length circuits), but to approximate the desired computation within an arbitrary accuracy at most. It is worth noticing that such constraint would not represent a substantial restriction to quantum computations since unavoidable noises would limit the possibility of distinguishing arbitrarily close unitaries anyway. How to find a strategy to determine the approximating sequence and understand which sets of gates could be exploited represents the main question addressed by the quantum compiling problem.

2.3.3 The Quantum Compiling Problem Informally, the quantum compiling problem consists of finding an optimal strategy to map quantum algorithms as circuits of quantum gates chosen by a given set. It is a fundamental problem in quantum computation theory, affecting different layers of abstraction, such as the logical and physical layers, depending on the set of gates taken into account and specific hardware constraints to be satisfied (see Sect. 2.2.2). At a high-level of abstraction, quantum compilers are usually exploited as quantum transpilers. The tasks typically consist of expressing the circuits into a different set of gates with a “similar level of abstraction” or optimizing a quantum circuit. In such a context, quantum transpilers can increase the performance by reducing the number of ancilla qubits or preferring some particularly optimized gates to decrease the run time and the overall noise. Some hardware constraints can be taken into account, because of the limited connectivity of the qubits. However, the

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resulting circuits are typically further compiled in later stages, as they are expressed by a high-level set of gates such as the Clifford+T library [36]. At a low level of abstraction, the quantum compilers take an arbitrary quantum algorithm (equivalently a quantum circuit expressed using high-level quantum gates) and approximate it, within a given tolerance, as a sequence of low-level transformations that can be implemented directly on quantum hardware such as the R/XX library on trapped-ion quantum computers [37] or U1, U2, and U3 on IBM QX architectures [38, 39]. Regardless of the layer of interest, three key features characterized quantum compilers [40] measuring their performance: Circuit depth. It represents the total number of quantum gates in the circuit. Pre-compilation time. It corresponds to the time taken by the compiler to be ready for use. It is usually performed once before exploiting the compiler. Execution time. It is the time that the compiler takes to return the sequence after the pre-compilation phase. Such quantities must scale optimally as a function of the accuracy requested, i.e., they do not grow exponentially. However, they would hardly scale optimally simultaneously [40], and we must choose a trade-off between speed and accuracy. For instance, the naive strategy of trying every possible sequence of gates would minimize the circuit depth and requires no pre-compilation time. Still, the execution time would explode very rapidly, making it unfeasible as a quantum compilation strategy. How to find a strategy to determine the approximating sequence, understand which sets of gates could be exploited, and balance those features remained unclear until the Solovay–Kitaev theorem in the late 1990s.

2.3.4 The Solovay–Kitaev Theorem The Solovay–Kitaev theorem [41] is a crucial result in quantum computation and a breakthrough in the quantum compilation problem. Robert M. Solovay first announced the results in 1995, but they were formalized and published independently a few years later by Alexei Y. Kitaev in 1997 on a review paper, including an algorithm to quickly approximate quantum gates [42]. The theorem roughly states that if we consider any quantum algorithm, i.e., a unitary transformation U ∈ SU (d), it is possible to find very quickly an approximating sequence of gates as long as they belong to a suitable set B. Such set needs to satisfy some requirements to be exploited: 1. All the gates in B need to be unitary matrix with determinant 1. 2. ∀Aj ∈ B the inverse operation A†j ∈ B. 3. B must be a universal set, i.e., it is possible to approximate any unitary operation as a finite sequence of gates from the set.

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It is worth highlighting that the second request is not strictly necessary approximate unitary transformations, but it is a condition required to prove the theorem only [43]. However, if the requirements are met, the theorem holds for every desired accuracy ε and the length of the resulting sequence scales efficiently. Theroem 1 (Solovay–Kitaev) Let ε > 0 be a desired fixed accuracy, then ∀ U ∈ SU exists a finite circuit C of gates driven by a set  (d) there   B such that  the distance d U, C < ε. The sequence S of gates has a length O logc (1/ε) , where c is a constant value. The circuit depth returned by the theorem and the execution time of its implementations scale polylogarithmically, even if distinct formulations and proofs achieve different values of the constants [44, 45]. For instance, the DawsonNielsen formulation [41] provides sequences of length O(log3.97 (1/ε)) in a time of O(log2.71 (1/ε)), while in [46] those quantities scale as O(log3+γ (1/ε)) and O(log3+γ (1/ε)), respectively, where γ is a positive constant that can be set at will. Additional boosts in performance can be gained by introducing some constraints such as restricting B a to diffusive set of gates [43] the space of unitary matrices efficiently or by employing ancilla qubits [46]. Geometrical proof [45] shows that despite of the strategy considered, no algorithm can return sequence using less than O(log(1/ε)) gates. Moreover, one critical issue that the theorem does not address is finding a universal set of gates, but fortunately, it can be proved that almost all sets of gates have such propriety [47, 48].

2.3.5 Beyond the Solovay–Kitaev Theorem The Solovay–Kitaev theorem provides an elegant and efficient classical algorithm for compiling an arbitrary unitary transformation into a circuit of quantum gates, balancing the sequence length, the pre-compilation time, and the execution time. However, algorithms based on the theorem do not represent the only potential strategies to approach the quantum compiling problem. An optimal quantum circuit for a general two-qubit gate requires at most 3 CNOT gates and 15 elementary one-qubit gates. In case of a purely real unitary two-qubit gate transformation, the construction requires at most 2 CNOTs and 12 one-qubit gates [49]. Such method, known as KAK decomposition, is used, for instance, by the IBM QX architectures to address two-qubits random SU (4) transformations [38]. For instance, the quantum fast circuit optimizer (Qfactor) optimizes the distance between a sequence of unitary gates and a target unitary matrix, using an analytic method based on the SVD operation. In the following, modern approaches are outlined. Machine Learning Approach to Quantum Compiling Machine learning and artificial intelligence approaches have been recently proposed as an alternative strategy [50]. Compiling of arbitrary unitaries into a sequence of gates native to

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a quantum processor has been, for instance, performed by an A* inspired algorithm. Such algorithm is conceived for Noisy-Intermediate-Scale Quantum devices era, as it aims to minimize the number of CNOTs used while accounting for connectivity [51]. Although machine learning approaches can return great quality results and optimized circuits, they usually have high execution times due to the limited precompilation steps that can be employed. In contrast, deep learning approaches exploiting artificial neural networks, which mimic human brains and can be trained like any other machine learning algorithm, seem to be of great promise. By exploiting a neural network, which is trained on a pre-compilation stage beforehand, it would be possible to considerably speed up the execution time. Such networks can be trained in a supervised fashion [52] by a generated training dataset containing optimal circuits only. The neural network, once trained, can decompose a unitary matrix into a product of quantum gates. The approach is limited by the quality of the data set and the size of the neural network, which scales sub-optimally in the number of qubits. Deep reinforcement learning [53–55] can represent an alternative approach [56]. The basic idea is to train a reinforcement learning agent to learn a suitable policy to approximate unitary transformations. The agent is not told how to learn such a policy, but it has to learn through interactions with an environment. It is a timeconsuming task, but it has to be performed once in the pre-compilation stage [57]. Although such a strategy could return a short circuit in minimal time, there is no guarantee that the agent will always find it. Hybrid approaches, where a planning algorithm such as A* is boosted by deep neural networks, could achieve even better performance [58]. However, the execution time is raised by the planning algorithm, which could scale sub-optimally for high accuracy. Hardware-Dependent Quantum Compilers Circuital quantum computers are based on several architectures and topology connections between qubits [39, 59, 60], both due to the different physical two-level systems exploited and the difficulty connecting qubits. Therefore, low-level quantum compilers are asked to consider the particular topology of a quantum computer architecture to map quantum circuit’s efficacy into instruction that can be run on quantum hardware (Fig. 2.2).

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Fig. 2.2 Different topologies for the IBM QX 5-qubit quantum computers. Ourense, Valencia, and Vigo share the same connectivity (left side), which is more limited than Melbourne (right side). Low-level quantum circuits have to meet hardware constraints: while it is possible to apply directly a CNOT gate between gates 1 and 3 on Valencia, it is impossible on Melbourne

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U

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Fig. 2.3 Example of circuit transpilation to meet LNN constraint. A circuit that does not meet the LNN constraint can be transpiled into a new circuit where every two-qubit gate acts on physically adjacent qubits by adding SWAP gates

For many physical realizations of quantum computers, the interaction distance between gate qubits is performed between adjacent qubits only, i.e., between nearest neighbors. A potential strategy to overcome the qubit’s limited connectivity is to ensure that the circuit satisfies a linear nearest neighbor LNN structure. In such a scheme, every two-qubit transformation must act on physically adjacent qubits. Although it is possible to meet the nearest neighbor constraints very quickly in a linear time by adding in front of each gates SWAP gates in a “cascade fashion,” such naive strategy implies a considerable increase in the circuit depth and therefore in the execution time. Many low-level quantum transpiler strategies have been proposed to decrease the number of additional SWAP gates exploiting a global [61] or a local [62–64] reordering scheme (Fig. 2.3). Additional approaches exploit different strategies and machine learning techniques [64] to map the circuits into physical ones for specific architectures such as IBM QX architecture or particular unitary sets [38]. IBM’s approach, which is implemented in its own SDK Qiskit, is based on Bravyi’s algorithm. It has limited performance mainly because it relies on random searches to meet the physical constraints. In contrast, look-ahead schemes [64, 65], which consider gates applied in the near future and exploit additional information on the circuit, explore a larger part of the search space, leading to increased performance.

2.4 Quantum Firmware for Adiabatic Quantum Computers The layered architecture of quantum computers can be used to conceptualize the different kind of quantum firmware and quantum control for making robust adiabatic quantum computers. In this section, the methods developed to address the five layers are discussed.

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2.4.1 Layer 1—Physical: The Annealing Process The starting point consists of considering an optimization problem that can be represented as the ground-state search of a spin–glass model of the general form: HP ≡ −

N 

hi σiz −



Jij σiz σjz ,

(2.1)

i= a|0 > +b|1 >,

(3.1)

where | A> is the state of the qubit; a and b are complex numbers, giving the probability amplitudes to measure “1” and “0,” respectively; and | 0> and | 1> are the basis states. The probability of measuring | 1> is b2 , while the probability of measuring | 0> is a2 . Needless to state that |a|2 + |b|2 = 1. This is the Born’s rule, given by Nobel Laureate Max Born, which states that a measurement of a quantum state gives as value one of the basis elements, with a probability given by the squared absolute value of the amplitude associated to that element. Thus, if there are n states with probabilities a, b, c. . . n, then: |a|2 + |b|2 + |c|2 . . . |n|2 = 1. Any quantum system is represented as a state in the complex Hilbert space. This means that any state is represented by a complex number. If | A> is the Ket representation of such a system, there exists a so-called Bra representation as well, which is the complex conjugate of the | A> Ket vector. Thus, . By using 2N complex coefficient, a generic joint  state can be written as | A> = a00 | 00> + a01 | 01> + a10 | 10> +a11 | 11>, where ij |aij |2 = 1 is the normalization condition. The state space of such a joint system is the tensor product of the state spaces of the component physical systems. In other words, if the component systems have states Aj , the composite system state can be written as |A >= |A1 >



|A2 >



|A3 >



. . . ..|An > .

The probability, P(x), of occurrence of the basis state x after a measurement of the state A can be calculated applying Born’s rule: P(|00 >) = |a00 |2 , P(|01 >) = |a01 |2 , P(|10 >) = |a10 |2 , P(|11 >) = |a11 |2 . In an n-qubit system, we can represent the state as the vertices of the ndimensional hypercube, {0, 1}n . In other words, such a state space of n-qubit system is a 2n -dimensional vector space over C and it is represented as a tensor product of n copies of single-qubit spaces. The measurement of such a system of qubits is not much different from the measurement of a single-qubit system. As an example, let us consider a two-qubit system. After any measurement, these qubits can be trivially decomposed into a product of two individual qubits or the product states. For example, let us apply the controlled NOT (CNOT) gate to the two-qubit state A defined above. A CNOT is a NOT gate with a control bit so it takes two qubits as input. The NOT gate does a bit flip if control bit is | 1>. Its operation can be represented as below: CNOT|A > = CNOT(a00 |00 > +a01 |01 > +a10 |10 > +a11 |11 >) = a00 |00 > +a01 |01 > +a10 |11 > +a11 |10 > .

3.2.3 Entanglement and Bell States There are some very interesting two-qubit systems that are called Bell pairs. To understand these qubit systems, one needs to comprehend entanglement. Entanglement is a very special quantum phenomenon that does not have any classical

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equivalent. It occurs when a pair of particles interact so that the quantum state of a particle of the pair cannot be described anymore independently of the state of the other. The entanglement is preserved even if, after the interaction, the particles are separated by a large distance: measurements of position, momentum, spin, etc., performed on entangled particles will be correlated. Bell pairs are four pairs of perfectly correlated (i.e., maximally entangled) qubits (Eq. 3.2): √ |+ = (|00 + |11 )/ 2, √ | + = (|01 + |10 )/ 2, √ |− = (|00 − |11 )/ 2, √ | + = (|01 + |10 )/ 2,

(3.2)

√ where the constant 1/ 2 is a normalization factor. Of these states, |+ > and |− > are very special states, for which any measurement of the first qubit will also collapse the state of the second qubit. Any effort to factorize these state to product states or tensor product is not possible. Let us try to appreciate the implication of such systems of qubits: any measurement will show that the outcomes are correlated and are stronger than could ever exist between classical systems. Quite clearly, these systems enable apparatus for communication that is far superior than that of classical systems. But they also have other interesting features such as being a new orthonormal basis where information about the complete system may be known, without knowing the states of the individual subsystems. In quantum networks, Bell pairs play an important role in sharing information between two nodes. With such a basis and entangled pair, it becomes easier to manage errors. This helps with entanglement distillation, as discussed in a later section.

3.2.4 No-cloning Theorem Any classical computer or communication system depends on memories and cloning of data (e.g., in amplifiers). Unfortunately, this is prohibited in quantum systems, due to the no-cloning theorem. The no-cloning theorem states that it is impossible to create an identical copy of an arbitrary unknown quantum state. This is a major challenge in quantum communications and makes it difficult to do longdistance communication since any physical channel is noisy. This means that the environment and qubits interact during the propagation along the channel leading to collapse of states and decoherence. This is the biggest challenge toward longdistance quantum communication.

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3.3 Entanglement in Quantum Networks 3.3.1 Entanglement-Based Quantum Network Applications There are two features that make entanglement fundamental to the operation of a quantum network. The first feature is its non-locality: two qubits will remain entangled until a measurement is performed, meaning that, in the absence of noise or other disturbances, one of them can be sent to a destination device located at any distance from the source, from the next room to a satellite distant thousands of kilometers while remaining correlated to the other qubit. The second feature is that, if two qubits are maximally entangled, it is impossible for any external system to share this entanglement, even partly: the entanglement establishes a dedicated and intrinsically secure link between two quantum objects. However, to entangle two qubits they need to interact and, if the qubits are located at two distant network nodes, this requires the capability to transmit quantum states between the nodes. A quantum network can just be defined as a collection of nodes that is able to exchange qubits and distribute entangled states among themselves [7]. Entanglement enables completely new network applications compared to classical communication networks, such as quantum cryptography, blind quantum computation (BQC), and distributed quantum computing. Quantum cryptography relies on fundamental physics laws to perform cryptographic tasks, different from classical cryptography, which assumes instead, based on mathematical or information theory arguments, that nobody can solve a certain difficult problem in an acceptably short time. Quantum cryptography and its most widespread implementation, quantum key distribution (QKD), are illustrated in a dedicated chapter and it will not be discussed here further. It is just recalled that, in addition to the so-called prepare-and-measure QKD protocols, such as the well-known BB84 [1], entanglement-based protocols exist, like E91[8] and its simplified version BBM92 [9]. In these protocols, the strong correlation given by the entanglement facilitates the coordination of multiple network nodes, exploiting the presence of a common node that prepares and distributes entangled states. For example, the √ common node could prepare the state (|00 > +|11 >)/ 2 and send it to two different network nodes that need to agree on the value of a single bit. When one of the nodes measures the state, it collapses either to | 00> or to | 11>. If first and second bits are destined to first and second network node, respectively, the two nodes will measure the same value. The approach can be extended to an arbitrary number of √ nodes, using states with multiple qubits (|00 . . . > +|11 . . . >)/ 2). In BQC, clients owning classical computers outsource to centralized quantum server tasks that quantum computers can perform exponentially faster than classical computers, so making expensive quantum computers available to the average user. For privacy and security reasons, BQC protocols [10] hide the details of the computation to the servers (hence the name “blind”). An example of BQC protocol is the Universal Blind Quantum Computation Protocol (UBQC) [11]. In the UBQC

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Fig. 3.3 Brickwork state

protocol, Alice models the quantum computation task as a measurement pattern on a brickwork state, Gnxm (Fig. 3.3). √ The qubits of the Gnxm state are originally all in the state (|0 > +|1 >)/ 2. They are first arranged in a matrix in the XY plane having n columns (also called layers) and n rows. Then, controlled-Z gates are applied between qubits linked by an edge in Fig. 3.3. The UBQC protocol has two stages: preparation and computation. In the √ first stage, Alice prepares and sends to Bob single qubits (|0 > +eiϕ |1 >)/ 2, where ϕ = k · π(k = 0, 1, . . . , 7) is a random variable. Bob receives all the qubits and entangles them in a brickwork state. This leads to disclosing partial information about the length of the computation input and the depth of the computation, but no other sensitive information about the task Alice has to perform is revealed. During the computation stage, for each qubit of the Brickwork state, Alice sends a message to Bob, over a classical channel, to indicate the basis in the XY plane he has to use to measure the qubit. Then, Bob communicates the results to Alice that, in the next iterations, set new values of the angle ϕ based on Bob’s results. The UBQC protocol ensures that Alice chooses quantum states and classical messages so that it is impossible for Bob to infer useful information about her measurement pattern. If Alice is calculating a classical function, the protocol ends when all qubits have been measured. If she is computing a quantum function, Bob returns to Alice the final qubits. As its classical counterpart, a distributed quantum computer consists of separate, interconnected computing units that coordinate with each other to execute a task, so acting as a single bigger virtual processor. Clearly, the capability to exchange qubits and distribute entangled states, i.e., the presence of a quantum network, is the key for achieving this goal. The need for a distributed computing infrastructure is more stringent for quantum computers than for classical ones, due to engineering and decoherence issues in quantum computers with a high number of qubits [12]. Today achievable hardware complexity limits the capability of current quantum computers to less than 100 qubits. Also, a significant number of these qubits must be used for quantum error correction, to face noise and occurrence of wrong operations, further limiting the computational power. Relying on many small interconnected quantum computers, a distributed quantum

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Fig. 3.4 CNOT telegate

infrastructure offers a natural path to linearly increase the number of qubits, ideally leading to an exponential increase of computational power. Not only distributed quantum computers move quantum information from a quantum device to another via the teleportation of an entangled photons pair (a concept referred as teledata). Entangled qubits pairs are also used to split a quantum logic gate between the two devices, realizing a so-called telegate. Since any kind of quantum computation can be performed by using CNOT gates in addition to single-qubit gates, it is sufficient to demonstrate that it is possible to implement the CNOT gate as joint operation between qubits belonging to two remote devices. A CNOT telegate [12] is shown √ in Fig. 3.4. The state |+ >= (|00 > +|11 >)/ 2, known as Einstein, Podolsky, and Rosen (EPR) pair or Bell pair, is a maximum entangled state consisting of two qubits shared by the source and destination nodes via the communication link.

3.3.2 Quantum Error Correction In the previous section, we defined a quantum network as a set of nodes connected with quantum links that allow them to share entangled states. However, transferring quantum states is not a trivial task. Any measurement of an unknown quantum state leads to its collapse, making it impossible to know what the state was originally. In addition, the transfer of quantum states is made difficult by the nocloning theorem, which states that it is impossible to create an identical copy of an arbitrary unknown quantum state. This prevents the use in a quantum network of signal amplification and regeneration techniques that are widespread in classical communications and allow to connect nodes distant even thousands of kilometers.

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Despite these issues, techniques for correcting errors in quantum communications exist, similar to forward error correction in classical communications, and allow, at the least in theory, the direct transmission of one of the two entangled qubits from a source to a destination node. Quantum error correction is a wide area that requires very specialized mathematical skills. Here we illustrate just the working principles, based on the following examples of single-qubit errors, where X, Y, and Z are Pauli’s matrixes and R(θ) is the rotation operator [13]: – – – –

Bit flip: X| 0>=| 1>, X| 1>=| 0>. Phase flip: Z| 0>=| 0>, Z| 1>=-| 1>. Rotation: R(θ)|0 >= |0 >, R(θ)|1 >= eiθ |1 >. Decoherence (or dephasing): |a >→ (|a > +Z|a > Z> )/2.

To correct bit flip errors, we can use a repetition code, as in classical communications. For example, 0→000 and 1→111. If there is a single bit flip error (e.g., 000→ 010), choosing the bit that occurs more times (0 in our example) will anyway give the correct decision. The method works if the error probability is low enough to neglect multiple errors. Let us now assume that a single bit flip error occurs in a quantum state α|0 > +β|1 >, encoded as α|000 > +β|111 >, resulting in α|010 > +β|101 >. Since any measurement would destroy the superposition, our goal is to realize that the second bit is different from the other two without knowing its actual value. This can be done with the circuit in Fig. 3.5, where the first bit of the syndrome indicates whether the first two bits of the state are equal or not and the second bit does the same for the second two bits. In our example, the syndrome for the state α|010 > +|β101 > is 11, indicating that the second qubit is different and can be recovered by applying to it an X operator. Remarkably, the syndrome does not depend on α and β. It is worth noticing that the code does not violate the no-cloning theorem since the state has been repeated only in the computational basis, while the superposition state, created by redundant encoding, has never been repeated. In mathematical terms: α|000 + β|111 = (α|0 + β|1 )⊗3 .

(3.3)

Phase flips errors can be corrected by moving to the Hadamard base, |+ >= √ √ H|0 >= (|0 > +|1 >)/ 2, |− >= H|1 >= (|0 > −|1 >)/ 2. A phase flip in the original basis results in a bit flip in Hadamard’s one. This suggests Fig. 3.5 Circuit for single bit flip error correction

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using a repetition code in the Hadamard basis to correct phase flips errors. Bit flip errors in the Hadamard basis are corrected by applying the Z operator: Z|+ >= |− >, Z|− >= |+ >. Correcting bit flip and phase flip errors at the same time is possible by identifying the two codes (nine-qubit code): α|0 + β|1 → (α|000 + β|111 )⊗3 + (α|000 − β|111 )⊗3 . Since a bit flip is given by the X operator and a phase flip by the X operator, the nine-qubit code is able to collect also an error given by their product, also called Y error because Y = iXZ and Y|0 = i|1 > and Y|1 >= −i|0 >. The previous error correction procedure can be easily extended to continuous rotation errors, using ancillary qubits, initialized to | 0>, as syndrome. This can be seen rewriting the Rotation matrix R(θ) as R(θ ) =

1 0 0 eiθ

= eiθ/2

−iθ/2

0 e = cos(θ/2)I − sin(θ/2)Z, 0 eiθ/2

(3.4)

where I is the identity operator. Measuring the syndrome, the state collapses giving no errors with probability cos2 (θ/2) and a Z error, which can be easily inverted, with probability sin2 (θ/2). Considering that a generic single bit error is given by an operator representable by means of a 2 × 2 matrix and that any 2 × 2 matrix can be written as a linear combination of I, X, Y, and Z, we conclude that any quantum error correction code (QECC) that corrects X, Y, and Z (and, trivially, I) single-qubit errors is able to correct every single-qubit error. In general, the following theorem holds [13]: If a quantum code corrects errors A and B, it also corrects any linear combination of A and B. In particular, if it corrects all weight-t Pauli errors, then the code corrects all t-qubit errors,

where A and B are operators giving the errors and the weight of an operator is defined as the number of tensor factors (of Pauli’s operators in our case) not equal to I. The majority of QEEC belong to a class known as stabilizer codes. The stabilizer formalism provides a general method for the synthesis of the correction circuit and the determination of the logical operations to be applied to the qubits [13, 14]. We illustrate here the working principle using the nine-qubit code as example: |0 >→ (|000 > +|111 >)(|000 > +|111 >)(|000 > +|111 >), |1 >→ (|000 > −|111 >)(|000 > −|111 >)(|000 > −|111 >).

(3.5)

In the stabilizer formalism, the operation of the code is summarized in a table (called stabilizer), where each row corresponds to an operator applied to the encoded

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data and each column corresponds to a qubit of the encoded word. Z I I I I I X I

Z Z I I I I X I

I Z I I I I X I

I I Z I I I X X

I I Z Z I I X X

I I I Z I I X X

I I I I Z I I X

I I I I Z Z I X

I I I I I Z I X

(3.6)

In our example (Eq. 3.6), first and second rows of the stabilizer apply the operator Z ⊗ Z ⊗ I to the first sequence of three qubits of the encoded data (Eq. 3.5), in order to check the parity of first and second qubits and the parity of second and third qubits, respectively. A similar operation is performed for the second (rows 3 and 4) and third (rows 5 and 6) sequences of three qubits. Finally, rows 7 and 8 detect Z errors (i.e., sign differences) of the first six and last six qubits. This gives a total of 8 operators. Noticeably, any detected error anticommutes with the measured operator. This relies on a general feature, which is the one used to correct the errors: each single-qubit error E anticommutes with a particular set of operators, which indicates what E is without any ambiguity. The previous discussion can be generalized by defining a quantum state | a> as stabilized by the operator K if it is an eigenstate of K with eigenvalue equal to +1. Among the possible operators, we consider those belonging to Pauli’s group (under matrix multiplication) P = ±I, ±iI, ±X, ±iX, ±Y, ±iY, ±Z, ±iZ. Then, we extend P to N-qubit states by means of the tensor product: PN = P ⊗N . An Nqubit stabilizer state, |a >N , is defined by the N generators of an Abelian subgroup (i.e., a subgroup whose all elements commute), G, of PN :   G − K i |K i |a = |a , [K i , K j ] = 0, ∀(i, j ) ⊂ PN .

(3.7)

The 8 stabilizers of the nine-qubit code are the tensor products of the operators on each row of the table in Eq. 3.6. A stabilizer code uses similar circuits for encoding and syndrome extraction, given by the structure in Fig. 3.6 [15], where we used the usual nomenclature [[n,k,d]] for a QECC: the double bracket indicates that the correcting code is a quantum correcting code, n is the codeword length, n-k is the number of ancillary qubits (redundancy), and d is the distance, i.e., the minimum correctable error size. A first codeword is obtained applying all qubits | 0> to the input of the circuit of Fig. 3.6. The other codewords are obtained by applying logical operators, specific of the code, to this first codeword. It is possible to write the state, |aout >, at the output of each of the n-k lines containing the Hadamard gates as |aout = 1/2(|ain + U |ain )|0 + 1/2(|ain − U |ain )|1 ,

(3.8)

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Fig. 3.6 Circuit for syndrome extraction and state preparation in stabilizer codes

Fig. 3.7 Circuit for syndrome extraction and state preparation for the 3-bit repetition code

where |ain > is the system input state, U is a unitary matrix given by the products of the stabilizers K in Fig. 3.6, and |0 > and |1 > refer to the considered ancillary qubit. After the measurement gate, if the result is |0 >, the state is projected onto |ain + U |ain , which is an eigenstate of U with eigenvalue +1. If it is |1 >, it is projected onto |ain − U |ain , which has instead eigenvalue equal to -1. The results of all measurements are then used as classical control input of a single-qubit Z gate, taking as input the output of the stabilizers chain in Fig. 3.6. This Z gate converts any −1 projected eigenstates into +1 eigenstates, i.e., it performs the parity check. The procedure for error correction is identical to the one just illustrated for state preparation and uses the same circuit. Error-free states will be a +1 eigenstate of all the stabilizers and no error correction will be performed; errors that anticommute with any of the stabilizers will flip instead the relevant eigenstate and, consequently, measuring the parity of these stabilizers will give | 1> as result. Figure 3.7 provides a simple example of the general circuit in Fig. 3.6, applied to the 3-bit repetition code discussed at the beginning of this section. So far, we have illustrated the basic design principles of a QECC, but there are several other aspects that make its implementation practical and compatible with

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actual hardware. Some of these aspects (i.e., efficient decoding, code threshold, fault tolerance, encoded computation) will be shortly discussed in the following. Efficient decoding is an issue for large size codes. A [[n, k, d]] code has indeed 2n−k possible syndromes. When n is small, it is possible to use lookup tables to list the error correction operations to perform for each syndrome. However, such a decoding strategy is impractical as the code size increases. Instead of lookup tables, large size QECCs use approximate inference techniques to deduce the most likely error that may occur for a given syndrome. Unfortunately, there is no standard design method that applies to a large class of QECC and these techniques must be tailored on the specific code. Many codes share the same underlying structure, so that it is possible to construct codes with improved distance (i.e., larger number of correctable errors) by scaling up the number of ancillary qubits and functional blocks in the decoding circuitry. At this purpose, estimating the error correction capability of a code is crucial. The so-called threshold theorem for stabilizer codes is of great help: it states that increasing the distance of the code results in a reduction of the logical error rate, provided that the physical error rate of the individual code qubits is below a certain threshold. QECCs working below the threshold can theoretically reduce the logical error rate to an arbitrarily small value. Upper bounds of the threshold are theoretically estimated by using statistical methods, assuming a given noise model, or by numerical simulations. Illustrating the operation of a QECC, we implicitly assumed that encoder and decoder operate without error, which is clearly an unrealistic assumption. A fault-tolerant QECC is defined as a code that can deal with errors occurring wherever in the code processing circuits. Various techniques exist to make encoder and decoder fault tolerant, e.g., by the adoption of transversal gate designs [14], ensuring that small initial errors do not propagate and amplify through the circuit. A drawback is that they require a bigger overhead. Another challenge is designing encoder and decoder so that they can be implemented with a universal quantum computer, i.e., a system that can perform any unitary operation to evolve an input quantum state. It can be demonstrated that any unitary operation can be expressed by using a finite set of elementary gates. An example of such a set is {X, Z, Y, H, NOT, T}, where T =



1 0 . 0 eiπ/4

(3.9)

Designing fault-tolerant universal encoded gates is a major challenge, since the so-called no-go theorem prevents the implementation of transversal universal gates [16]. Various alternatives have been proposed, but typically they require a high number of additional qubits. Summarizing this overview, QECC is a necessary feature to provide reliable connectivity in the Quantum Internet. Although a solid theory and standard design methods exist for QECC, ensuring computational efficiency, reliability, and compatibility with quantum computers hardware are still major challenges.

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3.4 Teleportation and Entanglement Swapping Apparently, the easiest way for the nodes of a quantum network to share an arbitrary entangled state is transmitting one of its qubits from a source node to a destination node (direct transmission). However, in the presence of noise or non-ideal implementation, this requires the application of complicated QECC techniques, as discussed in the previous section. Alternatively, a quantum network can rely on the distribution of only a particular type of entangled states, known as Bell pairs, reported in the following equation: √ |+ = (|00 + |11 )/ 2, √ | + = (|01 + |10 )/ 2, √ |− = (|00 + |11 )/ 2, √ | − = (|01 + |10 )/ 2.

(3.10)

The Bell pairs form a maximally entangled basis for two-qubit states, i.e., they are states having the strongest non-classical correlation among all possible two-qubit states. For two communicating nodes, sharing Bell pairs is easier than transmitting an arbitrary quantum state since there are only four possible known states that can be exchanged, leading to a simplification of the error correction algorithms. Moreover, each Bell pair can be transformed into another Bell pair by means of a local single-qubit operation. This makes Bell pairs especially useful in distributed quantum computing applications, where different nodes, a Bell pair has been distributed to, locally apply to it a sequence of single-qubit gates, transforming it into a different Bell pair. Distributing a Bell pair allows the distribution of any other arbitrary quantum state, as envisaged by the teleportation protocol, which is described in the following. Let us suppose that Alice wants to transmit a qubit, |c >= α|0 > +β|1 >, to Bob. Let us also suppose that Alice and Bob share the Bell pair |+ >AB , where the subscripts mean that the first qubit of the pair belongs to Alice and the second to Bob. This can be done preparing the qubits together in a source node, from where they are sent to Alice and Bob. Therefore, Alice owns the qubit she wants to transmit, C, and the first qubit of the Bell pair, A, while Bob owns the second qubit of the pair, B. The state of the whole system, |S >, can be written as √ |S = 1/ 2(α|0 C + β|1 C ) ⊗ (|0 A ⊗ |0 B + |1 A ⊗ |1 B ).

(3.11)

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|S > can be expressed in the Bell basis using the identities: √ |0 ⊗ |0 = (|+ + |− )/ 2, √ |0 ⊗ |1 = (| + + | − )/ 2, √ |1 ⊗ |0 = (| + − | − )/ 2, √ |1 ⊗ |1 = (|+ − |− )/ 2.

(3.12)

This results in the following superposition of four states: |S = 1/2[|+ CA ⊗ (α|0 B + β|1 B ) + |− CA ⊗ (α|0 B − β|1 B ) + | + CA ⊗ (α|1 B + β|0 B ) + | − CA ⊗ (α|1 B − β|0 B )].

(3.13)

Note that, so far, no operation has been performed: we just expressed |S > in a different basis. The teleportation of |c > occurs when Alice measures her two qubits, A and C, in the Bell basis, projecting |S > in one of the four states: |Φ + CA ⊗ (α|0 B + β|1 B ), |Φ − CA ⊗ (α|0 B − β|1 B ), |Ψ + CA ⊗ (α|1 B + β|0 B ),

(3.14)

|Ψ − CA ⊗ (α|1 B − β|0 B ). As shown in Eq. 3.14, after Alice measurement, her qubits, A and C, become entangled to each other while Bob’s qubit, B, loses its entanglement with A and evolves into one of the four possible states, all given by a unitary transformation of |c >. In order to allow Bob to recover the correct value of |c >, Alice tells him her result, sending two bits over a classical channel. Based on Alice’s results, Bob applies an operator to B to recover |c >, according to Table 3.2: Note that: (a) the teleportation protocol does not violate the no-cloning theorem as the original qubit owned by Alice is irreversibly destroyed as it becomes part of a maximally entangled state; (b) although the teleportation of |c > is instantaneous and does not absorb energy, the completion of the protocol requires the transmission of two information bits over a classical channel, so ensuring causality is not violated;

Table 3.2 Operators applied by Bob in the teleportation protocol Alice result |+ > |− > | + > − >

Bits sent by Alice to Bob (example) 00 01 10 11

Operator applied by Bob to B I (i.e. no operation) Z X ZX=iY

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(c) if an eavesdropper intercepts the two bits sent by Alice, she can infer the operator applied by Bob but this is useless information if she has no access to Bob’s qubit; (d) the teleported quantum state was never transmitted through the network: only Bell states and classical bits are distributed across the network. In a quantum network, any node should be able to exchange Bell pairs with any other node of the network, even the ones not directly connected to it. This feature is enabled by an extension of the teleportation protocol known as entanglement swapping: entanglement swapping creates a new Bell pair between two end nodes, Alice and Bob, starting from Bell pairs generated at one intermediate node, Charlie, along the path connecting them. This is, in summary, how it works: 1. Alice and Bob locally prepare known Bell pairs, resulting in the state |S = |+ A1 A2 |+ B1 B2 . 2. Alice and Bob send qubits A1 and B1 , respectively, to Charlie. 3. Charlie projects |S > onto one of the Bell states in Eq. 3.10 (Eq. 3.15). As result of Charlie measurement, Alice and Bob share an entangled state, consisting of qubits A2 and B2 . + |A1 B1 |S = |+ A2 B2 , − |A1 B1 |S = |− A2 B2 = ZB2 |+ A2 B2 ,  + |A1 B1 |S = | − A2 B2 = XB2 |+ A2 B2 ,

(3.15)

 − |A1 B1 |S = | − A2 B2 = XB2 ZB2 |+ A2 B2 . 4. Charlie sends two classical bits to Alice and Bob to tell the state he used for the measurement. 5. Optionally, Alice and Bob apply a Pauli operator to their qubit of the entangled state (correcting them with a Pauli operator, if they like to share a specific Bell State, e.g., |+ A2 B2 [see Eq. 3.15]). The whole process is summarized in Fig. 3.8, where wavy lines indicate entangled qubits. Note that at the end of the process the two local original entangled pairs are destroyed, and entanglement is created between remote qubits A2 and B2 .

3.5 Entanglement Distillation In describing entanglement teleportation and swapping, we assumed to work with pure, maximally entangled quantum states and that no errors occur during the generation and swapping of Bell pairs. However, due to channel losses, noise, and decoherence, the states in practical quantum systems are, in general, nonmaximally entangled. For example, the quality of an entangled state degrades at each swapping operation along the link between two end-nodes. It is possible, however, to create, from some pairs of non-maximally entangled qubits, a smaller number

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Fig. 3.8 Entanglement swapping

of pairs with a higher degree of entanglement. This process is called distillation. As is evident from the previous description, distillation introduces significant information overhead so it should be applied only when strictly necessary. There are several methods of entanglement distillation, such as Bernstein’s Procrustean method, the Schmidt Projection method, and the CNOT-purification method [17]. Since the qubits are, in general, separated in space, only local operations and classical communication (LOCC) are allowed in entanglement distillation. Here, we will illustrate the simplest procedure, i.e., Bernstein’s Procrustean method, following the experimental example in [18], not to make the discussion too abstract. A non-maximally entangled two-qubit state can be written as |a = (ε|00 + √ |11 )/ 1 + ε2 , with ε = 1. Bernstein’s method applies a non-unitary filtering process to |a >, measuring one of the qubits. The measurement process is conceived so that, at its end, |0 >→ |0 > and |1 >→ ε|1 >, “equalizing” such a way the two terms of |a > and generating a (quasi) maximally entangled state. In [18], qubits |0 > and |1 > are codified as horizontally and vertically polarized photon states, |H > and |V >, respectively. An entangled pair of infrared photons is generated by an ultraviolet photon (pump) incident onto a pair of adjacent non-linear crystals. An incident pump photon is either split into two horizontally polarized infrared photons in the first crystal, from pump’s vertical polarization component, or it is split in the second crystal from pump’s horizontal polarization component, in which case the photons are vertically polarized. In both situations, the two photons will be entangled. The value of the coefficient ε is simply set by rotating the polarization of the incident pump photon. The procrustean filtering is realized by inserting on

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Fig. 3.9 Symmetric scheduling algorithm

the photon’s path at the output of the crystals a series of coated glass slabs, at approximately the Brewster angle, so that the transmitted photons are preferentially horizontally polarized: as the victims of the ancient villain Procustes, photons are forced to fit, hence method’s name. The quality of the obtained quantum state is measured through its fidelity, a metric that indicates how close it is to a perfect Bell state. The fidelity is defined as the probability that an ideal measurement of the two qubits would result in the desired Bell state. To obtain a high-fidelity quantum state, usually several distillation steps are performed in sequence, according to some distillation or purification scheduling algorithm (here we use the two terms in interchangeable way). The most known scheduling algorithms [19] are the symmetric algorithm, the pumping algorithm, the greedy algorithm, and the banded algorithm. The symmetric algorithm (Fig. 3.9) attempts to purify only pairs having equal fidelity. In Fig. 3.7, circled dots indicate pairs with fidelity indicated by the underlying number (those numbers are merely for illustration purposes and do not correspond to any real process). In the example of Fig. 3.7, at step (1) the distillation of two pairs with fidelity 0.6 gives a pair with fidelity 0.7, while an attempt to purify two other 0.6 fidelity pairs fails. At step (2), another 0.6 fidelity pair is created and, this time, the distillation succeeds (step 3), resulting into a second 0.7 fidelity pair that is used to purify the first 0.7 fidelity pair (step 4). As result, a pair having fidelity 0.8 is obtained, which is judged to have enough quality. This example clearly shows the main disadvantage of the symmetric scheduling that is the required time needed to generate pairs with matching fidelity. This disadvantage is further emphasized by the fact that, waiting for the generation of a matched pair, the quantum state in memory degrades. The entanglement pumping algorithm (Fig. 3.10) tries to make a more efficient use of resources, using all created Bell pairs to gradually improve the fidelity of one Bell pair, called pumped pair. Even in this case, however, the fidelity improves too slowly, especially when the difference between the target and the original value is high.

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Fig. 3.10 Pumping scheduling algorithm

Fig. 3.11 Greedy scheduling algorithm

The greedy scheduling (Fig. 3.11) can be considered as a “parallelized version” of the pumping algorithm, where, at any step, all available pairs are used, never deferring immediate actions in favor of later operations that could potentially give a better fidelity. This is also the drawback of this method, since using low-fidelity pairs gives only a small increase in fidelity (when it succeeds). Thus, greedy scheduling requires high-fidelity initial pairs to work. Both the greedy algorithm and the pumping algorithm do not impose any constraint on the fidelity of the matched pairs. This results in low probability of success and slow operation if the quality of the pairs to distill is not sufficiently high. On the other hand, the symmetric algorithm is impractical, since it requires perfectly matched pairs, which is impossible to achieve, especially considering the state degradation in a quantum memory. Banded scheduling ([19], Fig. 3.12) was proposed as an approach to introduce more flexibility, compared to symmetric scheduling, while using resources efficiently, as in the greedy algorithm. The banded algorithm establishes ranges of fidelity, or bands, and allows only pairs within the

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Fig. 3.12 Banded scheduling algorithm

same range to distill with each other. In the example of Fig. 3.12, we assume two bands divided by a fidelity threshold of 0.65. At step 1, the greedy algorithm is performed on four 0.6 fidelity pairs, resulting into a 0.8 fidelity pair. With the banded algorithm, it is impossible to distill this pair with another 0.6 fidelity pair (step 2), because they belong to different bands. So, it is necessary to wait that the 0.6 fidelity pair is purified, for example, using again the greedy algorithm (step 3) and obtaining a 0.7 fidelity pair. This time, since 0.7 and 0.8 are both above the threshold, the two obtained pairs can be distilled, resulting into a 0.85 fidelity pair. Note that this last operation would be not allowed by the symmetric algorithm. Both symmetric and banded scheduling are probabilistic algorithms, where the sequence of distillation operations and required time depend on many factors, such as the fidelities of the input Bell pairs and the probability of success of the individual distillation operations. Moreover, both the banded and symmetric algorithms can fail and are potentially subject to deadlock, if no matching pair can be found (this happens in the banded algorithm if there is a single pair within each band). The banded algorithm solves this issue allocating a sufficient number of qubits, which is equal to the number of bands times the number of swapping operations to perform, plus one.

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3.6 Quantum Repeaters The distribution of quantum states over long distances is the key for the success of a Quantum Internet. However, quantum communication systems suffer from severe distance limitations compared to classical communication systems [20]. A record distance of 421 km has been reported in a lab experiment [21] with a quantum key distribution (QKD) system transmitting over ultra-low loss optical fiber but commercial QKD systems need to work on installed optical fibers and can guarantee acceptable performance only up to a much shorter distance, typically 80 km. This is far less than the thousands of kilometers that optically amplified classical communication networks can achieve today. Unfortunately, the no-cloning theorem imposes a fundament prohibition to the use of amplifiers in quantum networks. A solution for transmitting quantum states over long distances is the use of quantum repeaters. A quantum repeater can be defined as a device that allows to overcome the so-called Pirandola–Laurenza–Ottaviani–Banchi (PLOB) limit for the capacity, C, of a point-to-point QKD link [22]. For a pure-loss channel, the PLOB limit is given by C ≤ − log2 (1 − η) ,

(3.16)

where η is the channel transmissivity. As the optical amplifiers in classical networks, quantum repeaters potentially offer an economic and compact solution for long-distance quantum communication. The original proposal of a quantum repeater [23] was based on entanglement swapping, performed at each intermediate node of a link connecting two end-nodes. Entanglement distillation is also carried out, to recover the quality of degraded entangled states (see Sect. 3.3). More complex quantum repeater schemes were introduced later, to face implementation issues like the decoherence in quantum memories or errors in the entanglement swapping process. Regardless of the underlying scheme and despite promising research results (e.g., [24]), quantum repeaters are far from reaching a technology maturity suitable of industrialization: in the Strategic Research Agenda of the European Quantum Flagship [25], a “demonstration of a chain of physically distant quantum repeaters enabling quantum communication over at least 800 km using telecom fibers” is not expected before 6– 10 years from now. Basically, the first proposed scheme of a link with quantum repeaters extends the entanglement swapping procedure of Fig. 3.8 to the case of more intermediate nodes between Alice and Bob. The entanglement swapping process doubles indeed the distance between entangled nodes from L (distance between Alice and Charlie and Charlie and Bob) to 2L (distance between Alice and Bob). Iterating the procedure n times, in a nested way, the distance can be extended from L to 2n · L: Fig. 3.15 shows an example with two nested levels (Fig. 3.13).

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Fig. 3.13 Nested entanglement swapping in a chain of quantum repeaters

It can be demonstrated that the capacity limit for a quantum link with n equally spaced repeaters is [26]  √  C ≤ − log2 1 − n η .

(3.17)

A nice feature of quantum repeaters based on entanglement swapping is that they introduce an overhead in qubits, which is a polynomial function of the number of repeaters [27]. However, in a link with several repeaters, different entangled qubit pairs need to be processed at the same time. This is possible in principle storing in a quantum memory the quantum states that are generated first, waiting for the generation of the remaining states. Unfortunately, quantum memories able to store a quantum state for a sufficiently long time, while preserving its entanglement features, do not exist today: even a maximally entangled state will deteriorate sooner or later, due to decoherence. The coherence time required to quantum memories is further increased by the possibility that some swapping operation could fail due to link losses, so requiring a quantum state to be stored for a longer time before the link can be set up. The probability that the transmission from one node to the adjacent one will succeed is proportional to e−αL , where α is the optical fiber loss coefficient. Quantum repeaters that need to iterate some operation, due to errors caused by decoherence in quantum memories or link loss, are called probabilistic quantum repeaters. Decoherence and loss are not the only source of errors in a quantum repeater. The quantum states are usually encoded into photons, for both transmission into the link and for performing in linear optical circuits the Bell measurement needed for entanglement swapping. However, these circuits are not ideal, as well as the conversion between incoming photon states and internal memory states, resulting again into probabilistic operation and generation of imperfect, non-maximally entangled, states. Interleaving an entanglement swapping event with multiple occurrences of entanglement distillation is a solution to preserve the quality of the generated quantum state, but the distillation introduces: (a) further overhead, with the number of additional qubits increasing as a polynomial or logarithmic function

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of the distance [23], depending on the used algorithm; (b) latency, since it requires a handshake between the involved nodes, with an additional delay proportional to the link length. Latency is anyway an issue using probabilistic quantum repeaters, since it is not possible to swap entanglement at a certain nesting level until we know all the swapping operations at the previous nesting level succeeded. This requires exchanging classical information between the nodes, introducing a delay limited by the light propagation time along the link. In summary, the coherence time required to the quantum memories and the latency introduced by distillation and probabilistic operations are the main obstacles to the industrialization of quantum repeaters based on entanglement swapping. Advances in technology can help to mitigate those issues: quantum memories with a coherence time of about one second have been demonstrated [28] and technology platforms enabling deterministic entanglement swapping exist [29]. Other approaches try to avoid the time-consuming two-way communication needed for heralding the success of the distillation. One of these approaches uses one-way deterministic distillation algorithms based on QECC [30]. A second approach relies on qubit encoding across multiple nodes, entanglement swapping of the encoded qubits, and classical error correction techniques [31]. This approach is based on three steps: encoded generation, encoded connection, and Pauli frame. To illustrate the working principle of quantum repeaters with encoding, we will use the simple three-qubit repetition code for correcting bit flip errors discussed in Sect. 3.4. The encoded generation step is illustrated in Fig. 3.14, where grey dots indicate the three encoded qubits and white dots the other three ancillary qubits used by the procedure. The numbers at the top of the dots indicate the network nodes where the qubits are generated and stored. The encoded qubits at nodes 1 and 2 are prepared to |+ > and |0 >, respectively, and stored in a quantum memory. Then, physical Bell pairs are created between the ancillary qubits at the two nodes. Finally, applying a CNOT telegate (Fig. 3.4) between each pair of encoded qubits, the entangled state 12 is created between adjacent nodes 1 and 2.  The same procedure is repeated for all N adjacent nodes of the network 12 ,  23 , . . . Φ k−1k , . . . Φ N −1N . (Fig. 3.15a), creating entangled states  Then, a Bell measurement (dotted line in Fig. 3.15b) is performed on each pair of encoded qubits at each node, applying three pairwise CNOT gates. Note that each of the qubit of the pair is entangled with a different adjacent node. Each measurement produces 2 classical bits, which will be used for error correction. At the end of the

Fig. 3.14 Encoded generation in quantum repeaters with encoding

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Fig. 3.15 Operation of quantum repeaters with encoding

procedure, entanglement is created between the qubits at the end-nodes 1 and N. It is possible to demonstrate that this method offers protection versus incorrect entanglement swapping events, so that no distillation is necessary anymore. Consequently, the required storage time of the quantum memories increases only linearly with the number of nodes. The drawback of this scheme is the larger number of required multiple qubit operations, compared to the original proposal of quantum repeater [23], which was based on single-qubit operations. Other schemes of repeaters [32] use QECCs instead of shared entanglement followed by classical error correction. This eliminates the need for maintaining the quantum state coherence during the time necessary to the generation of the remote entanglement, so decreasing the storage time required to the quantum memories. In this kind of repeater, a QECC encodes logical qubits (i.e., the qubits that carry actual information) in a larger number of physical qubits (i.e., all the qubits used in the process, including the ancillary qubits used by the code overhead), which are typically mapped onto some photons’ feature. The photons are transmitted to the next node and their quantum state stored in a memory, so that the errors can be corrected, and the original logical qubits recovered and transmitted to the next link. As in classical communication networks with repeaters, the procedure is iterated until the receiving end-node is reached. In this type of repeater, the memory storage time only needs to be higher than the time required for error correction and is independent of the communication time between nodes. A disadvantage of this scheme is that one-way quantum communication has a theoretical loss limit of 50% [33], requiring the repeaters to be spaced more densely.

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3.7 Quantum Internet Architecture Quantum Internet can be defined as a network of distributed quantum systems [7]. Role of any network is to transmit information. In case of classical Internet, information packets are transmitted, which are then converted into bits and symbols for physical transmission. In the Quantum Internet the basic unit of transmission would be the qubit. Any network connects heterogenous systems, controlled by different kind of software. This suggests an inherent need to standardize the network architecture in a way that supports interoperability. This is a requirement also for the Quantum Internet, which has its own architecture and protocol stack. Any architecture development depends on the goals derived from its use cases. In case of new and evolving technology, where many technology and business aspects are still under research, this task is not easy and has to be done with certain degree of vagueness in the start. With technology evolution, use cases would be defined better and that would help in cementing the standards. What is sure is that the Quantum Internet will support a subset of existing as well as new applications, acting in conjunction with classical networks. The Quantum Internet Research Group (QIRG) at the Internet Engineering Task Force (IETF) started working on the Quantum Internet architecture, listing certain goals in [7] and [34]. This section would be mainly based on that work.

3.7.1 Goals of a Quantum Internet Architecture The primary purpose of a Quantum Internet, as per QIRG, is to support distributed quantum application protocols with highest fidelity. The Quality of Service goal is that the distributed quantum application protocols should run with highest efficiency. This also creates the necessity to define right metrics for the services and applications and domains. While this seems very fundamental and basic, it also illustrates an important use case difference between classical Internet and Quantum Internet. If we classify the applications based on the provided services, as defined in [7] and [34], it is natural to divide them into three categories, namely, quantum cryptography applications, quantum sensor applications, and quantum computing applications. Quantum cryptography applications refer to services that help in making the communications more secure. This typically relies on the stronger-thanclassical correlations and inherent secrecy of entangled Bell pairs. Examples of such applications include QKD protocols used in secure communication setup and fast Byzantine negotiation that can be used in blockchain applications. Quantum sensors are an interesting emerging area where quantum technology is used for supporting distributed sensors or Internet of Things (IoT) devices. That creates possibilities to use quantum technologies in cyber-physical systems, using critical infrastructure where clock distribution is secured using quantum technologies.

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Another interesting upcoming technology, at the time of writing, is sensors for ultra-wide radio frequencies with arbitrary frequency resolution. Finally, there is a lot of hype around quantum computing, where the Quantum Internet can support distributed quantum processing units by providing interconnect services. This also creates possibility of securing cloud computing, where source data can be kept secret. Such applications are also known as private, or blind, quantum computation. As a second goal, the evolution of the Quantum Internet architectures should be seamless, namely it will have to support any future new application and service without the need for forklift upgrades. The third goal is to support technology implementation heterogeneity. This is important for interoperability and standardization. As an example, today there are numerous approaches, proposed in various research works, to develop pragmatic and efficient Quantum Internet architectures including quantum repeater. All these quantum systems are quite fragile, due to noise leading to decoherence. This creates requirements around standardized error handling techniques, in terms of content and semantics of messages that cross boundaries, both for connection setup and real-time operation. The fourth goal for a Quantum Internet is the need for security. While quantum networks might be used to distribute keys or random numbers, it is also important that they are able to withstand cyberattacks. So, any Quantum Internet needs to be robust to malicious actors trying to disable connectivity. The fifth and extremely important goal for any network, including a quantum network, is the ability to manage it. Network management in classical networks include Fault, Configuration, Accounting, Performance, Security (FCAPS) aspects, as first defined by the International Organization for Standardization (ISO) for Telecom network management. In enterprises and not-for-profit organizations, where billing is not done, Accounting is sometimes replaced with Administration. Any user and service provider of the Quantum Internet would like to have a system that is easy to configure optimally, according to connection to be set up, type of service, quality of service, etc. This creates a requirement for configurability, as in the classical Internet. Any administrator would like to monitor the network performance and debug the network whenever required while protecting confidentiality and integrity of the devices connected to it. In a Quantum Internet this is a very complex task, since the qubits cannot be actively monitored as any monitoring would imply a measurement that irreversibly creates decoherence. Also, it is not possible to clone a qubit. Thus, measuring the fidelity of an individual pair is impossible. Any Key Performance Indicator (KPI) such as Bell Pair fidelity must be a statistical quantity that requires the collection of a big amount of data, which in turn requires the constant monitoring of the network and necessitates a parallel classical information signaling channel. Finally, availability and resilience are important KPIs of any network. This leads to the sixth goal for a Quantum Internet architecture. The network should be able to operate guaranteeing low levels of loss and failure probability. Of course, it is important to quantify the limits of such losses realistically, not to prevent the network implementation with the available technologies. It is expected that this

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aspect will be addressed later by some standards organization. As mentioned above, we would have at least two data planes and two channels in any possible Quantum Internet and availability and resilience requirements should apply to both. Having listed high-level goals and requirements for the Quantum Internet, let us now explore its functions, partitioning them in architectural layers. At the time of writing, it is understandable that the Quantum Internet will have at least four layers, namely, physical layer, link layer [71], network layer, and application layer. Some researchers have proposed the need for five layers [34], where there is separation between transport and application layers.

3.7.2 Physical Layer The lowest layer of any networking stack is the physical layer. The physical layer is expected to deal with the transmission of bits or qubits, mapped in some physical property of a particle, typically a photon, over a propagation medium, using a standardized interface. It is also responsible for the physical connectivity and channel compatibility issues such as spectrum frequency allocation, specification of signal strength, transmission bandwidth, etc. The physical layer is also responsible for timing and phase synchronization. Since quantum communication depends on quantum entanglement, the generation of entangled qubits is peculiar of the Quantum Internet physical layer. However, qubits are fragile and decohere quickly: one of the approaches to deal with this issue is to use heralding techniques. In a heralded photon source, a pair of photons in a highly correlated state is generated, so that one photon of the pair can be detected to “herald” the other and know its state before the detection. A typical model of the physical layer in the Quantum Internet involves a source and a destination node, linked by a chain of quantum nodes that attempt to generate and distribute entangled states. Synchronization with a common timing reference is necessary and, thus, having a distributed quantum clock is an additional requirement for a quantum physical layer. Based on the request from higher layers, the physical layer attempts to create Bell pairs but this action is not successful each time. The physical layer responds to the link layer about the successful generation of a Bell pair with sufficiently good fidelity.

3.7.3 Link Layer The link layer is above the physical layer and provides the service of robust entanglement creation to the communicating quantum nodes in the link. Designing the link layer in this fashion enables any higher layer, such as transport or application layer, to rely on entanglement generation in a uniform way. This also allows the higher layers to be architected and developed independently. Since the link layer

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Fig. 3.16 Architecture and layers for Quantum Internet

interfaces directly with the physical layer, it is in charge of scheduling requests, handling errors, and queue and memory management to ensure robust quantum entanglement generation. That apart, it can provide metrics for quantum nodes and channels to the network management software. Overall layers would look as Fig. 3.16.

3.8 Quantum Internet’s Applications and Use Cases The Quantum Internet can be thought as a standalone communication infrastructure empowered by a number of features classically impossible, capable of increasing its overall power, and enabling novel kind of interaction and dynamics within end users. It does not have to be thought as a completely independent infrastructure, while instead it will probably be implemented altogether to the classical Internet, with continuous interaction between the two. We try in this section to envision what will the Quantum Internet offer to a generic user, highlighting what has already been verified experimentally and what may be furtherly designed and implemented in the future. The Quantum Internet will so enable new frontiers of applications, reshaping the perception of what can be currently done in the Internet. To present the possible areas that will be impacted, it is useful to introduce some taxonomy, trying to simplify the huge number of applications. We illustrate first the empowering given on existing classical applications and infrastructures, describing then novel services that can exist only thanks to quantum phenomena that enable them. Additionally, as it is usually done in network architectures, we partition the applications between user plane applications, which are those applications oriented to the end user connected to the Internet, and control plane applications, which are applications that act within the network infrastructure, shaping indirectly the features perceived by the final user. Actually, most of the illustrated features will have a cascade impact on many of other Internet services: such a classification does

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not pretend to be absolute or complete, but it is useful for highlighting where the new services will be performed within the Quantum Internet.

3.8.1 Control Plane Applications The control plane applications represent the set of mechanisms, based upon quantum mechanics, that control and implement all the dynamics of infrastructure reconfiguration and network management. They include network logical topology configuration, information routing, packet forwarding, and any kind of system configuration.

3.8.2 Quantum Security A strong impact of the Quantum Internet will surely be on the aspects of cryptography, authentication, and cyber-security in general, empowering security level provided by classical encryption system. The quantum mechanics laws allow to completely reshape the approach to the cyber-security, moving from a “computationally difficult to solve” approach to a physically proved unconditional security, at least in theory not dependent on the amount of computational power an adversary could have. By using classical encryption systems, the only obtainable guarantee (neither theoretically proved) is that an intruder would require a huge amount of time to decrypt the messages with current technology. Moreover, in classical networks, nothing prevents a malicious listener from sniffing the channels and storing the data for future decryption, in a time where technology advances in quantum computing or information theory may facilitate his work and when current cryptograms may become vulnerable or obsolescent. To this end, quantum cryptography will instead enable a novel level of security due to its entirely different approach: the transmission is made over very sensitive links, such that any interaction with the channel can be easily identified by the two communicating parties, allowing immediately detecting an intruder. These techniques are proved to be unconditionally secure, i.e., their security is no more dependent on the computational power that the adversary possesses, ensuring security over technological advances and also against timedelayed attacks. In one of its possible implementations, the Quantum Internet will make a pervasive use of quantum encryption techniques, such as quantum key distribution, to continuously generate secure keys that will be distributed across the Internet, not only to the end users but also to secure infrastructure and devices. In such a network, all control nodes and connected devices will receive secure keys for authentication purposes and for constant monitoring of unauthorized users. The secure keys may be generated by dedicated nodes using quantum random generators and stored to be distributed on-demand. All the applications furtherly described in this section are conceived to be implemented in such secured environment.

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3.8.3 Fast Byzantine Negotiation Among network primitives, there exist a number of so-called consensus problems that are considered classically unsolvable but that could empower the network features by a huge amount. Consensus agreements encompass a broad class of problems related to determining multi-agent condition agreements, where it cannot be assumed that all agents are trustable. One well-known of those problem is the Byzantine Negotiation problem, which is proved to be unsolvable by classical information theory [35, 36]. The traditional, informal description of the problem is the following one [37]: “A number of divisions of the Byzantine army are camped outside an enemy city, each division commanded by its own general. The generals can communicate through messages to agree on a plan of action, which is finally decided by a commanding general. Unfortunately, some of the generals (also the commanding one) might be a traitor, trying to prevent the loyal generals from reaching an agreement for the plan. The problem stands in determining a protocol among the generals that, after its terminations, guarantees that: – All loyal generals agree on a plan – If the commanding general is loyal, then all loyal generals agree on the same plan.” The situation is depicted in Fig. 3.17.

Fig. 3.17 General Byzantine agreement problem

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That Byzantine problem can be formalized as: Byzantine Negotiation It is given a set of n + 1 users, among which one, S (the sender), is in charge of distributing a value x ∈ D among the other n partners, where D is a finite-sized dictionary. Upon the reception of such a value, every user Ui will decide a value yi ∈ D . The system is said to achieve the broadcast (condition agreement), if and only if for every honest user it holds that ∀i, j ∈ {1, n}. yi = yj = x, whenever S is honest. This task is fundamental for a distributed network of resources in all those contexts where faulty components or malicious users have to agree on a common view of shared information. One important example regards datacenters, where storage units are redounded and where their state consistency is shared and must be agreed, or, in general, any distributed data structure where the state consistency is fundamental. To give an idea of the importance of this problem, we have to consider that today around 7500 databases are stored in datacenters worldwide [38]. In cloud networks, the databases are physically located away from the place where the data are generated and may adopt a distributed architecture for both efficiency reasons and security purposes. This is the reason why reliable database interconnection is of utmost importance. The same concept applies to any use case relying on a distributed infrastructure, as it happens for crypto coins. The working principle is based on the absence of a central authority managing the transaction (bank) that is substituted by a distributed encrypted data structure of which the state is computed in a distributed manner by all the users. The users interact with the system, contributing to build its distributed state and so recording the set of transactions among users. The final goal is to achieve a fair monetary system, where no incumbent user can maximize its profits at the expenses of the others. However, any of the users, or partners, using the system may will to influence the distributed state for its own advantage, to inflate his or her saving account, to this end, so the capability to perform a fair and trusted negotiation of the network resources is fundamental. Unfortunately, this problem is known to be unsolvable classically in its general form. Instead, quantum algorithms exist to solve it, and a simpler version of the byzantine negotiation has been also implemented [39]. For example, let us consider the problem over a set of three users, one of them being the sender S that will broadcast over a channel one bit that the two other users, U1 and U2 , will receive. The goal is to end up with the three users agreeing upon the same value of the bit broadcasted by S. Only pairwise classical channels are assumed for the moment. We know that at most one user may be dishonest and may cheat on the bit value, flipping its own bit. By mutually exchanging this bit, the users can understand if someone is cheating or not, just by comparing the values received from different users. However, nobody will be able to understand who is actually cheating, if S, or the other user U. We already stated that this problem is unsolvable in its general form, according to the classical information theory. By introducing a

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quantum channel, however, we can solve a “slightly weaker” version of the problem, obtained by relaxing the agreement condition as follows: – If no user is dishonest (i.e., he does not flip bits), then the agreement is reached; – If one or more users are dishonest, then either the users reach an agreement or the honest users abort the protocol. This weaker problem remains unsolvable using only classical pairwise channels, but it can be solved by introducing also quantum channels. The idea is based on the distribution of entangled qutrits (i.e., 3-qubits) between the three users, which are used to verify malicious flipping of the bits. Though the approach is similar to that adopted by QKD protocols, there are significant differences. The essential part of the algorithm stays in testing the distributed entangled states, whose distribution is easily achieved through the available quantum channels. For the protocol to work, the three users have to share a high number of qutrits |ψ j each in the socalled Aharonov state, |A = (|0, 1, 2 + |1, 2, 0 + |2, 0, 1 ) √1 , where |0, 1, 2 , 6

|1, 2, 0, , |2, 0, 1 represent the tensor product between the components. The protocol starts with the sender S distributing via classical channels its decision bit, x, to U1 , U2 . The sender S then measures all his qutrits and distributes to the other users the state indexes for which the result was exactly x . The same is done by U1 , U2 that measure their qutrits and check the consistency with the indexes received from S. The users then share their results to check if some inconsistency exists between their measures. Such a way, they verify the loyalty of the sender S. Then, to crosscheck their own loyalty, they start sharing a high number of indexes for which the result is different from x , checking the rate of consistent measures. This allows one user to statistically check the honesty of the other user, similarly to the working principle of the BB84 protocol. If some inconsistency is detected, the users stop the process without trusting the broadcasted value. This is just one example of distributed agreement of a shared state, but many other protocols have been proposed [40–43].

3.8.4 Position Verification Mechanisms Location verification represents another new quantum application, not possible with classical-only channels [44]. It exploits the distribution of quantum entangled pairs in a network. The feature of position verification will be integrated in the Quantum Internet infrastructure, that is, quantum protocols will be put in place to unconditionally verify the geographical location of any user or equipment, also in the case they are dishonest or inaccurate in declaring it. This may be fundamental in many scenarios and may enable a range of new applications, such as positionbased cryptography, where authentication or encryption is computed starting from user location. For example, it will enable the possibility to discriminate or directly use the unique position of the target for authentication purposes.

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This problem is correlated to but slightly different from other cryptographical problems, such as QKD, coin flipping [68], and bit commitment. It can be treated using quantum entanglement, where using classical channels is instead not possible. Many implementation solutions to this task have been proposed [45–47], achieving also device independence [48], and all of them are fundamentally based on two main quantum principles: sharing of entangled pairs and quantum teleportation. Many are also the proposed scheme to use quantum position verification for quantum encryption protocols, the first of which, named Quantum Tagging, was investigated by Kent in 2002 [49] and many other protocols were proposed in 2010 [50]. All the implementations of position verification rely upon two main macro concepts: – Distance bounding [51], the unavoidable classical propagation limit of any communication, i.e., speed of light, that can be used to determine the lower bound of the distance from the target (but not the upper bound). The time needed to receive the information limits the radius of the circle in which the partner can be located. Intersecting the circles corresponding to multiple receivers, it is possible decrease the position uncertainty. – Figure 3.18) – Quantum position bounding, the determination of a quantum task, including entanglement sharing, that could deterministically be computed exclusively and uniquely in the authorized location declared by the user. The tremendous complexity related to this task lays in the possibility to have coalitions of multiple dishonest users who could try to generate messages from properly designed location from the verifiers, in order to convince the prover they are a unique user at a specific location, while none of them is really there. In its most simple form, the quantum protocol can be stated as follows in a 1-d space system, as follows.

Fig. 3.18 Position uncertainty reduction based on multiple receivers

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Quantum Verification Protocols (1D-QPVθ)

Given a Prover P at a certain position p, and a number of verifiers Vn where n ∈ [1, .., N] with minimum number depending on the space dimensionality of the system, the verifiers have the goal to verify if P is really in position p and to identify possible malicious users (or set of users) trying to declare a wrong position. To do so, the verifiers share 2 bits x, b ∈ {0, 1}. They then prepare, using x and b, the state |ψ = (Rθ )b |x in V1 , where Rθ = (cos cos θ − sin sin θ sin sin θ cos cos θ ) and θ ∈ [0, 2π ] (Fig. 3.19). At this point, upon synchronization between the verifiers, V1 transmits |ψ to P, while V2 simply transmits the bit b . Such transmission is performed in such a way that the information is received at the same time in P, based on the knowledge of their relative distance to P. Upon the reception of |ψ , b , the prover P carries out the measurement of |ψ on the proper basis that is dependent on b resulting on a value x . Prover then broadcasts x to the verifiers, which checks correctness and timing of the results. The time required to get back the response from P is limited by the speed of light, and the correct measure of the result determines that only a user exactly at position P may be able to perform measurement of |ψ , b in time, guaranteeing that he or she is the only participant to the protocol. More in general, considering a number of k verification stations, an analogous protocol can be employed, where N maximally entangled systems within the network of the k stations are considered, and where each system contains also k qubits for which any verification stations hold one of such qubits for every system. Solving this problem is possible to address similar problems that are as important as position verification; an example lays in the 2PC (two-party cryptography), where two partners, honest or not, try to cooperatively solve a task, pretending to be protected by malicious actions of the potential partner.

Fig. 3.19 Quantum verification protocol scheme

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The ability to offer a communication system capable of unconditionally guaranteeing its trust based on a function of the receiver location provides a range of new information security paradigms and applications: many are the industries and organizations that would be interested in delivering information content being sure that the recipient is at an a priori agreed location.

3.8.5 Increasing the Data Rate Between Nodes: Superdense Codes Another important feature of the Quantum Internet will be the communication performance empowering that can be enabled by quantum mechanics. Many are the works toward this direction and much advancement is still needed. Today, most of the results exploit the properties of superdense codes. Superdense coding is implemented by exploiting quantum mechanism effects such as entanglement swapping and generation of state superposition. It can be considered as the dual operation of quantum teleportation, where the transfers of a quantum state between two partners are made by communicating two classical bits, as long as the users pre-share a Bell pair. In superdense coding, it happens the opposite: assuming the pre-sharing of an entangled Bell pair, transmitting one classical bit between the users, it is possible to share two bits of information, by properly exploiting the measure of the entangled pairs [52]. The Quantum Internet will be characterized by continuous distribution of entangled pairs among partners, and it is reasonable to assume that parallel quantum states and entangled pairs will be distributed among users, equipment, and devices. In this way, once a user decides to transmit a classical packet to any other user in the network, assuming that they already share an entanglement pair, superdense coding can be used to achieve an exact doubling of the final data rate. In case two users want to establish a single duplex channel but no entanglement is shared, it has been shown a scheme in [53], where the backward link is used to generate and share a stream of entangled pairs (half of the pair are retained for the protocol to work); in this way, the channel may exploit the backward channel to empower the forward link doubling the rate in one direction, without the preemptive sharing of entanglement. Other protocols have been shown that achieve the same doubling of the rate, properly preparing superposed quantum states at the sender, without the need of an entangled pair to be shared [54]. This kind of protocols is extremely helpful in those scenarios where bursts of critical information need to be transmitted after a long silent period where entangled pairs were accumulated. We essentially accumulate entangled pairs to boost the communication when needed. The rate doubling is achieved at the bit level, for which, depending on the amount of the entangled pairs shared, a huge aggregated capacity boost can be achieved by exploiting many independent classical channels in the network, where in each classical channel superdense coding is applied. This is not the ultimate

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boosting achievable with superdense coding. Adopting higher-order entangled states, the boosting process due to superdense coding can be furtherly increased, as proved in [55], where they built a system adopting ququarts and achieved an information capacity boost of 2.09 +/− 0.01 bits per bit.

3.8.6 Empowered Channel Capacity and Noise Resilience Other than the specific case of the superdense coding, many are the results in the field of quantum information theory trying to determine the maximum information rate transmittable over a channel empowered by quantum mechanics. Classically, it is well-known, thanks to Shannon theory, what is the amount of information that can be transmitted over a channel [56]. However, the classical information theory operates on top of probability theory over distributions of symbols, for which Shannon’s results on capacity guarantee transmission rate only in average over the channel usage. The field of quantum information theory is a very wide theoretical branch, impacting the way a channel is defined and used. One of its outcomes is the zero-error capacity of the quantum channel, thanks to which it is possible to describe the amount of classical information that can be transmitted with zero error probability through a noisy quantum channel [57]. Classically, instead, the channel capacity can only be defined in the presence of an arbitrary small error probability, never vanishing to zero. The zero-error capacity could be exploited to design high priority communication links, where perfect transmission conditions have to be granted and no errors are allowed. An astonishing feature of zerocapacity channels is that information can be transmitted over a channel for which classically the achievable rate is zero. This fits to extremely noisy channels or channels preventing communications for some other reason, like interference. An example is given by one of the simplest and most famous classical channels, the binary symmetric channel (BSC) in Fig. 3.20: The BSC channel is described by a parameter, p , which is the probability that over the transmission of a bit, it would be flipped in value (0 → 1 or 1 → 0). This channel, in the worst case for the information rate (p = 12 ), is well-known to achieve a maximum information rate of zero, since totally random Fig. 3.20 Binary symmetric channel—bit flipping

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flipping is introduced. This could be compared to a real channel where the noise totally denies any chance of transmitting information. However, it has been proven that by using entanglement elements, it becomes possible in some scenarios to transmit information, enabling information transfer over classically zero-capacity channels [57]. This quantum information theory branch is still under study and limited to the theoretical results, so it is hard to state if and how it will impact the Quantum Internet. However, if implementable, these results may lead to design quantum-based communication protocols able to transmit information on extremely noisy channels, for example, in deep-space missions, underwater links, or in critical scenarios where communications may be interrupted by some adversary exploiting jamming or other techniques.

3.9 Quantum Empowered Optimization and Artificial Intelligence Under the assumption of having a number of quantum devices interconnected and capable of sharing entangled states, an ideally infinite number of optimization and artificial intelligence applications can be envisioned, enabling an extremely adaptive and resilient network. An example lays in anomaly detection and recovery planning tasks that are fundamental operations in any complex system and network, where critical assets are installed and cascade faults may be hardly predicted and recovered automatically. Fault tree analysis for preventing malfunctioning of hardware and software components, with the goal of predicting and recovering failures is a well-known and studied branch. Unfortunately, fault tree analysis over direct acyclic graphs, like most of the practical systems are modelled, is considered very hard to solve classically and, more specifically, cannot be solved in polynomial time (unless simplified assumptions are made on system’s behavior). However, this task is proved to be implementable through a quantum annealer, for which today commercial implementations exist, enabling accurate and pervasive detection of faults. This is only an example: other optimization problems could be boosted using quantum computing, such as resource usage and allocation, routing algorithms, protocol selection and optimization, control plane system parameter optimization, etc. Besides this, there are many machine learning, classification, and artificial intelligence applications that will empower the capacity of the network infrastructure, thanks to quantum technologies. As an example, unsupervised learning could be used to cluster data over similar features without the need of tagged input, highlighting datasets that manifest similar behaviors, e.g., to detect illegal or malicious dataflows. In the domain of clustering algorithms, many are the approaches that are classically unsolvable in polynomial time, which instead could be solved using quantum machines. An example are clustering algorithms based on Max-Cut problem [58]

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over graphs, for which an implementation has been proved to be implementable to a quantum device and some implementations have already been published [59].

3.9.1 Network Clock Synchronization Metrology deals with measurements and estimation of unknown physical parameters. Current metrological technologies already strongly exploit quantum mechanics, relying their foundation on atomic oscillators, resonators, and lasers. However, it has been proved how quantum mechanical effects, such as superposition and entanglement, can lead to big advances. A fundamental metrology related aspect in the Quantum Internet is given by time and frequency synchronization of nodes and devices in the network, providing them with a globally agreed reference time. This feature is essential to any kind of data networks, financial trading systems, airport traffic control, rail transportation networks, telecommunication networks, and navigation or localization system. It may enable unprecedented performance in navigation systems and earth sensing, as well as more accurate imaging systems [60] and new fundamental tests of quantum physics and on non-locality. To this end, synchronization and tuning of clocks is an essential task. Classically there are two approaches that are often used: Einstein synchronization and Eddington’s slow clock transport. The first is based on a line-of-sight pulse transmission forth and back between two nodes. Knowing the speed of light, it is possible to determine their time relation that can be used for local clock synchronization [61]. The second approach instead relies on the assumption that synchronizing two clocks in their proximity and then moving them away at low speed with respect to the reference time–space frame guarantee that the two clocks remain almost synchronous, being negligible the time dilatation effect. Recently, a number of schemes have been published where distributed entangled states and their measurement are exploited to synchronize distant clocks [28]. The known limit of classical synchronization scheme is the noise introduced by the channel where the time information is transmitted, which introduce unavoidable synchronization errors. The idea behind entanglement-based synchronization relies on sharing states that are mutually measured to obtain coherent results that are used for synchronization. No time information is directly transmitted over the channel with these schemes. This is similar to entanglement-based QKD protocols, where prior-entangled states are shared among the users, and where, by measuring the states and transmitting classical values, a secret key is built that does not exist at the start of the protocol. Similarly, time synchronization information is retrieved by the node after state measurement and classical information sharing among the nodes, increasing the chance of reducing synchronization errors. The main limit of this class of protocols in their basic formulation is their implicit assumption of a common phase reference between the nodes that is required to generate the entangled pairs. Indeed, the definitions of superposed states of qubits are based on a conventional phase defined locally. Dealing with this issue is as hard as

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synchronizing the nodes [62, 70]. This problem has been explored in [63], where it has been shown how entanglement distillation of states can be used to avoid this issue. Many protocols are also being studied and experimented to show multi-node and network quantum synchronization [64]. These approaches will enable incredible accuracy in time and frequency synchronization over extremely distant assets with high resilience to noisy channels.

3.9.2 User Plane Applications User plane applications represent the set of features that are directly exposed to the end users, facilitating their interaction with other users and dynamics in the Internet. Many of the features described in the previous section can be also provided to end users as primitives or may indirectly impact a feature offered to them. Examples are agreement protocols, position verification algorithms, quantum optimization primitives, and quantum random number generation, allowing the user to exchange secret keys with another user. It is relevant to describe, more than additional specific user plane applications, how will the end users access these services. Indeed, the users will be connected to the Quantum Internet in a number of different ways: with classical machines, using quantum as a service, or with their own quantum machine, directly participating to the quantum algorithms computation and interacting with other quantum nodes through the direct exchange of quantum states, possibly realizing distributed quantum machines.

3.9.3 Quantum Computing as a Service The Quantum Internet can be exposed as a service to users owning classical machines, offering its quantum infrastructure and computing resources as a service, as it happens in the classical Cloud. When we refer to end users, they have to be thought abstractly, since they could be human beings, but also automatic machines, networks of sensors, satellites, space vehicles, and so on: any relevant application domain will have the chance to perform quantum computation on demand and achieve incredible performance speedup. This is of foremost importance in all classical systems with specific bottlenecks that can be offloaded to a cloud quantum machine. It is to be noted that the users will use classical information transmissions and channels to communicate. So they need to adopt some classical formalism to represent the problem to be solved by the quantum computer. The formalism will then be interpreted by a remote quantum machine in terms of quantum circuits and logical gates. In this network architecture, classical applications will run on standalone machines but will have the chance to exploit quantum primitives on-demand, performing the computation in the closest

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Fig. 3.21 Classical User accessing the QI

quantum computer in the net. A network scheme representing the network operating principle is given in Fig. 3.21. In general, only a subset of the possible available quantum primitives will be exploitable by the user; indeed, classical machines cannot perform tasks that require a local manipulation of the quantum state. Possible available quantum primitives are: quantum optimization tasks, quantum artificial intelligence primitives, quantum simulations (i.e., simulating physical phenomena and processes down to the quantum level, without the need to introduce approximations), and quantum random generators.

3.10 Direct and Distributed Quantum Computing In different network architecture, users actively control a quantum device and interact with the Quantum Internet over quantum channels. Those users can potentially execute locally the already mentioned primitives, without the necessity to offload them to the quantum network. In this scenario, each user is a node of the Quantum Internet and can expose quantum computing resources to other users. Moreover, the users can adopt the Quantum Internet to increase the power of their local machines, distributing the computation tasks by sharing quantum states and entangled pairs with the network. This enables the full power of the Quantum Internet, including features and protocols that assume local quantum state manipulation. A network principle scheme is given in Fig. 3.22. Other than the local computation of the algorithms mentioned in the previous section for a classical user, the quantum user has access to the set of quantum protocols described in the previous sections such as:

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Fig. 3.22 Quantum User accessing the QI

• Privacy preservation (blind computing): The user will be capable, as already described, to cooperate or offload the computation to other quantum machines in the network that are not known or trusted, without the risk of exposing private information. This will enable high performance computation, by distributing the computation task, also for sensitive applications, where the privacy has to be preserved. • Leader Election (agreement algorithms): The user will be able to agree over distributed states shared among a number of possibly dishonest users, with the guarantee that an agreement will be reached only in the case that all the users are really honest. This will enable trusted distributed data structures in the network, enabling in cascade, a huge number of other applications, as described in Fast Byzantine agreement section. • Quantum Secret Sharing: The user will be able to share a secret information among a number of users, with the goal of guaranteeing that no single person or part of each department could read out the secret message, but only the set of all members of each group can. This means that for security to be breached, all people of one group must act together. Using quantum protocols, this could be achieved also in the scenario of dishonest users that cannot anyway sniff the secret also by stealing all the information of their own group [65, 66]. • Position verification: Verify the geographical positioning of a user with which you are interacting, as already described in the previous sections. This will enable the chance to authenticate the users by their geographical position. • End-to-end quantum key distribution.

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3.11 Quantum Internet in the Space Space communication is for sure one of the most important enabling technologies for the Quantum Internet. As already emerged in many sections of this book, one of the main limiting factors of the quantum communications and for the realization of a complex quantum network is the sensitivity of the quantum states to the propagation over long distances, and their capability to last in time. This is directly related to the complexity of implementing components such as quantum memories and quantum repeaters. Then interconnecting quantum nodes over long distances to build a wide-area Quantum Internet faces big technological hurdles. One of the possible solutions is taking a detour in space, a context that naturally and easily deal with long distances and where channel impairments are limited compared to the terrestrial scenario. With the incredible recent growth of the space market, satellitebased quantum networks will be an important asset for the implementation of the Quantum Internet, as well as the increasing number of satellite services. New frontiers of communication, navigation, and sensing will rely on massive number of objects flying in the atmosphere to assure high coverage and visibility of satellite systems. Many are the impacts of the Quantum Internet: the future Internet backbone may be orbiting around the earth, together with our personal data; observation of earth for critical alerting and catastrophe prediction and monitoring is already on going and will in future expand in more application fields. There are different reasons for which satellites represent an interesting asset for the Quantum Internet: • Satellite systems achieve global-scale coverage, which could be exploited to implement a global geographical Quantum Internet, using near-term technology. Today, the space represents the best domain to achieve global coverage with minor effort in terms of installation complexity and regulatory handling. • Satellites are a naturally secure system due to the physical difficulty in accessing them. They are located in free space, at hundreds of kilometers from the surface, traveling at speeds of thousands of kilometers per hour. By specifically designing them, they can represent the best way to implement the “trusted node” needed in quantum communications. Trusted node can be used to easily extend the quantum communication distances without the necessity of quantum repeaters, enabling intermediate ready-to-use solutions. • The most adopted medium for the transmission of quantum states makes use of photons in the visible to near-infrared frequency range. This is due to the easiness of their generation and of being transmitted between nodes. Free space is an optimal medium for photons since it is known they experience very low absorption and decoherence, while their detection is quite efficient. This is true in respect to ground-based free-space communications, where atmospheric noise undergoes an exponential decrease in transmission success probability with distance. Fiber optic networks instead suffer from high deployment complexity and cost to achieve global coverage or for updating existing infrastructures.

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A space-based Quantum Internet architecture will so employ space assets (satellites) as routing node interconnecting highly distant ground terrestrial network, enabling a hierarchical structure similar to those of core-metro-access networks in the classical Internet. Depending on the distance between nodes, the Quantum Internet will probably rely on different approaches. For distances of a few hundreds of kilometers, terrestrial fiber-based local area networks will be probably employed for direct connections. Scaling up to more than 500 km, quantum repeaters will be employed to increase the achievable distance. Moving up to communications beyond thousands of kilometers, satellite systems will start to become fundamentals. Low/medium orbit satellites will be used either to bridge distant terrestrial nodes or to transfer quantum states among geographical places, thanks to their incredible speed (they can travel thousands of kilometers in a few minutes and earth revisit time can be shorter than few hours). Finally, the very longest distances (beyond 5000 km) will likely require either more distant satellites, for example, geostationary (GEO) satellites. GEO satellites can exploit their stationary position to build static network topologies spanning continents, with high pointing quality and link stability. High altitude satellite will be used also as repeaters with low-earth orbit (LEO) satellite links. Figure 3.23 shows a schematic scheme of the hierarchical satellite network. Indeed, also between space assets there will be a hierarchy of satellites with different features: • Few GEO satellites, at high altitude and geostationary, will be used to connect areas over continental distances. The stationarity property will decrease the issue of pointing at ground and at other GEO satellites, since their relative position

Fig. 3.23 Quantum Internet in Space

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is fixed and can be optimized to maximize link efficiency. Moreover, they will be used as trusted nodes, similarly to quantum repeaters, to increase the overall distance achievable in transferring quantum states. • LEO and medium earth orbit (MEO) satellites, orbiting at very high speed, with a revolution period of few hours, will be used to store quantum states and to transfer them over long geographical distances at ground but also to connect distant metropolitan areas. They will transfer the quantum states above to higher satellites if longer communication is needed.

3.12 How Far Is a Quantum Internet from Reality? In the previous sections we described a very broad range of supposed features of a future Quantum Internet. It is to be noted, however, that a Quantum Internet will actually take shape depending on a huge number of factors, including economical ones [67]; and, of course, related to how the scientific progress will evolve in the various relevant branches and on the limits that may be discovered in the next years. What we did in this chapter is, starting from the current technology status, imagining how all these technologies may evolve and contribute to the Quantum Internet. Many others are the scenarios can be envisioned, more conservative or more creative depending on the assumptions made. At the current time, the experimental status of long-distance quantum networks is still at its bronze age: trusted-node networks have been demonstrated over metropolitan areas, but without the capability of end-to-end long-haul qubits transmission. A few QKD commercial systems are already available on the market. Further from this stage, scaling up quantum networks technology is an extremely hard endeavor, requiring sustained and concerted efforts in physics, computer science, and engineering to succeed. Nowadays, limited cases of wide entanglement distribution have been shown over reasonable distances, nodes numbers, and integrability with existing infrastructure [69]. But much better quantum memories are needed for guaranteeing robust storage of quantum states, needed for both distributed quantum computing, for which quantum states have to be also robust to quantum operations, and for entanglement-based protocols between end-nodes. The same holds for quantum repeaters, fundamental in extending the distance of communication protocols. Moreover, a very limited number of compatibility study with photonic communication hardware has been performed to prove the integration of quantum devices and networks in the existing network infrastructure, which is a fundamental step for an economically sustainable Quantum Internet. In summary, how the Quantum Internet presented in this chapter has to be considered as a best-case scenario, where all these problems are assumed to be solved, which may require long time.

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References 1. Perry Tekla, S. (2020). Move over, Moore’s law: Make way for Huang’s law. IEEE Spectrum. 2. Mims, C. (2020). Huang’s law is the new Moore’s law, and explains why Nvidia wants arm. Wall Street Journal. https://www.wsj.com/articles/huangs-law-is-the-new-moores-lawand-explains-why-nvidia-wants-arm-11600488001. 3. Wehner, S., Elkouss, D., & Hanson, R. (2018). Quantum internet: A vision for the road ahead. Science, 362(6412), eaam9288. 4. DiVincenzo, D. P. (2013). The physical implementation of quantum computation. Fortschritte der Physik, 48(9–11), 771–783. 5. Stanford Encyclopedia of Philosophy. (2020). The role of decoherence in quantum mechanics. 6. Thomas, R.A., Parniak, M., Østfeldt, C., Møller, C.B., Bærentsen, C., Tsaturyan, Y., Schliesser, A., Appel, J., Zeuthen, E., & Polzik, E.S. (2021). Entanglement between distant macroscopic mechanical and spin systems. Nature Physics, 17(2), 228–233. 7. Kozlowski, W., & Wehner, S. (2019). Architectural principles for a quantum internet. IETF Internet draft-irtf-qirg-principles-01. 8. Ekert, A. K. (1991). Quantum cryptography based on Bell’s theorem. Physical Review Letters, 67, 661. 9. Bennett, C. H., Brassard, G., Mermin, N. D. (1992). Quantum cryptography without Bell’s theorem. Physical Review Letters, 68, 557. 10. Fitzsimons, J. F. (2017). Private quantum computation: An introduction to blind quantum computing and related protocols. NPJ Quantum Information, 3, 23. 11. Broadbent, A., Fitzsimons, J., & Kashefi, E. (2009). Universal blind quantum computation. In 50th Annual IEEE Symposium on Foundations of Computer Science. https://doi.org/10.1109/ FOCS.2009.36 12. Cuomo, D., Caleffi, M., & Cacciapuoti, A. S. (2020). Towards a distributed quantum computing ecosystem. arXiv:2002.11808v2. 13. Gottesman, D. (2009). An introduction to quantum error correction and fault-tolerant quantum computation. arXiv:0904.2557v1. 14. Devitt, S. J., Munro, W. J., & Nemoto, K. (2013). Quantum error correction for beginners. Reports on Progress in Physics, 76(7), 076001. 15. Rofef, J. (2019). Quantum error correction: An introductory guide. Journal Contemporary Physics, 60(3), 226–245. 16. Eastin, B., & Knill, E. (2009). Restrictions on transversal encoded quantum gate sets. Physical Review Letters, 102(11), 110502. 17. Li, Y., Hua, S., Liu, Y., Ye, J., & Zhou, Q. (2007). Quantum repeaters: Fundamental and future. In Quantum Information and Computation (Vol. 6573). https://doi.org/10.1117/12.717206 18. Kwiat, P. G., Barraza-Lopez, S., Stefanov, A., & Gisin, N. (2001). Experimental entanglement distillation and ‘hidden’ non-locality. Nature, 409, 1014–1017. 19. Van Meter, R., Ladd, T. D., Munro, W. J., & Nemoto, K. (2008). System design for a long-line quantum repeater. IEEE/ACM Transactions On Networking, 17(3), 1002–1013. 20. Cavaliere, F., Prati, E., Potí, L., Muhammad, I., & Catuogno, T. (2020). Secure quantum communications technologies and systems: from labs to markets. Quantum Reports, 2(1), 80– 106. https://doi.org/10.3390/quantum2010007 21. Boaron, A., Boso, G., Rusca, D., Vulliez, C., Autebert, C., Caloz, M., Perrenoud, M., Gras, G., Bussiéres, F., Li, M. J. & Nolan, D. (2018). Secure quantum key distribution over 421 km of optical fiber. Physical Review Letters, 121, 190502. 22. Pirandola, S., Laurenza, R., Ottaviani, C., & Banchi, L. (2017). Fundamental limits of repeaterless quantum communications. Nature Communications, 8, 15043. 23. Tang, J.-S., Zhou, Z.-Q., Wang, Y.-T., Li, Y.-L., Liu, X., Hua, Y.-L., Zou, Y., Wang, S., He, D.Y., Chen, G., et al. (2015). Storage of multiple single-photon pulses emitted from a quantum dot in a solid-state quantum memory. Nature Communications, 6, 8652.

122

F. Cavaliere et al.

24. Krovi, H., Guha, S., Dutton, Z., Slater, J. A., Simon, C., & Tittel, W. (2016). Practical quantum repeaters with parametric down-conversion sources. Applied Physics B, 122, 52. 25. European Quantum Flagship. Strategic Research Agenda (2020). 26. Pirandola, S., Andersen, U. L., Banchi, L., Berta, M., Bunandar, D., Colbeck, R., Englund, D., Gehring, T., Lupo, C., Ottaviani, C., Pereira, J. L., Razavi, M., Shaari, J. S., Tomamichel, M., Usenko, V. C., Vallone, G., Villoresi, P., & Wallden, P. (2019). Advances in quantum cryptography. arXiv:1906.01645. 27. Rozpedek, F. (2019). Building blocks of quantum repeater networks. 10.4233/uuid:ed0af5137621-4007-9a34-1a3e17370952, Dissertation Thesis at Delft University of Technology. 28. Rancic, M., Hedges, M. P., Ahlefeldt, R. L., & Sellars, M. J. (2018). Coherence time of over a second in a telecom-compatible quantum memory storage material. Nature Physics, 14, 50. 29. Riebe, M., Monz, T., Kim, K., Villar, A. S., Schindler, P., Chwalla, M., Hennrich, M., & Blatt, R. (2008). Deterministic entanglement swapping with an ion-trap quantum computer. Nature Physics, 4, 839–842. 30. Dür, W., & Briegel, H. J. (2007). Entanglement purification and quantum error correction. Reports on Progress in Physics 70, 1381. 31. Jiang, L., Taylor, J. M., Nemoto, K., Munro, W. J., VanMeter, R., & Lukin, M. D. (2009). Quantum repeater with encoding. Physical Review A, 79, 032325. 32. Munro, W., Stephens, A., Devitt, S., Harrison, K., & Nemoto, K. (2012). Quantum communication without the necessity of quantum memories. Nature Photonics, 6, 777. 33. Gyongyosi, L., Imre, S., & Nguyen, H. V. (2018). A survey on quantum channel capacities. IEEE Communications Surveys & Tutorials, 20(2), 1149–1205. 34. Applications and Use Cases for the Quantum Internet, draft-irtf-qirg-quantum-internet-usecases-02 35. Lamport, L., Shostak, R., & Pease, M. (2019). The Byzantine generals problem. Concurrency: The works of Leslie Lamport (pp. 203–226). 36. Pease, M., Shostak, R., & Lamport, L. (1980). Reaching agreement in the presence of faults. Journal of the ACM 27(2), 228–234. 37. Lamport, L., Shostak, R., & Pease, M. (2019). The Byzantine generals problem. Concurrency: The works of Leslie Lamport (pp. 203–226). 38. Brands, S., & Chaum, D. (1993). Distance-bounding protocols. In Workshop on the theory and application of cryptographic techniques. Berlin, Heidelberg: Springer. 39. Fitzi, M., Gisin, N., & Maurer, U. (2001). Quantum solution to the Byzantine agreement problem. Physical Review Letters, 87(21), 217901. 40. Sun, X., Kulicki, P., & Sopek, M. (2020). Multi-party quantum byzantine agreement without entanglement. arXiv preprint arXiv:2003.09120. 41. Gao, F., et al. (2008). “Comment on “Experimental demonstration of a quantum protocol for Byzantine agreement and liar detection. Physical Review Letters 101(20), 208901. ˙ 42. Rahaman, R., Wie´sniak, M., & Zukowski, M. (2015). Quantum Byzantine agreement via Hardy correlations and entanglement swapping. Physical Review A, 92(4), 042302. 43. Luo, Q.-B., Feng, K.-Y., & Zheng, M.-H. (2019). Quantum multi-valued byzantine agreement based on d-dimensional entangled states. International Journal of Theoretical Physics, 58(12), 4025–4032. 44. Chandran, N., et al. (2010). Position-based quantum cryptography. arXiv preprint arXiv:1005.1750 45. Malaney, R. A. (2010). Location-dependent communications using quantum entanglement. Physical Review A, 81(4), 042319. 46. Malaney, R. A. (2010). Quantum location verification in noisy channels. In 2010 IEEE Global Telecommunications Conference GLOBECOM 2010. IEEE. 47. Malaney, R. A. (2010). Location-dependent communications using quantum entanglement. Physical Review A, 81, 042319. arXiv:1003.0949. 48. Ribeiro, J., Murta, G., & Wehner, S. (2016). Fully general device-independence for two-party cryptography and position verification. arXiv preprint arXiv:1609.08487.

3 The Future Quantum Internet

123

49. Kent, A. P., Munro, W. J., Spiller, T. P., Beausoleil, R. G. Patent US2006022832 (A1)—tagging systems. 50. Kent, A., Munro, W. J., & Spiller, T. P. (2011). Quantum tagging: Authenticating location via quantum information and relativistic signaling constraints. Physical Review A, 84(1), 012326 51. Brands, S., & Chaum, D. (1993). Distance-bounding protocols. In Workshop on the Theory and Application of Cryptographic Techniques. Berlin, Heidelberg: Springer. 52. Bennett, C. H., & Wiesner, S. J. (1992). Communication via one-and two-particle operators on Einstein–Podolsky–Rosen states. Physical Review Letters, 69(20), 2881. 53. Herbert, S. (2020). Increasing the classical data throughput in quantum networks by combining quantum linear network coding with superdense coding. Physical Review A, 101(6), 062332. 54. Kwiat, P., Bernstein, H., & Javadi, H. (2016). Entanglement-assisted communication system for NASA’s deep-space missions. 55. Hu, X.-M., et al. (2018). Beating the channel capacity limit for superdense coding with entangled ququarts. Science Advances, 4(7), eaat9304. 56. Shannon, C. (1948). A mathematical theory of communication. Bell Labs Technical Journal, 27, 379–423, 623–656. 57. Gyongyosi, L., Imre, S., & Nguyen, H. V. (2018). A survey on quantum channel capacities. IEEE Communications Surveys & Tutorials, 20(2), 1149–1205. 58. https://en.wikipedia.org/wiki/Maximum_cut 59. Willsch, M., et al. (2020). Benchmarking the quantum approximate optimization algorithm. Quantum Information Processing, 19, 197. 60. Gottesman, D., Jennewein, T., & Croke, S. (2012). Longer-baseline telescopes using quantum repeaters. Physical Review Letters, 109(7), 070503. 61. Einstein, A. (1905). Annalen der Physik, 17, 891. Einstein. The Swiss years: Writings, 1900– 1909 (Vol. 2, pp. 140–171). Princeton, NJ: Princeton University Press. 1989. 62. de Burgh, M., Bartlett, S. D. (2005). Physical Review A, 72, 042301. 63. Ilo-Okeke, E. O., et al. (2020). Entanglement-based quantum clock synchronization. In AIP Conference Proceedings (Vol. 2241. No. 1). AIP Publishing LLC. 64. Kong, X., et al. (2017). Implementation of multiparty quantum clock synchronization. arXiv preprint arXiv:1708.06050. 65. Zhang, Z.-J., & Man, Z.-X. (2005). Multiparty quantum secret sharing of classical messages based on entanglement swapping. Physical Review A, 72(2), 022303. 66. Hillery, M., Bužek, V., & Berthiaume, A. (1999). Quantum secret sharing. Physical Review A, 59(3), 1829. 67. Cavaliere, F., Prati, E., Potí, L., Muhammad, I., & Catuogno, T. (2020). Secure quantum communications technologies and systems: From labs to markets. Quantum Reports, 2(1), 80– 106. 68. Bennett, C. H., & Brassard, G. (1984). Quantum cryptography: Public key distribution and coin tossing. In Proceedings of IEEE International Conference on Computers, Systems and Signal Processing (Vol. 175, p. 8). New York. 69. Coecke, B. (2004). The logic of entanglement. arXiv:quant-ph/0402014. 70. Jozsa, R., et al. (2000). Quantum clock synchronization based on shared prior entanglement. Physical Review Letters, 85(9), 2010. 71. Dahlberg, A., et al. (2019). A link layer protocol for quantum networks. In Proceedings of the ACM Special Interest Group on Data Communication (SIGCOMM ’19) (pp. 159–173). New York, NY: Association for Computing Machinery. https://doi.org/10.1145/3341302.3342070

Chapter 4

Quantum Radar Kadir Durak, Zeki Seskir, and Bulat Rami

4.1 Introduction Quantum radar is a promising technology that has been getting attraction in recent years, with possible applications in military and civilian fields. In this chapter, the historical background, developments during the last two decades, proposals that are mainly referred to as quantum radar, and their experimental implementation cases are presented. Throughout this chapter, we will use the term radar to cover both radar and lidar, since quantum “radar” is not a radar in the strict sense of the meaning. It is used as an umbrella term to cover multiple proposals, some of which are capable of only detection and not ranging, with different techniques applicable both in the optical and microwave regimes. Therefore, when the term “quantum radar” is invoked, it is used for multiple purposes that are wider than just a quantum version of the classical radar systems. Within the historical context, maturation of the idea for a quantum radar is strongly related to the developments within the field of quantum sensing. These developments are strongly affected by the progress in quantum computation through spillover effects such as improvements in control electronics for quantum systems, modelling of quantum sensors as noisy quantum channels, and rapid progress in error correction techniques. In this sense, categorization of quantum radar types can also be associated with different quantum sensor types [1], which will be further elaborated in Sect. 4.2.

K. Durak () · B. Rami Istanbul, Turkey e-mail: [email protected] Z. Seskir Department of Physics, Middle East Technical University (METU), Ankara, Turkey © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Iyengar et al. (eds.), Quantum Computing Environments, https://doi.org/10.1007/978-3-030-89746-8_4

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Fig. 4.1 Number of newly published articles and proceedings with respect to year in the dataset

In the literature, when a search on quantum radar and quantum illumination related keywords is performed,1 it can clearly be seen from Fig. 4.1 that there is a growing interest since the inception of Lloyd’s quantum illumination paper [2]. However, the scholarly and practical studies on the applications of radar systems in the quantum limit goes back to early 2000s [3, 4] and there are two patent applications in 2004 titled “Entangled-photon range finding system and method” [5] and in 2005 titled “Radar systems and methods using entangled quantum particles” [6]. These are clear indications that the work on quantum radar has been an ongoing process for about two decades, and it has been catching attraction as there are more than 50 articles or proceedings have been published and 10 patent applications have been submitted in the last two years alone. It is also notable to understand on which literature this work on quantum radar and quantum illumination stands on. A co-citation analysis of the 167 articles and proceedings for commonly used references at 17 or more of the works in the data is provided below with co-citation accounts. It is important to note the number

1 Query used for this search in the ISI database is: (“supersensitivity” AND “fock state”) OR (“heisenberg limit” AND “fock state”) OR (“heisenberg limit” AND “squeezed state”) OR (“supersensitivity” AND “squeezed state”) OR “gaussian state illumination” OR “quantum Illumination” OR “entanglement illumination” OR “quantum radar” OR (“NOON state” AND (“standard quantum limit” OR radar)) OR (“standard quantum limit” AND heisenberg AND measurement AND entanglement) as of August 30th 2020. Afterwards, 203 articles and proceedings were checked manually and 167 of them were found to be acceptable to include into the dataset.

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Table 4.1 Commonly used references at 17 or more of the works in the dataset Cited references Lloyd S, 2008, Science, v321, p1463, https://doi.org/10.1126/science. 1160627 Tan SH, 2008, Phys Rev Lett, v101, https://doi.org/10.1103/physrevlett. 101.253601 Lopaeva ED, 2013, Phys Rev Lett, v110, https://doi.org/10.1103/ physrevlett.110.153603 Giovannetti V, 2004, Science, v306, p1330, https://doi.org/10.1126/science. 1104149 Barzanjeh S, 2015, Phys Rev Lett, v114, https://doi.org/10.1103/ physrevlett.114.080503 Guha S, 2009, Phys Rev A, v80, https://doi.org/10.1103/physreva.80. 052310 Shapiro JH, 2009, New J Phys, v11, https://doi.org/10.1088/1367-2630/11/ 6/063045 Audenaert KMR, 2007, Phys Rev Lett, v98, https://doi.org/10.1103/ physrevlett.98.160501 Lanzagorta M, 2011, Quantum Radar Zhang ZS, 2015, Phys Rev Lett, v114, https://doi.org/10.1103/physrevlett. 114.110506 Giovannetti V, 2006, Phys Rev Lett, v96, https://doi.org/10.1103/ physrevlett.96.010401 Shapiro JH, 2009, Phys Rev A, v80, https://doi.org/10.1103/physreva.80. 022320 Pirandola S, 2008, Phys Rev A, v78, https://doi.org/10.1103/physreva.78. 012331 Yurke B, 1986, Phys Rev A, v33, p4033, https://doi.org/10.1103/physreva. 33.4033 Mitchell MW, 2004, Nature, v429, p161, https://doi.org/10.1038/ nature02493 Sacchi MF, 2005, Phys Rev A, v72, https://doi.org/10.1103/physreva.72. 014305 Boto AN, 2000, Phys Rev Lett, v85, p2733, https://doi.org/10.1103/ physrevlett.85.2733 Caves CM, 1981, Phys Rev D, v23, p1693, https://doi.org/10.1103/ physrevd.23.1693 Zhang ZS, 2013, Phys Rev Lett, v111, https://doi.org/10.1103/physrevlett. 111.010501 Dowling JP, 2008, Contemp Phys, v49, p125, https://doi.org/10.1080/ 00107510802091298

Citations [92] [72] [50] [49] [43] [40] [33] [24] [24] [24] [23] [23] [20] [20] [19] [19] [18] [18] [18] [17]

of citations here are citations within the set of 167 scholarly work being utilized; hence, for instance, this indicates that 55% of the literature cited Lloyd’s quantum illumination paper [2] (Table 4.1). The top five cited references in the literature can be listed in the following manner. The first one is Lloyd’s paper on quantum illumination, as stated above.

128 Table 4.2 Organizations with five or more publications in the dataset

K. Durak et al. Organization MIT Naval Res Lab Def Res and Dev Canada Penn State Univ Chinese Acad Sci Univ Sci and Technol China China Ship Dev and Design Ctr Air Force Engn Univ Ranken Inst Univ Missouri Tamagawa Univ

Number of documents 20 14 9 7 6 6 6 6 5 5 5

Fig. 4.2 Relationship between organizations obtained from citation analysis on organization level from the dataset

The second one is the extension of quantum illumination to Gaussian states [20] by a group of authors, including Lloyd, Giovannetti, and Maccone. The third most cited article here is the experimental realization of quantum illumination, published in 2013 [21]. The fourth most cited article is the work on quantum-enhanced interferometric measurements [22], which is an earlier proposal than the quantum illumination technique. And finally, the fifth paper is a celebrated extension of quantum illumination to the microwave regime [23]. It is also important to note here that out of 96 papers on quantum illumination in the literature, 38 of them are studies regarding developments in microwave regime and 58 of them are in optical regime. The reason for this is assumed to be that experimental setups and groups operating in optical regime are more present in the field of quantum optics than microwave regime. Additionally, the work of Barzanjeh et al. [23, 24] is relatively recent (Table 4.2). The list and the relationship between different organizations involved in the aforementioned publications are also provided in order to enable a general understanding of interested parties in this line of research. In the table below, organizations with more than five documents are listed. Additionally, the citation relationship between organizations with more than three documents is visualized in Fig. 4.2. This analysis of the academic literature and commercial interest in the form of patenting signals that quantum radar is becoming a noteworthy field of research. Therefore, in this chapter, the concept of quantum radar is investigated both in

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theoretical and practical outlooks. Here, the focus is on the introduction of this emerging topic to the readers as concisely as possible without deliberating much on difficulties of different approaches utilized. For that, two recent critical presentations of the state of art on this topic are suggested for further reading [18, 25]. Likewise, readers who are more interested in this topic on an entry level should check Marco Lanzagorta’s book titled “Quantum Radar” [1] for ground-up discussions on the topic.

4.2 Theory Quantum radar theory is based on both classical radar theory and quantum sensing. Classical radar theory deals with utilization of classical electromagnetic waves and sensor systems for detection and ranging of objects. The main motivation of quantum radar research is to utilize principally new, quantum mechanical approaches to obtain higher sensitivity, target position resolution, stealthiness, etc. In this section of the chapter, the reader is presented with the following. Initially, a brief introduction on different types of quantum sensors and their corresponding application in quantum radar models are presented. Later, the differences between separable and non-separable states, and the literature on quantum entanglement (a key phenomenon in much of the discussions concerning quantum radar) is provided. These are followed by the introduction of coherent and squeezed states, effect of amplification on quantum noise, and description of fundamental limits on quantum measurements.

4.2.1 Quantum Sensor Types Different types of quantum sensors are identified with respect to their utilization differences of either entanglement or photo-detection procedures. A basic categorization of them is listed in Lanzagorta’s book (citing a technical report sponsored by Defense Advanced Research Projects Agency [1, 26]), which can be re-iterated in the following form: Type I Quantum Sensors: This type of quantum sensors deal with quantum states of light, which are not entangled (i.e., separable). Type II Quantum Sensors: Again, this type of quantum sensors deal with separable states of light; however, they utilize photo-detection models based on quantum mechanics such as photon counting. Type III Quantum Sensors: For this type, both the light and the detection method are quantum mechanical, and the sensor keeps some of the generated light that is entangled with the transmitted signal.

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Quantum radar proposals with different types of quantum sensors aim to utilize different physical phenomena to obtain advantages against their classical counterparts. Models that operationalize single photon emission and sensing methods can be accepted as Type I. Even if they use photons that are not entangled, there are some associated advantages like enhanced radar cross section [1]. Quantum illumination [2, 20, 23, 27, 28], and following proposals [16, 17, 19, 25, 29, 30] along this line can be considered as Type III. In the following subsection, the difference between separable and non-separable states is presented. This division can be understood as the main generator of major advantages (or novel differences) for some of the quantum technologies. For the case of quantum radar, the theoretical advantages are orders of magnitude higher than any foreseeably achievable practical advantages for the moment. Still, the emergence of actual differences in the theoretical realm through switching the model utilized is a clear demonstration that there is much to explore in the field of quantum technologies.

4.2.2 Separable and Non-separable States It is important to note that, as the first sentence of this subsection, separable does not necessarily imply classical. It is a much larger set of states that, although they have quantum mechanical properties, do not contain any non-local correlations. Nonseparable states are mostly called as entangled, and the most famous of these are Bell states [31], denoting bipartite two-leveled maximally entangled states. However, there is a whole range of other states between fully classical and maximally entangled ones. Bell inequality and its variations [14, 15, 32] are early methods developed to identify whether a quantum state is entangled or not under certain conditions. Further studies into the field later led to development of other methods such as the separability criterion [33, 34], which evolved into a wide array of entanglement witnesses [35]. One of the main reasons for focusing on the advantages of non-separable states is the fact that for separable states, it is not possible to surpass the standard quantum limit [22]. A short argument on this can be found below. In the general noiseless quantum metrology scheme using N probes, the N . Sending them through an independent sensing wavefunction is of the form |ψin parameter ϕ, and estimating the parameter from measurement outcomes with ν repetition, ϕ(x) ˜ is obtained, where x are outcomes of measurements MX . In this scheme, classical/quantum Cramer–Rao bound sets the ultimate limit on RMS error of the estimate in the asymptotic statistics regime (as ν → ∞). The classical and quantum fisher information bounds, FC and FQ , can be found below [36]: ϕ˜

≥ ν→∞

1 1 √ ·     ν FC p X N | ϕ ]

≥ ν→∞

1 1 √ ·   . ν FQ ρϕN

(4.1)

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The noiseless unitary parameter encoding in this case can be done through following steps:      ˆ ˆ    Uϕ = e−iϕ h ⇒ ρϕN = ψϕN ψϕN  with ψϕN = e−iϕ H and

Hˆ =

N  n=1

hˆ (n) ⇒

   N ψin

 !  FQ ψϕN = 2 Hˆ 

N ψin

.

 N N = |ψ N ⊗N ) gives F ∼ N, Using classical strategy here (i.e., |ψin Q ψϕ in which results in the standard quantum limit. However, using a quantum strategy (such as GHZ or NOON states of the form |ψin N = √1 (|λmin + |λmax ) provides 2   FQ ψϕN ∼ N 2 , which can reach to the Heisenberg limit. Quantum entanglement was first introduced as a nameless concept, an argument against quantum theory’s completeness by Einstein, Podolsky, and Rosen 85 years ago [37]. It was later coined by Schrödinger as “Verschränkung” [38], which is directly translated to entanglement in English. For almost 30 years, it stayed as a borderline metaphysical discussion related to the studies on the foundations of quantum mechanics rather than experimental physics. This changed following the monumental formulation of John S. Bell in 1964 [31], which was experimentally ratified in 1981 by Alain Aspect. However, even though it was a demonstrated physical phenomenon with many interesting applications, such as quantum cryptography [39], quantum teleportation [13], superdense coding [40, 41], and so on, it stayed as a hard resource to generate and manipulate. As of mid-2020, there are more than 35,000 academic articles and proceedings in indexed journals related in some form to quantum entanglement while there are less than 400 patent families. It is a highly developed theoretical construct with clear experimental demonstrations while being a difficult resource to utilize in real-life commercial settings. There are multiple well-structured and highly celebrated reviews on different aspects of quantum entanglement in the literature. The most used study with respect to citations is titled “Quantum Entanglement” and published in mid-2009 by the Horodecki family [42]. It is mainly focused on various concepts related to entanglement such as its characterization, detection, distillation, and quantification. This is followed by several more specialized review works, such as H. J. Kimble’s work on “The quantum internet” in 2008 [43]. It describes the uses of quantum systems to establish a quantum internet, a network capable of generating and characterizing quantum coherence and entanglement. On a similar note, “Entanglement in manybody systems” by Amico et al. [44] published in 2008 focuses on the zero and finite temperature properties of entanglement in distinct cases such as fermion, boson, and interacting spin models. For further reading on specific topics related to entanglement, the following references can be checked: entanglement purification and quantum error correction [45], manipulation of quantum entanglement with atoms and photons in a cavity [12], entanglement detection [46], entanglement

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measures [47], the sudden death of entanglement [11], entanglement entropy and CFT [48], and entangled states of trapped atomic ions [49]. In practice, entanglement of pure states is when a multi-partite or multi-mode [50] state cannot be written in the form of a product state: |ψ = |ψ1 ⊗ |ψ2 ⊗ · · · ⊗ |ψn .

(4.2)

Bipartite two-level qubit states are the most studied form of entangled states and maximally entangled states in this form are usually called Bell states. As the systems level increases, the concept of maximal entanglement becomes less clear and starting from tripartite qubit states there are maximally entangled classes such as GHZ and W states rather than single maximally entangled set of states. The relationship between and within these classes is also studied in the literature [51] and one important aspect is that any state within a given class must be obtainable from another state in the same class through Local Operations assisted by Classical Communication (LOCC). For mixed states, the condition of not being able to be represented as a nonproduct state is not enough. To be called as an entangled state, a mixed state should not be representable as a convex combination of product states [52]. ρ =



j

j

pj ρ1 ⊗ . . . ⊗ ρn .

(4.3)

j

This particularity of mixed state entanglement yielded many studies in identification [9, 10, 53] and quantification [54, 55] of such property in physical systems. Furthermore, an extension of quantum correlations to concepts beyond entanglement has been an ongoing discussion in the literature for quite sometime [56–58]. There are relatively recent reviews [59, 60] and books [61, 62] covering this topic for the interested readers.

4.2.3 Coherent and Squeezed States Coherent state can simply be defined as the quantum mechanical view of a classical monochromatic electromagnetic wave. The state can be expressed as a function of its amplitude and the phase as a dimensionless complex number α with dimensionless field quadratures: α = X1 + iX2 = |α|eiφ ,

(4.4)

 where the amplitude is |α| = X12 + X22 and the phase is carried in the quadratures as X1 = |α| cos φ, X2 = |α| sin φ. The quantized electromagnetic field states are

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the energy steps of harmonic oscillator, quantized with h¯ ω as:

1 hω. En = n + ¯ 2

(4.5)

√ Therefore, the field amplitude |α| is equal to the n, ¯ where n¯ is the average photon number. The 12 hω ¯ term represents the vacuum state. The uncertainties of X1 and X2 are equal: X1 = X2 =

1 . 2

(4.6)

The phasor diagram of quadratures with |α| as the amplitude and φ as the angle can be seen in Fig. 4.3a,b for classical light and coherent state, respectively. The uncertainties in amplitude and phase are equal to 1/2 and the amplitude is zero for vacuum state: α = φ =

1 . 2

(4.7)

The phasor diagram for both vacuum and coherent states is a circle with diameter of 1/2. The coherent state is therefore identical to the vacuum state with the origin of the phasor diagram shifted from origin to amplitude of the field α. This shift is clear, considering the harmonic oscillator given in (4.5). In Fig. 4.3, it can be seen that the uncertainty diameter of the circle can be approximated to the multiplication of the amplitude and the phase uncertainties: φ =

diameter . α

(4.8)

Since the coherent states follow the Poissonian distribution and the uncertainty in √ ¯ (4.8) can be re-written as: amplitude is determined by the shot noise n = n, nφ ≥

1 . 2

(4.9)

This equation is called the number-phase uncertainty relationship. It indicates that it is not possible to know the photon number and the phase precisely at the same time. The consistency√of the derivation can be checked by considering very large photon number |α| = n¯  1. The large number results in negligible phase uncertainty such that the quantum features cease to exist and classical picture of light becomes dominant as it can be seen in Fig. 4.3a that there is no phase uncertainty in classical case.

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Fig. 4.3 Phasor diagram for the (a) classical light, (b) coherent state, and (c) amplitude and (d) phase squeezed states. The amplitude of the state |α| is the length of the phasor and optical phase φ is the angle from the X2 axis. The quantum uncertainty is shown by a circle of diameter 1/2 for the coherent state. The uncertainty circle diameter is zero (a) as the uncertainty in field amplitude is zero classically. One of the quadrature fluctuations in time shows the amplitude and phase uncertainties in coherent state (f), while classical light does not have uncertainties (e)

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So far only the case of perfect uncertainty circle is considered. When this circle is forced to have elliptical shape, either number or phase uncertainty will be squeezed in a way that the area still remains the same. This is dictated by the number-phase uncertainty relationship given in (4.9). Such a state is called quadrature-squeezed or simply squeezed state. In Fig. 4.3c,d, the illustrations of number and phase squeezed states are shown, respectively. When the photon number is well defined such that the phasor diagram is amplitude squeezed, the phase uncertainty becomes very large. Then the photon statistics of such a field becomes sub-Poissonian. At its limit, when the photon number is perfectly known such that there n = 0, the phase becomes totally unknown. Such states are called photon number states. The generation of squeezed states is a rather broad subject such that fully covering it goes beyond the motivation of this book. However, the generation of squeezed vacuum state and detection method is briefly explained to provide clarity on certain concerns about the limitations of quantum radar implementations. The most common way to achieve the squeezed vacuum state is the use of optical parametric amplifiers (OPA). A crystal that exhibits χ (2) nonlinearity is pumped by a powerful laser with a frequency of ωp = 2ω. For the amplification to happen, the process has to be degenerate such that frequencies of signal and idler are equal ωs = ωi = ω. A weak signal beam is also provided to the nonlinear crystal to create idler photons via difference frequency mixing process: ωi = ωp − ωs .

(4.10)

Since the process is degenerate, the pump and idler photons drive the same process to create signal photons. The difference frequency mixing process happens with pump-signal and pump-idler photons consecutively. As a result, the amplification or de-amplification of signal happens depending on the relative phase of the pump and the signal. Figure 4.4a shows the schematic of OPA. The squeezed vacuum state generation part happens when no signal beam is provided at the input of the crystal. Instead the vacuum modes with ω frequency, from the randomly fluctuating vacuum

a)

b)

weak signal

D2

idler Nonlinear Crystal

strong pump

signal

LO i2 D1 i1

i1-i2

Fig. 4.4 A typical schematic for optical parametric amplifier (OPA) (a), which is commonly used for squeezed state generation, and a balanced homodyne detection setup (b). LO is the local oscillator, and D1 and D2 are photodetectors

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modes, are coupled to the system as signal. Depending on the relative phase of the nonlinear process, the vacuum mode is now amplified or de-amplified. By adjusting the phase to result in de-amplification, the quadrature squeezed vacuum state can be achieved. The measurement of the output can be only done via balanced homodyne detection scheme, where the photo-currents from two detectors are either subtracted or added, after being split by a beam splitter (BS). A generalized schematic of balanced homodyne detection setup is shown in Fig. 4.4b. When the signal in homodyne detection is the vacuum, subtraction of the photo-currents gives the shot noise. Similarly, the replacement of the vacuum with the quadrature squeezed vacuum can result in the measurement of squeezed vacuum depending on the phase of the local oscillator.

4.2.4 Effect of Amplification on Quantum Noise The amplification process is required for the systems where the signal level is either very low or it gets attenuated over lossy channels. A very well-known example of amplification is the optical amplification in telecommunication systems. An amplifier is characterized by its gain value G. The signal is amplified by the amplification process and so is the noise level. Therefore it is logical to consider the signal-to-noise ratio (SNR) for consideration of noise figure in amplification. In continuous systems the SNR is expressed as: I 2 #, I 2

SNR = "

(4.11)

# " where I 2 and I 2 represent the expectation values of the signal and noise amplitudes, respectively. For the quantum equivalent of the equation, the photocurrent I can be replaced by the photon number n: n 2 n¯ 2 #= , (n)2 n2

SNR = "

(4.12)

where n¯ is the average photon number. The amplification process is expected to decrease the SNR. The term that quantifies this decrease is called the noise figure in communication field. It is defined as the ratio of the input SNR to the output SNR: Noise figure ≡

SNRin 1 =2− . SNRout G

(4.13)

This equation indicates that after the amplification the state cannot be a minimum uncertainty state anymore. For very large gain values, the noise figure reaches 2 asymptotically. Figure 4.5 shows the amplified noise depending on the gain. It is

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X1

classical ampl.

quantum limited ampl. noiseless ampl.

squeezed state

X2 Fig. 4.5 Phasor diagram of amplification for squeezed states at different regimes

clear in the figure that after the amplification, the state is no longer a minimum uncertainty state. The diameter of the uncertainty circle depends on the gain with the equation: √ 2G − 1 . 2

(4.14)

However, by using degenerate phase-sensitive parametric amplifiers, it is possible to amplify one of the phase quadratures and letting the other one have the extra noise as a trade-off. These types of amplifiers are called noiseless amplifiers [63–66]. The use of coherent and/or squeezed state for ranging and detection purposes determines the fundamental limits that can be achieved in terms of sensitivity of the experimental setup and the quantum advantages over classical systems. It should be noted that the fundamental limits shape the future developments in the radar and lidar technologies. However, there are still other technical challenges to be addressed for the applications of quantum radar and lidar in practical scenarios.

4.2.5 Fundamental Limits on Quantum Measurements The measurement sensitivity of a sensing device is the most crucial concept of metrology and sensing technologies. While most of the sensors’ sensitivity is limited

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by the standard quantum limit (SQL), it is possible to beat it with extra care and convenient trade-offs. However, the ultimate limit is still set by the law of quantum mechanics, which is the Heisenberg uncertainty in this case. The sensors that have sensitivities better than SQL are called super-sensitive. In order to understand the fundamental measurement limits of different experimental configurations for quantum radar and lidar applications, the noise characteristics for light sources with different photon statistics should be analyzed. In (4.9) the Heisenberg uncertainty principle is shown in the form of number-phase uncertainty, where n and φ represent the photon number and phase fluctuations, respectively. The coherent state |α with Poissonian distribution can be expressed as: ∞

 |α = exp −|α|2 /2 n=0

αn |n , (n!)1/2

(4.15)

where |n represents the state of n photons. Probability of finding n photons in the system is P (n) = |n | α |2 =

e−n n n . n!

(4.16)

Since the number states are the eigenstates of the harmonic oscillator Hamiltonian, the creation operator aˆ † and annihilation operator aˆ can be used to determine the expectation values and uncertainties:       n = α aˆ † aˆ  α = |α|2 ,

(4.17)

ˆ = α|α . Similarly, where aˆ † a|α

        n2 = α aˆ † aˆ aˆ † aˆ  α = |α|4 + |α|2 .

(4.18)

This means the uncertainty in the photon number is given by the shot noise: n =

$

n .

(4.19)

1 φ ≥ √ . n

(4.20)

Then, the phase uncertainty becomes:

Here (4.20) indicates the uncertainty relation for the phase measurement in the SQL. The region of interest for super-sensitive measurement can be expressed as: 1 1 , > φs ≥ √ n

n

(4.21)

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1 where n

expresses the Heisenberg limit. This inequality indicates that the SQL is a result of unoptimized measurement on the photon number. It can be improved by applying certain methods to reach ultimate limit, which is independent of the measurement methods and set by quantum mechanical laws: Heisenberg limit. Entanglement can be exploited to reach this limit. However, the thermal noise, atmospheric attenuation, and other factors may limit the sensitivity of the system. On the other hand, even if there are no such perturbations or losses, the systems with separable states cannot beat the SQL. Alternative methods, including injection of squeezed states into transmitter side, are present in the literature and they work reasonably well on reaching the super-sensitive regime as well [67].

4.3 Proposals and Experiments on Quantum Radar There are multiple theoretical protocols being proposed as a conceptual quantum radar, which in essence is a radar-like system that uses quantum properties of physical systems, hence allowing new types of capabilities that cannot be obtained via classical means. The main literature on this topic can be divided into two main streams, the ones emanating from using quantum enhanced properties of positioning and clock synchronization [3], and enhanced sensitivity of photodetection via quantum illumination [2]. In this subsection, several proposals are introduced. Due to its wide adoption in the community, the approach mainly called as quantum illumination is going to be investigated and explained in detail in Sect. 4.3.2. The photonics applications of quantum sensing seem to recently develop faster than ever, similar to other quantum studies such as quantum computation and communication. Quantum communication experiments reached to inter-continental scale accomplishments. This brought the advances in entangled photon creation and detection technologies. Such developments lead scientists to look for solutions of classical problems in the quantum field. One of the problems of radar applications is to be operable in noisy environments. Quantum radar studies primarily aimed to find a solution to this problem. However, it should be noted that there are other advantages of quantum radar such as building a stealth radar, spoofing resistance, and enhanced target resolution. There are various methods as candidates of practical radar applications. The studies are all at proof-of-principle level; however, practical quantum radars/lidars are expected to be realized in the future. All proposed and tested proposals of quantum illumination use single photons or entangled photons for achieving quantum enhancements over classical ones. However, it must be noted that some protocols utilize quantum correlations instead of entanglement as it is not needed to exhibit the quantum advantage [2]. A generalized term called quantum discord is more convenient to use instead in this context. The approaches for quantum radar can be classified into two categories: interferometric, and quantum illumination-based quantum radars. While interferometric quantum radars monitor the interference between signal and idler paths

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depending on the relative phase difference, the quantum illumination idea utilizes the correlations of entangled photons for joint detection of signal and idler photons. The joint measurement in quantum illumination can be done via direct interaction of signal and idler photons, like upconversion phenomenon, or by storing the signal (idler) and looking for correlations with the detected idler (signal). Direct quantum illumination and post-processing-based quantum illumination methods have different advantages and disadvantages with respect to each other.

4.3.1 Interferometric Quantum Radars This approach takes advantage of quantum states in novel manners, through interferometric measurement processes. The light utilized in this method can be in either separable or non-separable states; hence, properties of quantum mechanics such as entanglement and squeezing can be tapped into to improve the precision of devices beyond the classical bounds imposed due to shot noise limit or the standard quantum limit [22]. In order to locate the position of any given object, the duration of light signal to travel between a known reference point and the object can be utilized. Measuring and averaging over the times of travel of single photons in a sent beam is the best approximate classical method that is bounded by limits provided earlier. Through this method, √ one can determine the travel time within an error range proportional to 1/(ω n), where ω signifies the signal bandwidth. In this approach, spread of the time of arrival of photons will be in direct relation of the signal bandwidth. However, it has been shown that [3] using n entangled photons in this scheme,√which causes all the photons to have the same frequency, provides an increase of n efficiency against the classical method. In this approach, to obtain the higher efficiency, system at hand starts with a highly entangled NOON state of the form [68]: 1 | = √ (|n0 + |0n ). 2

(4.22)

Then each half of this state passes through a different arm of the quantum interferometer, which causes a phase difference of φ to photons that pass through the second arm. Hence, the state becomes of the following form, where if the photons have passed through the first arm, there is no phase difference, and if they have passed through the second arm there is  1

| (φ) = √ |n 1 |0 2 + einφ |0 1 |n 2 , 2

(4.23)

which can be written in the second quantization form like following:

n  1

† n |0 1 |0 2 . aˆ 1 + einφ aˆ 2† | (φ) = √ 2n!

(4.24)

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After this, a measurement to identify phase shift of the form Aˆ D = |n0 0n| +    2 |0n n0| is performed where the noise is given as 2 AD = Aˆ 2D − Aˆ D = sin2 nφ  and the phase responsivity (the input–output gain of a detector system) is

d Aˆ D dφ

= −n sin nφ. These translate into the phase estimation error, which is   d Aˆ D

√ approximately given as AD /| dφ |, being n1 . That corresponds to a n increase in efficiency over its classical counterpart. In this type of quantum radars, the interference of signal and idler photons from two paths is measured in NOON state configuration for achieving the optical path dependency. An interferometer normally would have sensitivity of shot noise, usually referred to as standard quantum limit (SQL). However, an interferometer with NOON state can result in a sensitivity of phase measurement to the Heisenberg limit [3, 22]. NOON state can be simply defined as the entanglement in photon number, which is quantum analogous to Mach–Zehnder interferometer. There are other ways than entanglement to achieve Heisenberg limit and these methods use either squeezed or Fock states. While the SQL sensitivity is proportional to √1n , it is possible to reach sensitivities scaled with n1 using the above-mentioned methods [69–72]. Interferometric quantum radar ideas using NOON state can only beat the SQL at very low attenuations due to atmosphere. The atmospheric attenuation degrades the entanglement and results in decoherence. Such a decoherence should be carefully handled in order to implement NOON state-based quantum radars. Correction methods have been proposed for improving the robustness of the radar setup against decoherence sources [73]. The supersensitivity is defined as the sensitivity below the Heisenberg’s limit and above SQL. The interesting region for a NOON statebased interferometric quantum radar is within these limits. It has been shown that without any adaptive optics the maximum achievable target range is below 60 km for N = 2. However, with careful adaptive corrections against atmospheric attenuations the maximum achievable range can reach thousands of kilometers. It must be noted that the model in this calculation is idealistic and experimental ranges are expected to be much smaller. The idea of achieving supersensitivity is the main motivation of quantum radar applications. However, the practicality of proposed ideas requires further studies. Coupling the coherent state light into a NOON state interferometer provides a possible solution. When the coherent light interferes with squeezed states, the error in phase and therefore the estimation in range follow the form of N −3/4 . This sensitivity is still better than SQL, which follows the form of N −1 . Further discussions on this topic can be found in Section 5.2 of Lanzagorta’s book on Quantum Radar [1]. A neat method for achieving high NOON states is presented in the work of Afek et al., where a coherent state is mixed with SPDC photons to achieve NOON state of up to N = 5 experimentally [8]. Same setup can also be used for target detection and ranging by adding the two mirrors in one of the paths in the interferometer that acts as the phase controller. The schematic of such a setup

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a)

b)

target

D1

D1

j B2

SPDC

B1 coherent state

D2

B2

SPDC

D2

B1 coherent state

Fig. 4.6 Schematic of NOON state and measurement setup (a) and the phase control ϕ is replaced by extra path taken through the target for quantum radar ranging purpose (b)

is shown in Fig. 4.6. The part of the setup enclosed with a dashed line acts as a phase controller in a typical NOON state experiment. The detectors D1 and D2 are photon number resolving detectors, which can either be transition edge sensors (TESs) or a set of single photon detectors in Geiger mode operation constructed by utilizing beam splitters. The information gathered by the photon number resolving detectors are analyzed for estimation of the relative phase as a function of the optical paths.

4.3.2 Quantum Illumination-Based Quantum Radars Theoretical proposal of this method was first introduced in the literature by Seth Lloyd in 2008 [2]. In this approach, instead of relying on phase measurements, the joint measurement of signal and idler is performed. Quantum illumination method utilizes an entangled source of light for signal and idler beams (or photons) to interrogate a possible space region where a target could be located within. Either signal or idler mode is sent to space and the other mode is kept for reference. If there is an object at the space region being interrogated, a joint measurement is performed on the reference and the back-reflected light. The advantage of this joint measurement is not necessarily based on the quantum entanglement between signal and idler. Even if the entanglement is lost during the round trip from the target, the residual correlations are sufficient to create an advantage over the classical systems with the same photon rate and bandwidth. These correlations can be time, polarization, and frequency correlations. The joint measurement on the

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reference and back-reflected light is the crucial and challenging part of the quantum illumination experiments. The main purpose of quantum illumination experiments in quantum radar context is the detection of object with very low reflectivity at extremely noisy environments. It is also known that the classical radar applications require stealth detection schemes such that radar is not detectable by the other parties. This requires very low levels of search light. Quantum illumination provides a solution to this challenging problem by probing the target with very low light levels in a stealthy non-invasive way that is impossible to reproduce with classical means. Therefore, quantum illumination is very useful especially at very noisy and lossy environments for ranging and detection purposes. In order to show the advantage of quantum illumination with entangled state a comparison of single non-entangled photon case to the entangled case should be done. The comparison is done via the signal-to-noise ratio (SNR) that is the figure of merit for performance evaluation/comparison of radars. The SNR can be calculated via the ratio of the detection probabilities when the target is in range to the false positive when there is no target, where the latter is considered as the noise. When there is no target, the density matrix of the system for non-entangled light, for very small average number of photons per detection event (gb  1), can be expressed as:  ρnt ≈ (1 − gb)|0 0| + b

g 

% |k n k|n ,

(4.25)

k=1

where g is equal to the detection bandwidth W times the detection time window T : g = W T , NB is the number of background noise photons present in the system, |0 is the vacuum state, and |k n represents the noise photon mode. The coefficients before the states 1 − gb and b represent the probabilities of measurement of no signal and measurement of one out of g modes. Therefore, the probability of positive detection “+” of a target that is not within range, which is called false positive, is Pnt (+) = b.

(4.26)

Similarly, the probability of negative detection is Pnt (−) = 1 − b.

(4.27)

The sum of false positive and negative detection probabilities is unity. If there is a target within the range of the quantum illumination system, then there is a probability η that there will be a measurement of thermalized signal photon ρ, ˜ and a probability (1–η) that we will just measure noise ρnt . Therefore, when there is a target within range, the density matrix, in terms of no target case ρnt , can be

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expressed as: ρt = (1 − η)ρnt + ηρ˜  ≈ (1 − η) (1 − gb) |0 0| + b

g 

%

"

|k n k|n

" + η|ψ s ψ|s ,

(4.28)

k=1

where |ψ s indicates the signal state and η is the reflectivity coefficient of the target. The probability of positive measurement when there is a target is Pt (+) = (1 − η)b + η.

(4.29)

The false negative measurement probability is then Pt (−) = 1 − Pt (+) = (1 − η)(1 − b).

(4.30)

The probabilities of positive and false negative measurements when there is a target to be detected normalize to unity. Now it is possible to define the SNR for quantum illumination system with non-entangled states as the ratio of the positive measurement probability when there is a target to the false positive measurement probability when there is no target: SNRne =

(1 − η)b + η Pt (+) = . Pnt (+) b

(4.31)

For quantum illumination with entangled states when there is no target at the search zone, the density matrix can be expressed as:  e ρnt

≈ (1 − gb)|0 0| + b

g  k=1

% |k n k|n ⊗

&

' g 1 |k a k|a , g

(4.32)

k=1

where the last term is the state of the reference light, which can be signal or idler. Here for the generalization we use the term ancilla denoted as a. The first part of the equation, on the other hand, describes the absence of the target in the interrogated space region. Since there are no restrictions on quantum illumination to work on any particular frequency, in theory it can work on any band required. The first experiments on quantum illumination have been made in optical band. Later, the methods to enable quantum illumination in the microwave regime have been proposed [23, 24]. In this method, entangled photons are generated and one photon is sent as signal, while another one is kept as idler. Signal photons take a round trip to target and back, and both halves of the entangled pair are now at hand to be compared. It is mainly argued that entanglement is necessarily broken in this process, though it is argued in the literature that this might not be the case as the polarization correlations are

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preserved during the transmission and detection of the photons [29]. Nevertheless, correlation between the stored and received beams in frequency, timing, and photon counts is compared. Two main methods here are upconversion and using crosscorrelogram. Further elaboration on the general topic of quantum illumination in continuous and discrete variable quantum systems is provided below. Thus, instead of dividing the studies depending on the wavelength regime as quantum radar and lidar for microwave and optical bands, respectively, the discrete and continuous nature of the systems are considered.

4.3.2.1

Introduction on Continuous Variables and Gaussian States

It is known that electromagnetic field can be decomposed into different radiation modes characterized by the wave number vector and polarization. For defining polarization, a discrete space of quantum variables with two orthogonal vectors is usually used, while the wave number vectors form a space of continuous variables. Entanglement in the case of the discrete variables has been discussed above. Below we give a short introduction on a class of entangled states described by continuous variables (see for more detailed description, e.g., [18, 74, 75]). A continuous variable system describing states of the electromagnetic quantum field with N modes is described by the tensor product of N infinite-dimensional Fock spaces, that is (N defined by a global Hilbert space H = k=1 Hk . Mode of electromagnetic field |nk means a quantum state of the electromagnetic field containing nk photons with spatial and temporal profile is defined by a function f (r, t) (e.g., a plane wave f (r, t) = exp [−i(k · r − ωt)]) [18]. Any Fock state |nk is an eigenstate of photon number operator nˆk , i.e., nˆk |nk = nk |nk . Creation operator aˆ † and annihilation operator aˆ are related to photon-number operator as follows: nˆ = aˆ † a. ˆ A specific type of the continuous variable states, in which entanglement can be encoded, is the Gaussian states [25]. These quantum states are fully characterized by the second-order moments of the quadrature operators (Eq. 4.6). Each mode nk (k = 1, 2, . . . , N) is characterized by the quadrature operators qˆk and pˆk (which are analogues of the “position” and “momentum” operators or canonical variables) [75, 76]. These operators in terms of the creation and annihilation operators of the mode are defined as follows:2

2 For

qˆk = (ak + ak† ) ,

(4.33)

pˆ k = (ak − ak† )/ i.

(4.34)

convenience, in these formulas natural units with h¯ = 2 are accepted, as in Ref.[74].

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The quadrature operators define the phase space of each mode. We can group together the quadrature operators in the vector:   xˆ = qˆ1 , pˆ 1 , · · · , qˆN , pˆ N .

(4.35)

The commutation relations between the quadrature phase operators: [xˆk , xˆl ] = 2ikl , where  is 2N × 2N matrix known as the symplectic form: ⎛ =

N ) i=1

⎜ ω=⎝

⎞

ω ..



0 1 with ω = . −1 0

⎟ ⎠,

. ω

(4.36)

In quantum optics a mixed state is characterized by a density operator ρ, which defines a set of probabilities for the field to be found in the pure states. The space Hk is spanned by the Fock basis {|n k } of eigenstates of the number operator nˆ k = aˆ k† aˆ k . For each mode k, there exists a different vacuum state |0 k ∈ Hk such that aˆ( k |0 k = 0. The vacuum state of the global Hilbert space will be denoted by |0 = k |0 k . In the single-mode Hilbert space Hk , the eigenstates of aˆ k form a complete set of coherent states [18]. Coherent states result from applying the single-mode Weyl displacement oper† ∗ ator Dˆ k to the vacuum |0 k , |α k = Dˆ k (α) |0 k , where Dˆ k (α) = eα aˆ k −α aˆ k , and the coherent amplitude α is eigenstate of aˆ k : aˆ k |α k = α |α k . We note, here, that the coherent states are the quantum states that most closely resemble a classical electromagnetic field, e.g., the state of light in lasers [18, 25]. A coherent state may be written through the Fock basis of mode k: 1

|α k = e− 2 |α|

2

∞  αn √ |n k . n! n=1

(4.37)

Expanding this to the global Hilbert space of N different modes, we obtain a coherent state in this space by applying the N-mode Weyl operators Dˆ ξ to the global vacuum |0 : |ξ = Dˆ ξ |0 . Defining operator Dξ in terms of the canonical operators x, ˆ we have

 D(ξ ) = exp i xˆ T ξ , (4.38) where T denotes the transpose, ξ ∈ R2N . The density operator of an arbitrary quantum state has an equivalent representation in terms of a quasi-probability distribution (Wigner function) defined over a real symplectic space (phase space). That is an arbitrary ρˆ is equivalent to a (Wigner) characteristic function [76]: χ (ξ ) = tr [ρD(ξ )] ,

(4.39)

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and its Fourier transform known as the Wigner function:  W (x) =

2N ξ R2N

(2π )2N )

 exp −ixT ξ (χ ξ ).

(4.40)

The Wigner function has the meaning of quasi-probability distribution [76]. In the phase space the quantum states are described by the quasi-probability distributions instead of wave functions or density matrix in the Fock space. The quantities that characterize the Wigner representations (i.e., the characteristic function or the Wigner function) are the statistical moments. These are the mean vector (displacement vector): x¯ = x = tr (ρx) ,

(4.41)

and the covariance matrix V, whose elements are defined as: Vi,j =

1 {xˆi , xˆj } , 2

(4.42)

where xˆi = xˆi − xˆi and { } means the anti-commutator. The diagonal elements of this matrix give the covariances of the quadrature operators Vii = (xˆi )2 = xˆi2 − xˆi 2 , while the off-diagonal elements reflect the correlations between the different modes [18]. It is the characteristic feature of Gaussian states that they are completely described by the first two moments (the mean vector and the covariance matrix). The Gaussian states are defined by the Gaussian function of the following form [76]:   ¯ ¯ T V−1 (x − x)/2 exp −(x − x) W (x) = . √ (2π )N det V 4.3.2.2

(4.43)

Two-Mode Squeezed Vacuum Beam

It has been proposed firstly in Ref.[20] to realize the quantum illumination with the Gaussian beam in a two-mode entangled state created by spontaneous parametric down-conversion (SPDC), that is using the two-mode squeezed vacuum state [18]: |ξ =

∞  n=1

*

Nsn |n s |n i , (Ns + 1)n+1

(4.44)

where Ns is the mean photon number for both signal and idler modes: nˆ s = nˆ i = Ns . The state (4.44) is a Gaussian state [76] with mean vector x¯ = (0, 0, 0, 0) and

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covariance matrix ⎛

S 0 ⎜0 S V =⎜ ⎝Cq 0 0 −Cq

⎞ Cq 0 0 −Cq ⎟ ⎟, S 0 ⎠ 0 S

(4.45)

where S = 2Ns + 1 is the variance of the mode quadratures, while Cq = √ 2 Ns (Ns + 1) indicates the correlations between the signal and idler modes. The more general form of covariance matrices used in the quantum illumination can be given in the form: ⎛

A ⎜0 V =⎜ ⎝C 0

0 A 0 −C

C 0 B 0

⎞ 0 −C ⎟ ⎟. 0 ⎠ B

(4.46)

(See, e.g., the covariance matrix of two-mode squeezed thermal state given in the work [77].) As it was mentioned above, the off-diagonal elements of the V matrix contain information on the signal–idler correlations, a criterion that tells us if it is entangled or not by comparing the off-diagonal terms of the covariance matrix with the diagonal ones. For a covariance matrix of a Gaussian state given by (4.46), an entanglement criterion may formulate as [18]: C>

√

1 − A − B + AB.

(4.47)

From the expressions for Cq and Cc , we can make a very important conclusion on the regime where we have advantage in quantum illumination. Although the two-mode squeezed vacuum is always entangled, we can see that the correlation enhancement due to entanglement (Cq > Cc ) becomes less and less important as we increase the number of signal photons Ns [18]. In conclusion, we note that two-mode squeezed vacuum in optical domain is usually generated in SPDC (see Fig. 4.4). As we have seen in the previous section, it is a nonlinear process, which should match both energy and momentum conservation conditions (for the SPDC process sometimes mentioned as phasematching condition). In the SPDC, a pump beam with frequency ωP and wave vector kP is converted into correlated pairs of photons signal and idler modes characterized by the frequencies ωS , ωI and the wave vectors kS , kI . The SPDC at optical frequencies is realized in nonlinear crystals (i.e., optical parametric amplifiers [OPAs]), while at microwave frequencies Josephson parametric amplifier (JPA) (sometimes called as Josephson parametric converter) as nonlinear analogues of an OPA are used [18].

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Quantum Illumination with Gaussian States

Soon after the first work of Lloyd on quantum illumination, Tan et al. theoretically discussed a quantum illumination with Gaussian states [20]. Motivation of his work is based on the Lloyd’s adoption of a microcanonical noise model, which is, however, too restrictive. In reality, thermal-noise baths are in Gaussian states with Bose–Einstein distributed photon numbers. Besides, the Lloyd’s QI was restricted to photon pulse carrying a single photon manifold, effectively reducing to detect only one photon during the measurement interval, independent of whether the target is present or absent. In the work [20], a full Gaussian state treatment of quantum illumination detection of the target, which is embedded in a high noise bath has been proposed. In the Gaussian QI scheme, a signal and an idler can be obtained from continuous-wave laser pumped SPDC. The state of generated signal and idler photons with m mode has the joint state |ψm si =

∞  n=0

*

Nsn )n+1 |n sm |n im . (Ns + 1

(4.48)

As we have noted above, the Gaussian QI shows quantum enhancement only for a low brightness SPDC source, that is Ns  1. Furthermore, theory of Gaussian quantum illumination states that the conditions Ns  1, NB  1, and η  1 should be satisfied to reveal a quantum enhancement over any classical detection scheme. We refer readers to the original works for details of the calculations of the detection error probabilities (e.g., the mean error probability of the false alarm and the miss probability) for the cases of classical and quantum illuminations [18, 20]. It has been shown that coherence (classical illumination) in the Chernoff or the Bhattacharyya bound is given by: P (e)CI ≤ e−MNs η/4NB /2,

(4.49)

where M is the number of photons sent to the target region. The same probability for a Gaussian quantum illumination [20], obtained from an analytical expression for the Bhattacharyya bound in the limit η  1, Ns  1 and Nb  1, is given by: P (e)QI ≤ e−MNs η/NB /2.

(4.50)

Comparing these expressions, we see that a 6 dB advantage in the error probability exponent could be achieved compared to a coherent state illumination (this is, of course, an ideal case of application of the best possible detection scheme). Here also, the entanglement is expected to be lost entirely because of the high noise background. Therefore, the temporal correlation of signal and idler photons is explored to achieve quantum enhancement. We have seen in the previous section that the signal and idler correlations are mathematically explained using the covariance matrix. In a hypothetical problem of target present (H1 ) or absent (H0 ), one of the

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temporal correlated signals is sent out to find the target. According to the Gaussian detection model, if the target is absent, the annihilation operator of the return signal would be aR = aB . When it comes to target present in a background noise, the return signal annihilation operator is aˆ RM aˆ I M Hi ,

Hi = 0, 1

√

$ 1 − ηaˆ BM .

aˆ RM =

ηaˆ SM +

(4.51)

The target present hypothesis (H1) is solved when the non-negative value of aˆ SM and aˆ I M joint measurement is made. The positive operator-valued measurements (POVMs) associated with aˆ SM and aˆ I M can be realized by ideal (quantumlimited) optical heterodyne detection. Although heterodyne detection provides information about both quadratures, the Heisenberg uncertainty principle forces this measurement to incur extra noise on each quadrature that is not present in homodyne measurement of a single quadrature. In this context, this problem limits the demonstration of a quantum advantage system in the Gaussian scheme of quantum illumination. The quantum illumination and Gaussian illumination protocols require a measurement scheme, which automatically corrects for the extra time travelled by one of the temporally correlated photons. This means that the prior knowledge of the target range is required. Currently, no effective solution of this problem has been proposed. For that reason, some of researchers proposed an alternative scheme of the so-called quantum enhanced noise radar. In this concept, a temporal profile of the idler photon is detected and digitized immediately. Later this “fingerprint” is stored to make a matching filtering with received photons.

4.3.2.4

Quantum Illumination with Gaussian States in Optical Domain

The experimental scenario of Gaussian quantum illumination in optical domain can be described as follows. A continuous-wave (CW) pump beam is used to generate the entangled pairs in the SPDC process. Experimental conditions, revealing a quantum advantage of the Gaussian illumination, are as follows: Nb  1, Ns  1, and M = TW  1, where W is the SPDC’s phase-matching bandwidth. The generated signal and idler pairs are in an identical entangled two-mode squeezed state with a Fock basis representation as expressed in (6). The advantage of Gaussian illumination comes from the phase-sensitive cross-correlations of the returned signal with the retained idler mode. It is difficult to demonstrate a quantum advantage in a normal detection scenario as explained in the theory section. Therefore, an optical parametric amplifier (OPA) is inserted where returned signal photons are mixed with its retained one before making the measurement. Further, noise is added at the same wavelength than the signal beam. The total number of output photons are counted at one port of the OPA. The deciding factor whether the target is present or absent is by count rate N—if N < Nth (Noise count), the target is absent, present otherwise.

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In the scheme with the OPA receiver, which is not optimal one (it is known as “suboptimal” strategy for the discrimination of two mixed quantum states [18]), a 3dB advantage is expected. Experimental results with the OPA receiver [78] have shown only 20% (0.8 dB) advantage of Gaussian QI compared to coherent light in scheme due to experimental imperfections. Gaussian QI has a practical limitation because this approach reveals better performance over classical scheme only when Nb  1. However, this condition cannot be satisfied in optical domain at normal sky light conditions. Therefore, a practical implementation is of no use, unless a special condition arrives where a strong jamming signal is externally shot. In contrast, at microwave frequencies the condition Nb  1 holds; therefore, Gaussian QI at these frequencies is expected to be advantageous in both normal and jammed situations.

4.3.2.5

Quantum Illumination with Gaussian States in Microwave Domain

Due to established quantum-optical approaches, the first experiments on quantum illumination have been done at optical frequencies [21, 78]. Initially it has been proposed to generate entangled photons at optical frequencies and use quantum frequency converters to transduce a signal photon in microwave (MW) domain and back. In this scheme, after application of down frequency conversion the signal beam is sent to a target, while another photon is stored in idler or quantum memory device. For detection of returned photons, a quantum converter in a reverse direction (up-converter) is applied to transduce the detected photon back in the optical domain. Finally a phase-sensitive optical detector for detection of cross-correlations between a signal and idler photons is applied to receive a quantum advantage (see, e.g., [23]). Later, it has been realized that entangled photons in microwave regime can be realized with the use of Josephson parametric amplifiers (JPAs). The recent developments with JPA as sources of two-mode squeezed microwave photons enabled the first quantum radar experiments in these frequency domains [19, 24, 79]. It should be noted, however, that these publications are not based on the Gaussian quantum illumination scheme discussed in the previous section, first of all due to the fact that they do not implement the joint measurements on the stored idler and the signal photons. Instead, they apply heterodyne detection separately to both the idler photon generated by JPA and to the return photon. The time profiles of detected wave forms are digitally filtered and compared to reveal cross-correlation between them (Fig. 4.7). In this case there is no need to store idler photon. An important problem of all these works that they apply amplification of both the signal and the idler, introducing an additional noise. Because of the important differences with respect to classical quantum illumination scheme, they are usually referred to as hybrid quantum illumination radar, quantum two-mode squeezing (QTMS) radar, or quantum-enhanced noise radar. Despite these important limitations, they do show experimentally that the quantum-enhanced noise (or QTMS) radar works better than the classical counterpart. However, the theoretical work of Shapiro et al. [80] insists on the fact that classical radar schemes, compared

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Fig. 4.7 Simplified block diagram of the quantum two-mode squeezing (QTMS) radar

in these works, are not optimal ones; therefore, a quantum advantage has not been demonstrated in the hybrid quantum illumination approach. The main idea of the Shapiro work is that for the optimal target detection, the signal and idler intensities should not be the same in the classical case, while in the quantum illumination these intensities have to be the same due to a way of their generation. An important objection of Shapiro et al. is that the heterodyne measurements of the return and idler modes introduce additional noise, deteriorating the correlations between these two modes [18, 76]. Besides, there exist very efficient classical strategies to counteract the noise, which however are not applicable to the quantum case [80]. Thus, in the situation where the signal and return mode are measured separately, it is possible to propose an optimal classical radar having better performance than the quantum one. In this respect we have to note that currently there are no works applying joint measurements of the signal and idler photons in microwave domain. Apparently it is a non-trivial task due to very small energies of MW photons. Another issue of hybrid quantum illumination approaches is related to a need to amplify signal and idler beams prior to their transmission and detection due to a very small power of MW photons generated by JPA. As was already discussed above, amplification increases the noise variance reducing an entanglement of two-mode squeezed light. Two possible solutions, yet technically challenging, for resolving this issue have been proposed. First one is to use a quantum-limited amplifier instead of a conventional low noise amplifier. Another possible solution is to increase the power of the entangled source by combining many JPA sources in a single device, i.e., source multiplexing. In the Shapiro’s analysis the quantum two-mode squeezed (QTMS) radar is called as quantum correlated noise (QCN) radar, while its classical counterpart is called as classically correlated noise (CCN) radar (two-mode noise [TMN] radar in the work of [79] and classical illumination in the work of [24]). Analyses applied

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in these works are based on correlation matrices (similar to the covariance matrix introduced above, see Eq. (4.46)). In the analysis a difference between the signal and the idler is introduced in the case of CCN radar. The correlation matrices can be re-written in the form (compare with the form discussed in [17]): ⎡

⎤ √ √ NR + NF 0 ηNS cos(θ ) − ηNS sin(θ ) √ √ ⎢ ηNS sin(θ ) ηNS cos θ ⎥ 0 N + NF ⎢√ ⎥ √ R ⎢ ⎥ NF −1 ηNS sin(θ ) 1 + NS +1 0 ⎣ ηNS cos(θ ) ⎦ √ √ −1 0 1 + NNFS +1 − ηNS sin(θ ) ηNS cos(θ ) for the quantum correlated noise radar. For the classical correlated radar, the correlation matrix is of the form: ⎡ ⎤ √ √ NR + NF 0 ηNS cos(θ ) − ηNS sin(θ ) √ √ ⎢ ηNS sin(θ ) ηNS cos θ ⎥ 0 N + NF ⎢√ ⎥ √ R ⎢ ⎥ NF ηNS sin(θ ) 1 + NI 0 ⎣ ηNS cos(θ ) ⎦ √ √ − ηNS sin(θ ) ηNS cos(θ ) 0 1 + NNFI . In these expressions, GA is the pre-amplifier’s gain coefficient [80], η is the reflectivity coefficient in the interval 0 ≤ η  1, NS is the average number of the signal photons per mode, NI is the average value of the idler photons, NR = η NS + NB , and NF is the noise figure (NF ≥ 1). The correlation matrices for the absence of target are defined by η = 0. We note that in these matrices a more general form of covariance matrix (compare with (4.46)) is introduced, which contains extra non-diagonal terms. These terms are absent in the previous analysis, because in the Gaussian quantum illumination approach the stored idler is accepted to be matched in time delay and phase with the signal photon returned from the target. For this generalization, the phase θ (0 ≤ θ ≤ 2π ) is introduced (note that the matrix (4.3.2.5) will have the same symmetry as in (4.46) if we accept θ = 0). Following Shapiro’s analysis, NI and NS are accepted as independent parameters and for NF > 1 the CCN radar provides a better performance than QCN radar if the idler intensity matches a condition of NI > NF

NS + 1 . NF − 1

(4.52)

Despite the strong criticism of Shapiro’s work, there exist counterarguments against its main conclusions. Firstly, the experimental results of [24] and [17] have been received with the use of a low intensity idler. Therefore, these results in fact do not contradict Shapiro’s work, revealing although a modest but detectable enhancement using hybrid quantum illumination. Besides even we accept the argument from Shapiro’s work, hybrid quantum illumination with heterodyne detection could bring a technical advantage over classical approaches for specific tasks. The hybrid protocols would be advantageous when the use of coherent light for illumination

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or ranging is not possible. For instance, its implementation in short-range security applications or for non-invasive scanning with ultra low powers may be of interest. For these applications, the situation when the power of the idler does not satisfy the condition (4.52) is possible. In this respect, recently published work of [81] is very interesting. This work attracts attention to single photon statistics of fluctuations of the quantum-enhanced noise (QTMS) radar. It is stressed that simultaneous generation of signal and idler photons is only possible from an entangled source [81]. These quantum noise correlations explain an enhancement with respect to classical correlated noise scheme despite the fact that all measurements are classical ones. We have to note that this work follows the ideology proposed in the work [29]. Computer simulations performed in this work reveal that a single photon correlated (binary) waveform leads to SNR advantage only in the low photon level regime in agreement with the literature. Surely, extra experiments are needed to study the quantum-correlated noise radar and benchmark its performance against alternative classical radar approaches expected to provide a comparable SNR.

4.4 Discussion and Comparison of Quantum Radar Techniques There are four general approaches to quantum radar that has been identified in the most recent review paper by Torrome, Bekhti-Winkel, and Knott [25]. With an addition to these, the total list of five quantum techniques to be covered in this section is provided below: • • • • •

Interferometric quantum radar Quantum radar based on quantum illumination protocol Hybrid quantum radar systems Maccone–Ren theoretical quantum radar protocol Quantum illumination with Maccone–Ren’s quantum radar protocol

Discussions on interferometric quantum radar systems were presented in 4.3.1. Similarly, introduction of quantum illumination proposal was given in 4.3.2 and elaborated for continuous and discrete systems. The hybrid quantum radar systems’ classification of the authors is a more recent idea named as quantum two-mode squeezing (QTMS) radar system [17, 19]. Fourth approach is the Maccone–Ren theoretical quantum radar protocol [16], which is a recent proposal that utilizes entangled photons to investigate a spatial location expected to contain potential targets. The final proposal is another recent proposal, which is presented by Torrome, Bekhti-Winkel, and Knott [82], which is a combination of Lloyd’s quantum illumination idea and Maccone–Ren quantum radar protocol. The interferometric quantum radar proposal is the one that has been around for the longest time, and therefore it has many variations. The constant point

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√ among these proposals is the N efficiency increase in sensitivity against classical approaches. However, it has been discussed in the literature that using highly entangled NOON states cannot reach the Heisenberg limit in the presence of attenuation [1, 83] and only be able to surpass the standard quantum limit under special conditions (i.e., very low attenuation). This is a major limitation of utilizing NOON state-enabled interferometric quantum radar applications under atmospheric attenuation conditions [25, 84]. Possible approaches to overcome this obstacle can be using other forms of entangled states [22] or squeezed states [85]. The increase of interest (and hype to a certain extent) to the field following Lloyd’s quantum illumination proposal in 2008 [2] was shown in the introduction through Fig. 4.1. This is understandable as the abstract of the arxiv version [27] contains the following sentence: Quantum illumination with e bits of entanglement increases the effective signal-to-noise ratio of detection and imaging by a factor of 2e , an exponential improvement over unentangled illumination.

Coming from a veteran of the quantum information field like Seth Lloyd, such a promise carries a lot of weight. Of course, it is promised under ideal circumstances and special conditions. First of all, an entangled photon source with high time– bandwidth product is required. Secondly, lossless quantum memory systems are essential, which have been steadily being developed but nowhere near to the required efficiencies [86, 87]. Finally, optimal detection of single photon pairs is assumed, which is again a developing field with a long way ahead to reach demanded levels of optimal operational capabilities [88, 89]. As a bonus to all these technical assumptions, approximate location of the target is also needed to be known beforehand; hence, it is a deviation from the concept of a true radar, which is also capable of performing ranging operations. In this sense, quantum illumination is a method that promises orders of magnitude advantage at detection over classical methods while requiring previous knowledge on where to look at. Regardless of the last point, this work is extended to compare coherent light against quantum illumination and found out that in the optical regime, coherent beams outperform the latter [28]. A further improvement upon this work utilizes Gaussian states that have less requirements with a lower (though still significant) advantage against its classical counterparts [20], 6 dB against classical illumination under the same conditions. Finally, an expansion of quantum illumination into the microwave regime by Barzanjeh’s group [23, 24] is promoted to deal with issues arising from operating in the optical regime. The quantum two-mode squeezing (QTMS) radar system proposal builds upon the quantum illumination approach, but it alters the signal detection and filtering methods to enable classical techniques. This method compares the effectiveness of using entangled light for QTMS versus two-mode noise (TMN) radar systems [17] and proposes some advantages (such as reduced operation time) when entanglement is utilized. However, this is still lower than even the 6 dB improvement [20] let alone an exponential improvement promised by Lloyd [27]. At this point, current achievable proposals depending on quantum illumination offer not so significant

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(but still real) improvements over classical systems, but they mostly require a good beforehand information on the spatial region of a potential target. Fourth proposal investigated by Torrome, Bekhti-Winkel, and Knott [25] is a method called as the Maccone–Ren protocol for quantum radar, based on the work of Maccone and Ren in 2020. In this protocol, instead of keeping half of the photons as idler, all N of the entangled photons are sent to the suspected region in the following form and later detected to identify the location of an object with a high precision, which is probably why they name this approach as a quantum estimation protocol to estimate the location of a target in three dimensions [16].  |ψN ≡

N   a † (ω, k)  dωd kψ(ω, k) |0 .

(4.53)

Two main difficulties of this approach are generation of this entangled state and detection of it in the described form of Maccone and Ren. However, even in the original paper, extension of the method to other types of quantum states (such as squeezed states [90]) is proposed. In this sense, the entangled state used for this protocol can be accepted as a method of demonstration rather than an actual proposal for practical implementation. The last approach to be discussed here is the quantum illumination with Maccone–Ren’s quantum radar protocol proposed in [25, 30]. This method builds on Lloyd’s quantum illumination with utilizing the entangled states provided in Maccone–Ren protocol. Instead of using N entangled photons, they propose to use a 3-photon entangled state of the following form:  |ψ3 ≡

3   a † (ω, k)  dωd kψ(ω, k) |0 .

(4.54)

Out of this state, one photon is kept as idler while the other two are sent as signal. Through this, the idler photon provides a baseline for timing of observing two photons in the spectrum range of the signal, providing a criterion for positive detection. This is obtained via joint measurements between the signal and idler photons. Again, the main limitations of this approach are the generation of above entangled state, and the issues with quantum memories that are required to keep the idler photon (Table 4.3).

4.5 A General Comparison of Quantum Radar and Classical Radar In this section, a brief and general comparison of quantum and classical radar systems is presented. This comparison is carried out on two levels: theoretical and practical.

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Table 4.3 Quantum radar techniques and proposed advantages Quantum radar technique Interferometric quantum radar [3, 22] Quantum Illumination [2, 20, 23, 27, 28]

Proposed advantage √ N reduction in estimation error 2n or 3 dB–6 dB increase in signal-to-noise ratio of detection Lower increase than quantum illumination A reduction in the range detection error of √ order 2N per measurement

Hybrid quantum radar [17, 19] Maccone–Ren quantum radar [16] Torrome, Bekhti-Winkel, and Knott [25, 30]

–

Classical radar systems utilize a large number of photons, and therefore they can be represented through Maxwell’s equations. Furthermore, a classical radar system can be neatly described by what is called as the radar equation [1, 91] of the following form: Pr =

Pt Gt Ar σ F 4 , (4π )2 R 4

(4.55)

where Pr is the power of the receiver, Pt Gt is the outgoing power, σ is the radar cross section, Ar is the aperture of the receiver, F is a factor that gives the transparency of the medium, and R is the estimated range to the target. This is a compact formula that allows other properties of a classical radar such as maximum range under certain conditions to be deduced from. For the concept of a quantum radar, no such clear description is present that is applicable to all proposals covered above. There is not a single formula that is valid for both quantum enhanced interferometric measurements, quantum illumination variants, or an alternative method utilizing novel properties. A proposal by Lanzagorta derived from the classical description is presented below: Q

PrQ =

Pt Ar σQ , (4π )2 R 4

(4.56)

where PrQ is the received power at the quantum radar, PtQ is the transmitted power of the quantum radar, σQ is the quantum radar cross section. There are several studies conducted in the literature on how to calculate or determine the quantum radar cross section of different configurations [7, 92, 93]. There are still much to explore in this line of research on the path towards mainstreaming a general approach to the concept of quantum radar. Particular technical limitations of different protocols were covered on the previous section. In this part, a more general list of shortcomings is going to be listed. These shortcomings can be divide into three categories: conceptual, technical, and operational. The main conceptual downside of most quantum radar proposals is that they require a beforehand knowledge on where to look at, which is a significant

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shortcoming compared to classical radar concept. Although some of the more recent proposals [16, 29, 30] offer alternative methods to incorporate ranging as well, its lack at the original proposals makes it not a straightforward process, which in return results in notable reductions in advantages. Second category is the technical shortcomings, mainly the troubles of storage for idler (or signal) photons, not being able to generate required entanglement characteristics with high fidelity, and lack of high efficiency quantum sensors. These are all technical issues that are expected to be solved in the following years, however, since their effect on the potential usefulness of a quantum radar system is crucial, currently they render most of the quantum radar proposals technically not applicable. Finally, on an operational level, field ready implementation of quantum radar systems are out of question with the current state of the art. Any realistic comparison on operational capacity between classical and quantum radar systems would require development of a field ready system, which does not seem to be the case for near term and depends on many pre-requisite developments, which may or may not occur in the middle term. By keeping these considerations in mind, the benefits that a quantum radar technology could have over conventional technologies are still appealing. In this perspective, the biggest technical challenges preventing the quantum radar technology today should be addressed explicitly. The first and the greatest challenge is the target ranging problem. The joint measurement requirement of quantum illumination and hybrid protocols is biggest obstacle on the path. For the joint measurement, one of the photons (either signal or idler) is delayed by a known time to fit the time of flight of the other photons. This require the preliminary knowledge of the extra optical path taken by one of the photons, which is against the idea of ranging. Although there are proposed techniques utilizing classical digital methods or matched filtering [16, 24, 29, 82] to this problem, it is still far away from practical use, and it is an open debate whether the mentioned post-processing methods fully eliminate the entanglement properties or not. A parallel problem to the above-mentioned one is the idler (or signal) reference photon storage. Quantum memories have been proposed for efficient storage of idler to allow joint measurements [17, 23]. Another challenge of building practical quantum radars is the efficient generation and detection of entangled light. The entangled beams can be generated via spontaneous parametric down conversion (SPDC) or through Josephson parametric amplifiers (JPAs) in optical/near-infrared and microwave domains, respectively. Alternatively microwave–optical frequency converters can be used. While the optical/near-infrared band entanglement generation is rather straightforward and robust, the microwave entangled beams are hard to create via JPA as it requires cryogenic temperatures [23, 24, 94, 95]. The microwave–optical converters also have very low conversion efficiencies. The detection methods in microwave domain are also rather challenging compared to optical/near-infrared domain. In optical/nearinfrared domain the single photon detectors work quite efficiently; however, the maximum count rate of detectors limits the maximum achievable range of quantum radars and lidars [21, 25, 96]. This is a rather mild challenge as it is always possible to do multiplexing of detectors [29]. A related challenge to the detection is the low signal level due to the low reflectivity of target and atmospheric losses.

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This challenge is valid for all types of quantum radar proposals. The maximum achievable range for quantum radars in optical/near-infrared domain is less than a kilometer, which is due to the combined effect of all above-mentioned challenges and limitations [29]. On the microwave domain the signal level is not sufficient for target detection and exhibiting quantum advantage when NS  1.

4.6 Conclusion and Outlook: Potential of Practical Applications of Quantum Radar The aim of this chapter was to cover the historical and theoretical background of quantum radar as a concept, introduce different proposals on how to realize this concept, present the current situation in the field, and to provide a critical view on implementation claims in the near term. To this regard, most conclusions of reviews on quantum radar propose further integration with other types of quantum technologies would be mutually beneficial [1, 25, 80]. Two ideas are explicitly noted in Shapiro’s review on Quantum Illumination. First one is introduced as an intellectual grandchild of the quantum illumination idea, which allows QKD secretkey rates with Gbps levels called floodlight QKD [97]. The second is a more recent method developed for decreasing the chance of success for an eavesdropper attack titled quantum low probability of intercept (QLPI) [98]. Another potential area of research that is in close relations with quantum radar and that can benefit from these studies is noted as quantum metrology [99, 100], demonstrating that research on quantum radar does not only progress the narrow field of radar applications in the quantum regime. Reverse ordering of this relation can also be discussed. Recent developments in the fields of quantum memory, entanglement generation, and photon detection schemes can help accelerate the field of quantum radar and may even allow operational deployment in the future. This requires collaboration between different groups, and adoption of novel techniques. During the literature analysis performed for this chapter, a trend of two-fold approach was taken notice. Academic research oriented groups mainly try to utilize continuous variables on microwave regime, meanwhile industry oriented research is focused on discrete variables and the optical regime. Diversification of research and expanding the definition of the problem to cover solving necessary requirements through collaborative projects may hasten the development for all. It is suggested here that quantum radar should be acknowledged less as a near-term countermeasure for stealth, but as an exciting field of scientific research with many possible spillover applications. Finally, it should be noted that not looking at quantum radar as an alternative to classical radar but a complementary or even enabling technology would expand the potential for research oriented at producing end products that are more likely to be operationalized for field-ready purposes. Quantum radar proposals are relatively resistant to spoofing; furthermore, due to utilization of low number of photons

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for detection, they themselves can be considered as stealth. Additionally, high resolution and ability of detection of smaller objects make quantum-enhanced radar systems more versatile than their classical counterparts. Since range is a major technical limitation for the current quantum radar proposals, aiming to use these devices for short-range purposes may solve or relax this limitation up to a certain extent just by changing the definition of the problem. Similar re-framings of the technical limitations may help expand the potential uses of quantum radar, instead of demanding it to be an alternative for classical radars capable of long-range ranging and detection purposes, which it may never be able to perform.

References 1. Lanzagorta, M. (2011). Quantum radar. Synthesis Lectures on Quantum Computing, 3(1), 1– 139. https://doi.org/10.2200/S00384ED1V01Y201110QMC005. 2. Lloyd, S. (2008). Enhanced sensitivity of photodetection via quantum illumination. Science, 321(5895), 1463–1465. ISSN: 0036-8075. https://doi.org/10.1126/science.1160627. https:// science.sciencemag.org/content/321/5895/1463.full.pdf. https://science.sciencemag.org/ content/321/5895/1463. 3. Giovannetti, V., Lloyd, S., & Maccone, L. (2001). Quantum-enhanced positioning and clock synchronization. Nature, 412(6845), 417–419. ISSN: 1476–4687. https://doi.org/10.1038/ 35086525. 4. Steinhardt, A., & McCrae, J. (2003). Radar in the quantum limit. In IEEE International Symposium on Phased Array Systems and Technology (pp. 31–34). 5. Zaugg, T. (2004). Entangled-photon range finding system and method 6. Edward, H. (2005). Allen and Markos Karageorgis. In Radar systems and methods using entangled quantum particles. 7. Brandsema, M. J., Narayanan, R. M., & Lanzagorta, M. (2016). Theoretical and computational analysis of the quantum radar cross section for simple geometrical targets. Quantum Information Processing, 16(1), 32. ISSN: 1573–1332. https://doi.org/10.1007/s11128-0161494-6. 8. Afek, I., Ambar, O., & Silberberg, Y. (2010). High-NOON states by mixing quantum and classical light. Science, 328(5980), 879–881. 9. Eisert, J., & Plenio, M. B. (1999). A comparison of entanglement measures. Journal of Modern Optics, 46(1), 145–154. https://doi.org/10.1080/09500349908231260. https:// www.tandfonline.com/doi/pdf/10.1080/09500349908231260. https://www.tandfonline.com/ doi/abs/10.1080/09500349908231260. 10. Horodecki, M., Horodecki, P., & Horodecki, R. (1998). Mixed-state entanglement and distillation: is there a “Bound” entanglement in nature? Physical Review Letters, 80, 5239–5242. https://doi.org/10.1103/PhysRevLett.80.5239. https://link.aps.org/doi/10.1103/PhysRevLett. 80.5239. 11. Yu, T., & Eberly, J. H. (2009). Sudden death of entanglement. Science, 323(5914), 598– 601. ISSN: 0036-8075. https://doi.org/10.1126/science.1167343. https://science.sciencemag. org/content/323/5914/598.full.pdf. https://science.sciencemag.org/content/323/5914/598. 12. Raimond, J. M., Brune, M., & Haroche, S. (2001). Manipulating quantum entanglement with atoms and photons in a cavity. Reviews of Modern Physics, 73, 565–582. https://doi.org/10. 1103/RevModPhys.73.565. https://link.aps.org/doi/10.1103/RevModPhys.73.565. 13. Bennett, C. H., et al. (1993). Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Physical Review Letters, 70, 1895–1899. https://doi.org/ 10.1103/PhysRevLett.70.1895. https://link.aps.org/doi/10.1103/PhysRevLett.70.1895.

4 Quantum Radar

161

14. Mermin, N. D. (1990). Extreme quantum entanglement in a superposition of macroscopically distinct states. Physical Review Letters, 65, 1838–1840. https://doi.org/10.1103/PhysRevLett. 65.1838. https://link.aps.org/doi/10.1103/PhysRevLett.65.1838. 15. Clauser, J. F., et al. (1969). Proposed experiment to test local hidden-variable theories. Physical Review Letters, 23(15), 880–884. https://doi.org/10.1103/PhysRevLett.23.880. 16. Maccone, L., & Ren, C. (2020). Quantum radar. Physical Review Letters, 124, 200503. https://doi.org/10.1103/PhysRevLett.124.200503. https://link.aps.org/doi/10.1103/ PhysRevLett.124.200503. 17. Luong, D., et al. (2020). Receiver operating characteristics for a prototype quantum two-mode squeezing radar. IEEE Transactions on Aerospace and Electronic Systems, 56(3), 2041–2060. 18. Sorelli, G., et al. (2020). Detecting a target with quantum entanglement. eprint: arXiv:2005.07116. 19. Chang, C. W. S., et al. (2019). Quantum-enhanced noise radar. Applied Physics Letters, 114(11), 112601. https://doi.org/10.1063/1.5085002. 20. Tan, S.-H., et al. (2008). Quantum illumination with Gaussian states. Physical Review Letters, 101, 253601. https://doi.org/10.1103/PhysRevLett.101.253601. https://link.aps.org/ doi/10.1103/PhysRevLett.101.253601. 21. Lopaeva, E. D., et al. (2013). Experimental realization of quantum illumination. Physical Review Letters, 110, 153603. https://doi.org/10.1103/PhysRevLett.110.153603. https://link. aps.org/doi/10.1103/PhysRevLett.110.153603. 22. Giovannetti, V., Lloyd, S., & Maccone, L. (2004). Quantum-enhanced measurements: beating the standard quantum limit. Science, 306(5700), 1330–1336. ISSN: 00368075. https://doi.org/10.1126/science.1104149. https://science.sciencemag.org/content/306/ 5700/1330.full.pdf. https://science.sciencemag.org/content/306/5700/1330. 23. Barzanjeh, S., et al. (2015). Microwave quantum illumination. Physical Review Letters, 114, 080503. https://doi.org/10.1103/PhysRevLett.114.080503. https://link.aps.org/doi/10. 1103/PhysRevLett.114.080503. 24. Barzanjeh, S., et al. (2020). Microwave quantum illumination using a digital receiver. Science Advances, 6(19). https://doi.org/10.1126/sciadv.abb0451. https://advances.sciencemag.org/ content/6/19/eabb0451.full.pdf. https://advances.sciencemag.org/content/6/19/eabb0451. 25. Torromé, R. G., Bekhti-Winkel, N. B., & Knott, P. (2020). Introduction to quantum radar. arXiv: 2006.14238. http://arxiv.org/abs/2006.14238. 26. Harris Corporation. (2009). Quantum sensors program. https://apps.dtic.mil/dtic/tr/fulltext/ u2/a506209.pdf. 27. Lloyd, S. (2008). Quantum illumination. eprint: arXiv:0803.2022. 28. Shapiro, J. H., & Lloyd, S. (2009). Quantum illumination versus coherent-state target detection. New Journal of Physics, 11(6), 063045. https://doi.org/10.1088/1367-2630/11/6/ 063045. 29. Durak, K., Jam, N., & Dindar, C. (2019). Object tracking and identification by quantum radar. arXiv:1908.06850. 30. Torromé, R. G., Bekhti-Winkel, N. B., & Knott, P. (2020). Quantum illumination with multiple entangled photons. arXiv:2008.09455. 31. Bell, J. S. (2004). On the Einstein Podolsky Rosen Paradox. In Speakable and unspeakable in quantum mechanics (pp. 14–21). Cambridge University Press. 32. Greenberger, D. M., et al. (1990). Bell’s theorem without inequalities. American Journal of Physics, 58(12), 1131–1143. https://doi.org/10.1119/1.16243. 33. Horodecki, M., Horodecki, P., & Horodecki, R. (1996). Separability of mixed states: necessary and sufficient conditions. Physics Letters, A 223(1), 1–8. ISSN: 0375-9601. https:// doi.org/10.1016/S0375-9601(96)00706-2. http://www.sciencedirect.com/science/article/pii/ S0375960196007062. 34. Horodecki, P. (1997). Separability criterion and inseparable mixed states with positive partial transposition. Physics Letters A, 232(5), 333–339. ISSN: 0375–9601. https:// doi.org/10.1016/S0375-9601(97)00416-7. http://www.sciencedirect.com/science/article/pii/ S0375960197004167.

162

K. Durak et al.

35. Chru´sci´nski, D., & Sarbicki, G. (2014). Entanglement witnesses: construction, analysis and classification. Journal of Physics A: Mathematical and Theoretical, 47(48), 483001. https:// doi.org/10.1088/1751-8113/47/48/483001. 36. Braunstein, S. L., & Caves, C. M. (1994). Statistical distance and the geometry of quantum states. Physical Review Letters, 72, 3439–3443. https://doi.org/10.1103/PhysRevLett.72. 3439. https://link.aps.org/doi/10.1103/PhysRevLett.72.3439. 37. Einstein, A., Podolsky, B., & Rosen, N. (1935). Can quantum-mechanical description of physical reality be considered complete? Physics Review, 47, 777–780. https://doi.org/10. 1103/PhysRev.47.777. https://link.aps.org/doi/10.1103/PhysRev.47.777. 38. Schrödinger, E. (1935). Die gegenwärtige situation in der quantenmechanik. Naturwissenschaften, 23(48), 807–812. ISSN: 1432–1904. https://doi.org/10.1007/BF01491891. 39. Ekert, A. K. (1991). Quantum cryptography based on Bell’s theorem. Physical Review Letters, 67, 661–663. https://doi.org/10.1103/PhysRevLett.67.661. https://link.aps.org/doi/10.1103/ PhysRevLett.67.661. 40. Bennett, C. H., & Wiesner, S. J. (1992). Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Physical Review Letters, 69, 2881–2884. https://doi.org/ 10.1103/PhysRevLett.69.2881. https://link.aps.org/doi/10.1103/PhysRevLett.69.2881. 41. Mattle, K., et al. (1996). Dense coding in experimental quantum communication. Physical Review Letters, 76, 4656–4659. https://doi.org/10.1103/PhysRevLett.76.4656. https://link. aps.org/doi/10.1103/PhysRevLett.76.4656. 42. Horodecki, R., et al. (2009). Quantum entanglement. Reviews of Modern Physics, 81, 865–942. https://doi.org/10.1103/RevModPhys.81.865. https://link.aps.org/doi/10.1103/ RevModPhys.81.865. 43. Kimble, H. J. (2008). The quantum internet. Nature, 453(7198), 1023–1030. ISSN: 1476– 4687. https://doi.org/10.1038/nature07127. https://doi.org/10.1038/nature07127. 44. Amico, L., et al. (2008). Entanglement in many-body systems. Reviews of Modern Physics, 80, 517–576. https://doi.org/10.1103/RevModPhys.80.517. https://link.aps.org/doi/10.1103/ RevModPhys.80.517. 45. Dür, W., & Briegel, H. J. (2007). Entanglement purification and quantum error correction. Reports on Progress in Physics, 70(8), 1381–1424. https://doi.org/10.1088/0034-4885/70/8/ r03. 46. Gühne, O., & Tóth, G. (2009). Entanglement detection. Physics Reports, 474(1), 1–75. ISSN: 0370–1573. https://doi.org/10.1016/j.physrep.2009.02.004. http://www.sciencedirect. com/science/article/pii/S0370157309000623. 47. Plenio, M. B., & Virmani, S. (2007). An introduction to entanglement measures. Quantum Information and Computation, 7(1), 1–51. ISSN: 1533–7146. 48. Calabrese, P., & Cardy, J. (2009). Entanglement entropy and conformal field theory. Journal of Physics A: Mathematical and Theoretical, 42(50), 504005. https://doi.org/10.1088/17518113/42/50/504005. 49. Blatt, R., & Wineland, D. (2008). Entangled states of trapped atomic ions. Nature, 453(7198). https://doi.org/10.1038/nature07125. 50. Heaney, L., & Vedral, V. (2009). Natural mode entanglement as a resource for quantum communication. Physical Review Letters, 103, 200502. https://doi.org/10.1103/PhysRevLett. 103.200502. https://link.aps.org/doi/10.1103/PhysRevLett.103.200502. 51. de Vicente, J. I., Spee, C., & Kraus, B. (2013). Maximally entangled set of multipartite quantum states. Physical Review Letters, 111, 110502. https://doi.org/10.1103/PhysRevLett. 111.110502. https://link.aps.org/doi/10.1103/PhysRevLett.111.110502. 52. Werner, R. F. (1989). Quantum states with Einstein-Podolsky-Rosen correlations admitting a hiddenvariable model. Physical Review A, 40, 4277–4281. https://doi.org/10.1103/PhysRevA. 40.4277. https://link.aps.org/doi/10.1103/PhysRevA.40.4277. 53. Thirring, W., et al. (2011). Entanglement or separability: the choice of how to factorize the algebra of a density matrix. The European Physical Journal D, 64(2), 181–196. ISSN: 1434– 6079. https://doi.org/10.1140/epjd/e2011-20452-1.

4 Quantum Radar

163

54. Mintert, F., Ku s´, M., & Buchleitner, A. (2004). Concurrence of mixed bipartite quantum states in arbitrary dimensions. Physical Review Letters, 92, 167902. https://doi.org/10.1103/ PhysRevLett.92.167902. https://link.aps.org/doi/10.1103/PhysRevLett.92.167902. 55. Cramer, M., Plenio, M. B., & Wunderlich, H. (2011). Measuring entanglement in condensed matter systems. Physical Review Letters, 106, 020401. https://doi.org/10.1103/PhysRevLett. 106.020401. https://link.aps.org/doi/10.1103/PhysRevLett.106.020401. 56. Bennett, C. H., et al. (1999). Quantum nonlocality without entanglement. Physical Review Letters, 59, 1070–1091. https://doi.org/10.1103/PhysRevA.59.1070. https://link.aps.org/doi/ 10.1103/PhysRevA.59.1070. 57. Ollivier, H., & Zurek, W. H. (2001). Quantum discord: a measure of the quantumness of correlations. Physical Review Letters, 88, 017901. https://doi.org/10.1103/PhysRevLett.88. 017901. https://link.aps.org/doi/10.1103/PhysRevLett.88.017901. 58. Henderson, L., & Vedral, V. (2001). Classical, quantum and total correlations. Journal of Physics A: Mathematical and General, 34(35), 6899–6905. https://doi.org/10.1088/03054470/34/35/315. 59. Bera, A., et al. (2017). Quantum discord and its allies: a review of recent progress. Reports on Progress in Physics, 81(2), 024001. https://doi.org/10.1088/1361-6633/aa872f. 60. De Chiara, G., & Sanpera, A. (2018). Genuine quantum correlations in quantum many-body systems: a review of recent progress. Reports on Progress in Physics, 81(7), 074002. https:// doi.org/10.1088/1361-6633/aabf61. 61. Streltsov, A. (2014). Quantum correlations beyond entanglement: And their role in quantum information theory. SpringerBriefs in Physics. Springer International Publishing. ISBN: 9783319096568. https://books.google.com.tr/books?id=-rQjBQAAQBAJ. 62. Fanchini, F. F., de Oliveira Soares Pinto, D., & Adesso, G. (2017). Lectures on general quantum correlations and their applications. Quantum Science and Technology. Springer International Publishing. ISBN: 978-3319534121. https://books.google.com.tr/books?id= G3opDwAAQBAJ. 63. Yuen, H. P. (1986). Amplification of quantum states and noiseless photon amplifiers. Physics Letters A, 113(8), 405–407. 64. Adnane, H., Teklu, B., & Paris, M. G. A. (2019). Quantum phase communication channels assisted by non-deterministic noiseless amplifiers. JOSA B, 36(11), 2938–2945. 65. Abram, I., & Levenson, J. A. (1994). Quantum noise in parametric amplification. In Nonlinear spectroscopy of solids (pp. 251–287). Springer. 66. Zavatta, A., Fiurášek, J., & Bellini, M. (2011). A high-fidelity noiseless amplifier for quantum light states. Nature Photonics, 5(1), 52–56. 67. He, H., et al. (2020). Non-classical semiconductor photon sources enhancing the performance of classical target detection systems. Journal of Lightwave Technology, 38, 4540–4547. 68. Boto, A. N., et al. (2000). Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit. Physical Review Letters, 85, 2733–2736. https://doi.org/10. 1103/PhysRevLett.85.2733. https://link.aps.org/doi/10.1103/PhysRevLett.85.2733. 69. Israel, Y., Rosen, S., & Silberberg, Y. (2014). Supersensitive polarization microscopy using NOON states of light. Physical Review Letters, 112(10), 103604. 70. Wang, W., et al. (2019). Heisenberg-limited single-mode quantum metrology in a superconducting circuit. Nature Communications, 10(1), 1–6. 71. Luo, C., et al. (2017). Heisenberg-limited Sagnac interferometer with multiparticle states. Physical Review A, 95(2), 023608. 72. Zhou, Z.-Y., et al. (2017). Superresolving phase measurement with short-wavelength noon states by quantum frequency up-conversion. Physical Review Applied, 7(6), 064025. 73. Smith III, J. F. (2009). Quantum entangled radar theory and a correction method for the effects of the atmosphere on entanglement. In Quantum information and computation VII (Vol. 7342). International Society for Optics and Photonics, 73420A.

164

K. Durak et al.

74. Adesso, G., & Illuminati, F. (2007). Entanglement in continuous-variable systems: recent advances and current perspectives. Journal of Physics A: Mathematical and Theoretical, 40(28), 7821–7880. ISSN: 1751–8113. https://doi.org/10.1088/1751-8113/40/28/S01. https:// iopscience.iop.org/article/10.1088/1751-8113/40/28/S01. 75. Wang, X., et al. (2007). Quantum information with Gaussian states. Physics Reports, 448(1–4), 1–111. ISSN: 03701573. https://doi.org/10.1016/j.physrep.2007.04.005. https:// linkinghub.elsevier.com/retrieve/pii/S0370157307001822. 76. Weedbrook, C., et al. (2012). Gaussian quantum information. Reviews of Modern Physics, 84(2), 621–669. ISSN: 00346861. https://doi.org/10.1103/RevModPhys.84.621. eprint: 1110.3234. 77. Barzanjeh, S., et al. (2019). Stationary entangled radiation from micromechanical motion. Nature, 570(7762), 480–483. ISSN: 0028–0836. https://doi.org/10.1038/s41586-019-13202. http://www.nature.com/articles/s41586-019-1320-2. 78. Zhang, Z., et al. (2015). Entanglement-enhanced sensing in a lossy and noisy environment. Physical Review Letters, 114(11). ISSN: 10797114. https://doi.org/10.1103/PhysRevLett. 114.110506. arXiv:1411.5969. 79. Luong, D., Rajan, S., & Balaji, B. (2020). Quantum two-mode squeezing radar and noise radar: correlation coefficients for target detection. https://doi.org/10.1109/JSEN.2020. 2971851. arXiv: 1911.09062. 80. Shapiro, J. H. (2019). The quantum illumination story. eprint: arXiv:1910.12277. 81. Brandsema, M. J., Narayanan, R. M., & Lanzagorta, M. O. (2020). Correlation properties of single photon binary waveforms used in quantum radar/lidar. In A. M. Raynal & K. I. Ranney (Ed.), Radar sensor technology XXIV (p. 35). SPIE, Apr. ISBN: 9781510635937. https:// doi.org/10.1117/12.2560184. https://www.spiedigitallibrary.org/conference-proceedings-ofspie/11408/2560184/Correlation-properties-of-single-photon-binary-waveforms-used-inquantum/10.1117/12.2560184.full. 82. Torromé, R. G., Bekhti-Winkel, N. B., & Knott, P. (2020). Quantum illumination with multiple entangled photons. arXiv preprint arXiv:2008.09455. 83. Gilbert, G., & Weinstein, Y. S. (2008). Aspects of practical remote quantum sensing. Journal of Modern Optics, 55(19–20), 3283–3291. https://doi.org/10.1080/09500340802428314. https://doi.org/10.1080/09500340802428314. 84. Gilbert, G., & Hamrick, M. (2000). Practical quantum cryptography: a comprehensive analysis (part one). eprint: arXiv:quant-ph/0009027. 85. Barnett, S. M., Fabre, C., & Maítre, A. (2003). Ultimate quantum limits for resolution of beam displacements. The European Physical Journal D – Atomic, Molecular, Optical and Plasma Physics, 22(3), 513–519. ISSN: 1434–6079. https://doi.org/10.1140/epjd/e2003-00003-3. 86. Simon, C., et al. (2010). Quantum memories. The European Physical Journal D, 58(1), 1–22. ISSN: 1434-6079. https://doi.org/10.1140/epjd/e2010-00103-y. 87. Heshami, K., et al. (2016). Quantum memories: emerging applications and recent advances. Journal of Modern Optics, 63(20), 2005–2028. PMID: 27695198. https://doi.org/10.1080/ 09500340.2016.1148212. 88. Hadfield, R. H. (2009). Single-photon detectors for optical quantum information applications. Nature Photonics, 3(12), 696–705. ISSN: 1749-4893. https://doi.org/10.1038/nphoton.2009. 230. 89. Chunnilall, C. J., et al. (2014). Metrology of single-photon sources and detectors: a review. Optical Engineering, 53(8), 1–17. https://doi.org/10.1117/1.OE.53.8.081910. 90. Maccone, L., & Riccardi, A. (2020). Squeezing metrology: a unified framework. Quantum, 4, 292. ISSN: 2521-327X. https://doi.org/10.22331/q-2020-07-09-292. 91. Skolnik, M. (2002). Introduction to radar systems. McGraw-Hill Education. ISBN: 0072881380. 92. Liu, K., et al. (2014). Analysis of quantum radar cross section and its influence on target detection performance. IEEE Photonics Technology Letters, 26(11), 1146–1149. 93. Fang, C., et al. (2018). The calculation and analysis of the bistatic quantum radar cross section for the typical 2-D plate. IEEE Photonics Journal, 10(2), 1–14.

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94. Chang, C. W. S., et al. (2018). Generating multimode entangled microwaves with a superconducting parametric cavity. Physical Review Applied, 10(4), 044019. 95. Messaoudi, N., et al. (2020). Quantum-enhanced noise radar. Bulletin of the American Physical Society, 65. 96. England, D. G., Balaji, B., & Sussman, B. J. (2019). Quantum-enhanced standoff detection using correlated photon pairs. Physical Review A, 99(2), 023828. 97. Zhang, Z., et al. (2017). Floodlight quantum key distribution: demonstrating a framework for high-rate secure communication. Physical Review A, 95, 012332. https://doi.org/10.1103/ PhysRevA.95.012332. https://link.aps.org/doi/10.1103/PhysRevA.95.012332. 98. Shapiro, J. H., et al. (2019). Quantum low probability of intercept. Journal of the Optical Society of America B: Optical Physics, 36(3), B41–B50. https://doi.org/10.1364/JOSAB.36. 000B41. http://josab.osa.org/abstract.cfm?URI=josab-36-3-B41. 99. Berchera, I. R., & Degiovanni, I. P. (2019). Quantum imaging with sub-Poissonian light: challenges and perspectives in optical metrology. Metrologia, 56(2), 024001. https://doi.org/ 10.1088/1681-7575/aaf7b2. 100. Gregory, T., et al. (2020). Imaging through noise with quantum illumination. Science Advances, 6(6) (2020). https://doi.org/10.1126/sciadv.aay2652. https://advances.sciencemag. org/content/6/6/eaay2652.full.pdf. https://advances.sciencemag.org/content/6/6/eaay2652.

Chapter 5

Quantum Finance Do Ngoc Diep

Money problem has a long history. There are some attempts to create cheques with qubits inserted for security. There are also some essays by using blockchains to have some kind of currencies. Another attempt are some online cheques and also the problems of risk management and forecasting of time series and portfolio creation. We will discuss these problems in the this chapter and will take a special attention to Sect. 5.4 about Quantum Cheques and Quantum E-Cheques, which are in our opinion important for currency management.

5.1 Quantum Currency We introduce the notion of quantum currency in this section and then discuss the main problems of the theory. Quantum money is a proposed design of bank notes that are hundred percent secured. It is based on the obtained results of the study of quantum key distribution protocols in quantum cryptography. In the history, this idea was firstly appeared from Stephen Wiesner from Columbia University (1970). Later, this work was published in 1983 [1]: The main idea is as follows. At each transaction with the bank, the unique serial number of the note is supposed to be verified. The serial number is printed on the note; everybody can see and can use but the state (i.e., polarization) of the secret qubits is secret. If the user chooses a wrong quantum measurement (resp., polarization) to measure the secret qubits, then as a result the

D. N. Diep () Thang Long Institute of Mathematics and Applied Sciences, Thang Long University, Ha Noi, Vietnam Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Iyengar et al. (eds.), Quantum Computing Environments, https://doi.org/10.1007/978-3-030-89746-8_5

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serial number should be changed, because the no-cloning theorem for those qubits cannot be copied. Quantum money is not yet practical to implement with the current technology because the quantum bank notes require to store the quantum states in a quantum memory. Quantum memories can currently store quantum states only for a very short time. Recently, several other schemes for quantum currencies have been proposed, all relying on the assumption that the quantum source device acts according to its specification. This makes several known quantum money protocols vulnerable to the hardware Trojan horse attacks. In [2], the authors studied the following problem: to what extent quantum money schemes can be made independent from the inner working of source and verification devices used by the honest parties (bank and mint) in creating and processing the quantum money? Drawing inspirations from the semi-device-independent quantum key distribution protocol, the authors introduce the first scheme of quantum money with this assumption partially relaxed, along with the proof of its unforgeability and a quantum analog of the Oresme– Copernicus–Gresham’s law of economy. Recently, Bozzio et al. [3] have demonstrated experimentally how to replace challenging quantum verification with a classical channel and a quantum retrieval game (QRG). The procedure consists of the following steps: 1. The bank encodes quantum money (QM) by using a secret sequence (token) of pairs consisting of one qubit pure state |0 or |1 and one qubit mixed state |± = √1 (|0 ± |1 ) from 2

|0 + := |0 ⊗ |1 + := |1 ⊗ | + 0 :=

√1 |(|0 + |1 ), 2 √1 |(|0 + |1 ), 2

√1 (|0 + |1 ) ⊗ |0 , 2 √1 (|0 + |1 ) ⊗ |1 , 2

|0 − := |0 ⊗ |1 − := |1 ⊗ | − 0 :=

√1 |(|0 − |1 ), 2 √1 |(|0 − |1 ), 2

√1 (|0 − |1 ) ⊗ |0 , 2 √1 (|0 − |1 ) ⊗ |1 . 2

| + 1 := | − 1 := 2. The tokens and the public serial numbers are stored in the quantum credit cards, and the bank gives the quantum credit cards to their clients. 3. Upon payment, the quantum credit card is inserted into the vendor’s terminals. The terminal provides an authentication of the credit card a projection measurement on these pairs in measurement basis requested by the bank and sends the classical result to the bank. 4. The bank knowing the specified encoded state checks the result and sends back to the terminal the decision to accept or deny the payment. This brings QM significantly closer to practical realization, but still thorough analysis of the revised scheme QM is required before it can be considered secure. We address this problem by presenting a proof-of-concept attack on QRG-based QM schemes, where one shows that even imperfect quantum cloning can, under some circumstances, provide enough information to break a QRG-based QM scheme; see [4], Figure 6:

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An honest implementation of the bank note’s state in the scheme is the same as in the original Wiesner’s scheme. However, because the source device is untrusted, in many cases an arbitrary state is of the form:  y=(y0i ,y1i )∈{0,1}2n

|y y|Bank s branch ⊗

1 n  i=1

2 ρy(i)

, Alice

(i)

where ρy are the arbitrary states of Alice’s qubits. Therefore, there are some difficulties with the quantum money problem.

5.2 Quantum Coin Coin problem has a long history and appeared as the money problems. There are a lot of researches in this domain, concentrating on the problem to provide some algorithm with high security. The classical bitcoin appeared in a work of the anonymous genius creator Shatoshi Nakamoto [5, 6]. He introduced a scheme for producing bitcoins. This scheme consists of schemes for: - transactions, - timestamp server, - proof-of-work, - network, - incentive, - payment verification, - privacy, and calculations. We refer the readers to these works for details. The idea of quantum coin was initialized but it was not yet effective, since it uses the quantum states immerged in the coin [7].

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The quantum key distribution method of Bennett and Brassard [IBM Tech. Discl. Bull. 28, 3153 (1985)] by exploiting a nonconjugate coding scheme was extended by showing that the original method of Bennett and Brassard gives optimal security [8]. Quantum economics and finance uses quantum mathematical theories to model phenomena including cognition, financial transactions, and the dynamics of money and credit through the quantum coin. Unlike a classical coin toss, which can be either heads or tails, a quantum coin can be in a superposition of states. And one uses this property to simulate everything like from the prisoner’s dilemma, to the credit relationship, to the pricing of options. It is explained in the book “Quantum Economics and Finance: An Applied Mathematics Introduction”, cf. [9]. A quantum money scheme can be private in the sense that only the bank can verify the money states, or public, meaning anyone can verify. In the work [10], the authors propose a way to lift any private quantum coin scheme, which is known to exist based on the existence of one-way functions, to a scheme that closely resembles a public quantum coin scheme. Verification of a new coin is done by comparing it to the coins the user already possesses, by using a projector on to the symmetric subspace. It is the first construction that is very close to a public quantum money scheme and is provably secure based on standard assumptions. The lifting technique when instantiated with the private quantum coins scheme gives rise to the first construction that is very close to an inefficient unconditionally secure public quantum money scheme. Let us first remind some basic notion related to classical coin (money) and the corresponding quantum ones [11]. The classical bitcoin protocols are well-known, and a classical blockchain is a random access ordered array with times stamped dictionary entries those can be used to solve some proof-of-work puzzle blocks. They can be added to the end of the chain. A classical distributed ledger scheme L is a classical algorithm AppendL that provides as an input a pair (s, kpub ) of a classical serial number s and a public key kpub . The algorithm gives an output fail if the inputted serial number is different from the serial number that is kept in the ledger and gives a pass output if the puzzle is solved at the time the ledger is added a new block. An algorithm LookupL is a polynomial time that corresponds the input s and the output kpub if it is found in the ledger. AppendL is applied continuously until it passes, if another miner solves a proof-of-work puzzle. A classical public signature scheme D consists of 3 pieces: 1. The key generator keyGenD that takes as input a secured number and a randomly chosen number to generate a pair (kprivate , kpublic ). 2. The sign procedure SignD that takes kprivate as input and a message M to generate code σ = SignD (kpublic , M). 3. The verification procedure VerifyD in quantum bitcoin QBC that lets to derive the kprivate from kpublic .

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Use random oracles to implement the following state generator G(r) and a serial number verifier H(s): The state generator G(r) that takes randomly n-bit string r and returns (sr , Ar ) of 3n-bit string sr and a set of linearly independent generators {x1 , . . . , xn/2 } for a subspace Ar ⊆ Fn2 . Required that the serial number are distinct for all r. The serial number verifier H(s) that takes as input a serial number and verifies, and it passes if the serial number is valid, s = sr for some Ar and fails otherwise. A hidden subspace mini-scheme M consists of two polynomials in time algorithms MintM and VerifyM : 1. MintM (n) takes as input a security number n and randomly generates a secret nqubit string r, which passes to the generator G(r), that returns the value (sr , Ar ) to produce a quantum bitcoin c = (sr , ρ = |Ar ). 2. VerifyM takes as input the alleged quantum bitcoin c = (sr , ρ) with classical serial number sr and quantum state ρ and performs: (a) Form check : Accept the quantum bitcoin c if it has the form (s, ρ) where s is a classical serial number and ρ is a quantum state. (b) Serial number check : accept the QBC if the verifier H(s) accepts. (c) Apply POVM : Apply the quantum Fourier transform over Fn2 : Apply VA := H2⊗n PA⊥ H2⊗n PA (ρ) and accept if VAr (ρ) = 0. The distributed quantum bitcoin Q as sequence of the following: Q = (KeyGenQ , MintQ , VerifyQ ), where: • KeyGenQ is a polynomial algorithm, which takes the input as a parameter n and randomly generated key pair (kprivate , kpublic ) consisting of a private key kprivate and a public key kpublic . • MintQ is an algorithm that takes a security parameter n and a private key kprivate and generates a quantum bitcoin $. This procedure consists of 6 steps: (1) Call the key generation procedure KeyGenQ to randomly generate a pair of keys (kprivate , kpublic ). (2) Generate quantum bitcoin candidate by calling MintM , which returns a pair (s, ρ) of a classical serial number s and a quantum state ρ. (3) Sign the serial number by σ = SignD (kprivate , s). (4) Call AppendL (s, kpublic ) to append the serial number s and the public key kpublic . (5) If the AppendL (s, kpublic ) fails, restart from the step (1). (6) If the serial number was successfully appended, put the serial number s, the quantum state ρ, and the signature σ together to create the quantum bitcoin $ = (s, ρ, σ ). • VerifyQ is a polynomial time algorithm, which takes as input an alleged quantum bitcoin c and a corresponding public key kpublic and a decision either to accept or reject, by using the previous mini-scheme M = (MintM , VerifyM )

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with a digital signature scheme D = (keyGenD , SignD , VerifyD ) and a distributed ledger scheme L = (AppendL , LookupL ). The verification algorithm VerifyQ consists of 4 points: (1) Check if the quantum bitcoin is of the form c = (s, ρ, σ ) with a classical serial number s, a quantum state ρ, and a classical digital signature σ . (2) Call the procedure LookupL (s) to retrieve the public key. (3) Call the procedure VerifyL (kpublic , s, σ ) to verify the digital signature of the quantum bitcoin. (4) Call the procedure VerifyL (s, ρ) from the mini-scheme and decide to pass if and only if the four steps pass. In order to avoid malicious miner to undermine the payment at any time the following proposal is supposed to do. Introduce a new security parameter m and a time maximum Tmax , and the situation becomes: 1. Quantum shard miner uses intuitive minting scheme to find out quantum shard s, ρ, σ ). 2. The quantum shard miners sell the quantum shards in a market. 3. Another quantum bitcoin miners percharge m quantum shards {(si , ρi , σi )}1≤i≤m , fulfill the conditions: VerifyM (s, ρi , σi ) accepts and the difference of the current time t and the timestamp T (of the quantum shards in the quantum shard ledger L) fulfills t − T ≤ Tmax . 4. The quantum bitcoin miner calls KeyGenL to randomly generate key pair (kprivate , kpublic ). 5. The quantum bitcoin miner takes the serial numbers of m quantum shards and compiles s = (s1 , . . . , sm ) and signs it as s0 = SignD (kprivate , s). 6. The quantum bitcoin miner takes m quantum shards and σ0 to produce (s1 , ρ, σ1 , . . . , sm , ρm , σm ). 7. The quantum bitcoin miner calls AppendL (s, kpublic ) to attempt to pair the quantum bitcoin miner’s public key kpublic with classified descriptor s in the ledger. This Append fails if and only if any of m quantum shards have already been combined into a quantum bitcoin existing in the ledger. In order to avoid the situation one supposes to modify the verification process VerifyQ as follows. Introduce a new security parameter λ > 0, and the verification passes if and only if at least (1 − ε − λ)m of the invocations of VerifyM pass and do the following steps: 1. Check that the quantum bitcoin c is of the form (s1 , ρ, σ1 , . . . , sm , ρm , σm ). 2. Call LookupL ((s1 , . . . , sm )) to retrieve kpublic of the quantum bitcoin miner associated to the classical descriptor (s1 , . . . , sm ). 3. Call VerifyD (kpublic , (s1 , . . . , sm ), σ0 ) to verify the digital signature of the quantum bitcoin. 4. For any i, 1 ≤ i ≤ m, call LookupL (si ) to retrieve the corresponding public key kpublic,i from the quantum shard miners.

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5. For each 1 ≤ i ≤ m, call VerifyD (kpublic,i , si , σi ) to verify the digital signature of each of the quantum shard. 6. For each 1 ≤ i ≤ m, call VerifyM (si , ρi , σi ) to verify the quantum bitcoin states. The authors of [10] proposed the following public quantum coin scheme Pk-QC: Aplgorithm (Public key quantum bitcoin Pk-QC) 1. procedure key-gen (∅, sk) ← Pr-QC.key-gen(λ) return (∅, sk) End procedure 2. procedure mint(sk) κ ≡ log(λ)c for some constant c |m ⊗κ ← ((Pr-QC.key-gen))⊗κ returm |C = |m ⊗κ end procedure 3. Init: ω ← mint(sk) 4. Verify(ρ) ω˜ = combined (maybe entangled) wallet state ω and new coin ρ measure ω˜ with two-outcome measurement {Symκ.(1+m) , I − Symκ.(1+m) } Denote the post measurement state by new wallet state ω if Outcome is Symκ.(1+m) then accept else reject end if end procedure 5. procedure Count|C ((ρ1 , . . . , ρm )) Set Counter ← 0 Run Init to initialize the wallet ω ← |C = mint(sk) for i = 1 to m do Run Verify(ρi ) if Verify(ρi ) = accept then Counter = Counter +1 end if end for Output Counter. end procedure

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6. procedure verifybank (sk, ρ) k ← Pr − QC.Count(sk, ρ) Accept with probability κk , reject with probability 1 − end procedure

D. N. Diep

k κ

7. procedure Countbank (sk, (ρ1 , . . . , ρm )) Set Counter ← 0 for i = 1 to m do Run verifybank (ρi ) if verifybank (ρi ) = accept then Counter = Counter +1 end if end for Output Counter end procedure

5.3 Quantum Blockchain and Post-Quantum Blockchain Blockchain appeared as some decentralized banking with security guaranteed from the blockchain protocols. We will explain the main points of the theory in this section. Blockchain and other distributed ledger technologies (DLTs) have evolved significantly in the last years, and their use has been suggested for numerous applications due to their ability to provide transparency, redundancy, and accountability. Blockchains are provided through public key cryptography and hash functions. However, the quantum computing has opened the possibility of performing attacks, namely based on Grover’s Search and Shor’s Prime Factoring algorithms by threatening both public key cryptography and hash functions, forcing to redesign blockchains to make use of cryptosystems that withstand quantum attacks, thus creating which are known as post-quantum, quantum-proof, quantum-safe, or quantumresistant cryptosystems. The most relevant post-quantum blockchain systems are studied, as well as their main challenges. Furthermore, extensive comparisons are provided on the characteristics and performance of the most promising postquantum public key encryption and digital signature schemes for blockchains; see [12]. Bitcoin is a digital currency and payment system based on classical cryptographic technologies, which works without a central administrator such as in traditional currencies. It has long been questioned what the impact of quantum computing would be on Bitcoin, and cryptocurrencies in general. Here, we analyze three primary directions that quantum computers might have an impact in: mining, security, and forks. The authors [13] find that in the near term the impact of quantum

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computers appears to be rather small for all three directions. The impact of quantum computers would require considerably larger number of qubits and breakthroughs in quantum algorithms to reverse existing hash functions. The authors [14] propose a conceptual design for a quantum blockchain. Our method involves encoding the blockchain into a temporal GHZ (Greenberger– Horne–Zeilinger) state of photons that do not simultaneously coexist. It is shown that the entanglement in time, as opposed to an entanglement in space, provides the crucial quantum advantage. All the subcomponents of this system have already been shown to be experimentally realized. Furthermore, our encoding procedure can be interpreted as nonclassically influencing the past. Cracking of digital signatures is the most imminent threat. A wrongdoer equipped with a quantum computer could use Shor’s algorithm to forge any digital signature, impersonate that user, and appropriate their digital assets. Most specialists think that this feat would require a universal quantum computer (one capable of performing a wide variety of calculations), and physics stipulates that quantum states cannot be copied or measured without being altered. Any eavesdropper will be immediately uncovered. Quantum cryptography can be used to replace classical digital signatures and to encrypt all peer-to-peer communications in the blockchain network. Our group has demonstrated such a simple system. However, the complexity and cost of quantum cryptography networks will limit their adoption. In particular, current protocols require that each node in the network be connected to every other through optical fiber channels, because there is no trust in any intermediary node and hence all communications must be direct. Protocols will be needed to maintain secure communications even when information flows through untrustworthy nodes; these systems have been developed but need to be made more accessible for consumers. Photon losses in optical fibers is another challenge. These limit the range of modern quantum key distribution systems to a few tens of kilometers. The solution is to develop a quantum repeater, which uses quantum teleportation and quantum optical memory to distribute entangled states between the communicating parties. Research is progressing, but it is a long way from delivering a practical device. In the interim, one-way functions should be tightened. Some alternative encryption functions have been proposed that should be equally difficult to reverse using conventional or quantum computers. Although not completely secure, these could be run on existing hardware and would buy time, but they, too, could be deciphered in the long term. The blockchain business needs to update its existing software to use oneway cryptographic functions that are equally hard to reverse using conventional or quantum computers. Until these post-quantum solutions are established or standardized, platforms must be flexible and capable of changing cryptographic algorithms on the fly. The longer-term answer is to develop and scale up the quantum communication network and, subsequently, the quantum internet. This will take major investments from governments. However, countries will benefit from the greater security offered; see [15].

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5.4 Quantum Cheques and Quantum E-Cheques We analyze the procedure providing quantum cheques of S. R. Moulick and P. K. Panigrahi [16] to produce quantum e-cheques, based on multiparty quantum telecommunication between customer and cooperated branches of bank. The problem of providing a quantum code of classical cheques is a central problem of the so-called quantum money problem. The question is to provide a scheme of quantum code in such a way that it should be similar to the classical ones but with absolute high secrecy. In the work [16], the authors gave an adequate survey of development of the problem and constructed a scheme for quantum cheques. The scheme covered the classical version of cheques: The quantum cheque will use the schemes of the form  = (Gen, Sign, Verify).

5.4.1 Quantum Cheques Some customer Alice and bank make initialization by the Gen Scheme Namely, Alice came to some bank branch to open an account with secret key as a binary L-digit number k ∈ {0, 1}L to provide an electronic signature in the future, by using some secret key generation scheme for Alice and bank. The bank later gives her a cheque book serial number s. For secrecy, Alice produces some public key pk and stores a secret key sk. The bank produces 3 entangled qubits in GHZ states  1

|φ (i GH Z = √ |0(i) A1 |0(i) A2 |0(i) B + |1(i) A1 |1(i) A2 |1(i) B , 1 ≤ i ≤ n 2 and send two of them, namely |φ A1 and |φ A2 to Alice. Therefore Alice holds (id, pk, sk, k, s, {|φ (i) A1 , φ (i) A2 }) and the bank branch holds (id, pk, sk, k, s, {|φ (i) B }). The next step is the sign scheme. Sign Scheme Alice chooses a random number r using a random number generation procedure r ← U{0,1}L , a numeration i = 1, . . . , n of orthogonal base |φ (i) , and certainly an amount M she likes to make some transaction with bank (debit or credit) and then evaluates the one-way function f : {0, 1}∗ × |0 → |ψ (i)

at the concatenation x||y of the data as k||id||r||M||i to provide a state ψ (i) = αi |0 +βi |1 . Alice encodes the data |ψ (i) with the |φ (i) A1 , making them entangled and measuring the Bell states: 1 |φ ± = √ (|00 ± |11 ) , 2

1 |ψ ± = √ (|10 ± |01 ) . 2

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The system is in the states of form: |φ (i) = |ψ (i) ⊗ |φ GH Z = 13 + |φ A1 (αi |00 A2 B + βi |11 A2 B ) + |φ − A1 (αi |00 A2 B − βi |11 A2 B ) 2 4 +|ψ + A1 (αi |00 A2 B + βi |11 A2 B ) + |ψ − A1 (αi |00 A2 B − βi |11 A2 B ) . Then Alice performs Pauli transforms |φ + → I =

10 , 01

01 , |ψ → σX = 10 +

|φ − → σZ =



1 0 0 −1



0 −i |ψ → σY = i 0 −

and makes correction to |φ (i) A2 . Alice makes signature by using the procedure (i) signpk (s) and produces the quantum cheque χ = (id, s, r, σ, M, {|φA2 }) and then publicly send through Abby to an arbitrary of the valid branches of the bank. The final step is the verification. Verify Scheme A valid bank branch after receiving the cheque informs to the main branch in order to check the signature Vf ry(σ, s). For this, one uses, namely, the well-known Fredkin gate ([16], Picture 1). If the (id, s) or σ is invalid, the bank destroys the cheque; otherwise, the bank continues the measurement in Hadamard basis |φ B . If the result is |φ + or |φ − , the main branch communicates with the acting branch to continue. The acting branch performs transformation |φ + → I and |φ − → σZ . The bank accepts the cheque if it passes the swap test and then destroys it. The schemes are summarized as in Fig. 5.1 of [16]: We remark that the quantum cheque is produced and used quite similar to the classical one. We propose therefore to use the multipartite quantum key distribution to make quantum cheques become quantum e-cheques of high secrecy [17]. Our main result is Theorem 5.4.2 stating that a code for quantum e-cheques can be

Fig. 5.1 Quantum cheques transferring code

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provided with high secrecy by the multipartite public key distribution of quantum share. The feature of our approach is that (i) Alice does not need to go to a bank branch to do a transaction but divides her data to a disjoint union of parts and connects with acting branches to send to each one part of her data, (ii) the bank can record the Alice’s data only if all acting branches cooperate together, and therefore (iii) they defense the origin of data and Alice prevents some dishonest branches to change the data. We separately consider the problem of e-cheque transferring in the situation of absolute secured channel in order to point out the main idea of e-cheque transaction. The more complicated problem of e-cheque transferring in the presence of eavesdroppers in a nonsecured channel or dishonest participants will be separately considered in a subsequent section.

5.4.2 Quantum e-Cheques as Multiparty Quantum Secrete Sharing Consider the following modified problem for the situation when Alice does not send the quantum cheques via Abby but could online connect with acting branches of a bank. To prevent the fact that some distrusted branches could change the cheque. The bank could discover the information from the quantum cheques only if all acting branches cooperate together and in that case the other branches prevent some untrusted branches to change the contents of the cheque. The quantum cheques in that case are what we call e-cheques. Solution to this problem is the following code. After the first step Gen scheme, in the second step Sign Scheme One keeps the same as in the previous section, only now, Alice divides the provided concatenated information k||id||r||M||i into n parts, (i) (i) D1 , . . . , Dn , where n is an appropriate number of branches in action. Then she produces the corresponding states by using a one-way function f to have f (Dj(i) ) = ψj(i) , for all j = 1, . . . , n. Following the multiparty secret sharing, when the branches cooperate together and inform to the main branch, one discovers the (i) states |φj . Theorem 5.4.1 The quantum cheques could be with higher secrecy electronically transferred from Alice to the acting bank branches by a code of multiparty quantum telecommunication problem of secret sharing with quantum public key distribution. Proof The theorem is proved by the following procedure, which is similar to the one in the 3-person case by Cabello [18], following which the system states are changing as follows: |ψi  −−−−→ |ψii  −−−−→ |ψiii  −−−−→ |ψiv 

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Let us consider it in more detail. Transfer Step 1. Initialization of 3n Qubits For a fixed i, Alice uses n + 2 qubits, (i) (i) named: 1, 2, 3, D1 = D1 , . . . , Dn−1 = Dn−1 : qubits 1 and 2 are entangled in a Bell state and qubit 3, D1 , . . . , Dn−1 are entangled in GHZ state with n − 1 acting bank branches: Branch 1, Branch 2, . . .., Branch n-1, where each has 2 entangled qubits i + 3, Ci , i = 1 . . . , n − 1, namely in null state. Alice produces a Bell state measurement on qubits 1 and 2 and a Fourier measurement Fn on n qubits 3, D1 , . . . , Dn−1 . Each of acting branches makes a Bell state Fourier measurement F2 of entangled i + 3, Ci , i = 1 . . . n − 1. At the end of this step 1, the system is in the state |ψi = |0 . . . 0 3D1 ...Dn−1 ⊗ |00 12 ⊗ |00 4C1 ⊗ · · · ⊗ |00 n+2,Cn−1 . Transfer Step 2. Entangled Bell State Measurements Alice sends each qubit Di of her GHZ state out to each i th acting bank branch of the other n − 1 branches. The system is in the state |ψii = |AP 3D1 ...Dn−1 ⊗ |BP 1C1 ⊗ |CP 2C2 ⊗ · · · ⊗ |NP n+2,Cn−1 . Transfer Step 3. Secret Bell State Measurements Next, Alice and each user perform a Bell state Fourier measurement F2 on the received qubit and one of their qubits. After these measurements, the state of the system becomes |ψiii = |AP 3D1 ...Dn−1 ⊗ |AS 2,3 ⊗ |BS 4,D1 ⊗ · · · ⊗ |NS n+2,Dn−1 , where |AP is n-qubit GHZ state of the standard orthonormal basis. Transfer Step 4. Secret Sharing The n−1 acting branches send a qubit (the one they have not used) to Alice, and she performs a Fourier measurement Fn to discriminate between the 2n GHZ states and publicly announces the result |AP 1C1 ...Cn−1 . After these measurements, the state of the system becomes |ψiv = |AP 1C1 ...Cn−1 ⊗ |AS 2,3 ⊗ |BS 4,D1 ⊗ · · · ⊗ |NS n+2,Dn−1 . The result AP and the result of their own secret measurement allow each legitimate acting branch to infer the first bit of Alice’s secret result AS. To find out the second bit of Alice’s secret AS, all users (except Alice) must cooperate. In case n = 3 there is an illustration of Cabello [18] as in Fig. 5.2. The proof therefore is achieved.  The 4-step scheme of public key secret sharing distribution can be generalized to the case of two levels grouped secret sharing as illustrated in the work of A. Jaffe et al. [19] in Fig. 5.3:

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Fig. 5.2 Quantum cheques transfer code with n = 3

After discovering the e-cheque, bank continues to proceed with the same procedure Verify Scheme as in the quantum cheques scheme above to verify the validity of the e-cheque and accept or destroy it.

5.4.3 Transferring Quantum E-Cheques in Nonsecured Channels In the previous part, we considered the scheme of transferring quantum e-cheques such that the bank branches should cooperate together in order to prevent the dishonest branches from changing the information, but we supposed that we could transfer in the absolutely secured channels. Let us now consider the scenario where the transferring channels are unsecured and there may appear some eavesdroppers. Therefore Alice should check the channels before sending information of cheques.

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Fig. 5.3 n-Party-sharing problem and corresponding BVK code

Theorem 5.4.2 The quantum cheques could be with high secrecy electronically transferred from Alice to the acting bank branches by a code of multiparty quantum telecommunication problem of secret sharing with quantum public key distribution in unsecured channels possibly with eavesdroppers.

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Proof The theorem is proved by the following procedure: |ψi  −−−−→ |ψii  −−−−→ |ψiii  −−−−→ |ψiv 

Let us consider it in more detail. Transfer Step 1. Checking Channels and Initialization Before sending the information, Alice should check security of the channels. She uses 2(n − 1)L qubits A A {qiA1 , . . . qi n−1 , piA1 , . . . pi n−1 }, i = 1, . . . , L to check the channels, where each Aj ˜ = √1 (|00 + |11 ), |1

˜ = q is randomly in one of the four states |0 , |1 , |0

i √1 (|01

2

2

Aj

+ |10 ) and each pi

˜ |1 . ˜ For is randomly in one of the two states |0 , A

A

each j = 1, . . . , n − 1, Alice sends the qubits Aj , qi j , pi j to bank branch Aj in a secretly random manner, i.e., the order, that Alice should keep for later use. After Branch 1, . . ., Branch (n-1) confirm that they have received the qubits, A Alice reveals the position of the checking qubits {qi j } and asks every participant Branch 1, . . ., Branch (n-1) to measure them in the appropriate bases (i.e., in those they have been prepared by Alice) and then announce her their results. Through a careful statistical analysis of the measurement outcomes for the checking qubits A {qi j }, Alice is able to assess the error rate of secure sharing of the quantum channel. If it exceeds a predetermined threshold, she decides to abort the scheme. Otherwise, she proceeds to the next step; see [19] for more details. (i) For a fixed i, Alice uses n + 2 qubits, named: 1, 2, 3, D1 = D1 , . . . , Dn−1 = (i) Dn−1 : qubits 1 and 2 are entangled in a Bell state, qubit 3, D1 , . . . , Dn−1 is entangled in GHZ state with n − 1 acting bank branches: Branch 1, Branch 2, . . .., Branch n-1, where each has 2 entangled qubits i + 3, Ci , i = 1 . . . , n − 1, namely in null state. Alice produces a Bell state measurement on qubits 1 and 2 and a Fourier measurement Fn on n qubits 3, D1 , . . . , Dn−1 . Each of the acting branches makes a Bell state Fourier measurement F2 of entangled i + 3, Ci , i = 1 . . . n − 1. At the end of this step 1, the system is in the state |ψi = |0 . . . 0 3D1 ...Dn−1 ⊗ |00 12 ⊗ |00 4C1 ⊗ · · · ⊗ |00 n+2,Cn−1 . Transfer Step 2. Entangled Bell State Measurements Alice sends each qubit Di of her GHZ state out to each i th acting bank branch of the other n − 1 branches. The system is in the state |ψii = |AP 3D1 ...Dn−1 ⊗ |BP 1C1 ⊗ |CP 2C2 ⊗ · · · ⊗ |NP n+2,Cn−1 . Transfer Step 3. Secret Bell State Measurements Next, Alice and each user perform a Bell state Fourier measurement F2 on the received qubit and one of their qubits. After these measurements, the state of the system becomes

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|ψiii = |AP 3D1 ...Dn−1 ⊗ |AS 2,3 ⊗ |BS 4,D1 ⊗ · · · ⊗ |NS n+2,Dn−1 , where |AP is n-qubit GHZ state of the standard orthonormal basis. Transfer Step 4. Checking Security and Secret Sharing Alice appoints one of the branches, namely the Branch 1, as the leader branch and she reveals Branch 1 the position of C1 , C2 . Before the branches 1, . . ., n-1 sending the qubits to Alice, she do again checks the channels. She asks branches to measure the remaining entangled A qubits 3 + j, pi j in Bell states and publish the results. If Alice discovers that the error rate exited a predetermined threshold, she discards the transferring cheque process. Otherwise, she knows that branches are not cheated and then asks leader branch 1 to make the state |φi1 ,j1 ,...,in−1 ,jn−1 C1 ,C2 = ξi1 ,j1 ,...,in−1 ,jn−1 |00 + ζi1 ,j1 ,...,in−1 ,jn−1 |01 + ηi1 ,j1 ,...,in−1 ,jn−1 |10 + γi1 ,j1 ,...,in−1 ,jn−1 |11 , the coefficients are known from measurement public results; see [16] for more details. The n − 1 acting branches send a qubit (the one they have not used) to Alice, and she performs a Fourier measurement Fn to discriminate between the 2n GHZ states and publicly announces the result |AP 1C1 ...Cn−1 . After these measurements, the state of the system becomes |ψiv = |AP 1C1 ...Cn−1 ⊗ |AS 2,3 ⊗ |BS 4,D1 ⊗ · · · ⊗ |NS n+2,Dn−1 . The result AP and the result of their own secret measurement allow each legitimate acting branch to infer the first bit of Alice’s secret result AS. To find out the second bit of Alice’s secret AS, all users (except Alice) must cooperate. The proof therefore is achieved. 

5.4.4 Conclusion We show [17] that the quantum cheques can be electronically transferred with higher secrecy by a code of multiparty quantum telecommunication problem of secret sharing with quantum public key distribution. We show that the quantum cheques can be electronically transferred with high secrecy by a code of multiparty quantum telecommunication problem of secret sharing with quantum public key distribution in nonsecured channel [20].

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5.5 Quantum Risk Management The risk management is an important problem of finance. We analyze the main points of the theory in this section. In the classical decision theory the decision maker chooses between several lotteries of gambles each of them consisting of finite set of outcomes with some objective probability distribution by von Neumann and Morgenstern theory [21]. The main difficulty is that there are a lot of packages of outcomes with unknown probability distribution. The decision is often difficult to make. Savage [22] generalized the theory to the case of subjective probabilities, which has been demonstrated the major flexible but making risk and uncertainty. The classical decision theory had many successful applications but also lead to enormous paradoxes. To overcome these paradoxes, one proposed some theories like cumulative-prospect theory or reference-point theory when the decision making is based on relative reference point. Nevertheless the reference point is not exactly defined. The basic difficulty is to evaluate the risk, when deciding under uncertainty. Risk is always related to emotions and the question is how one could describe emotions within a quantitative framework suitable for decision making? One constructed quantum algorithms by using quantum amplitude estimation to price securities to evaluate risk measures such as Value at Risk and Conditional Value at Risk on a gate-based quantum computer and then implemented these algorithms and hence how to trade-off the convergence rate of the algorithms and the circuit depth; see [23]

5.6 Forecasting of Time Series One proposes a quantum learning scheme approach for time series forecasting, through the application of the new non-standard qubit neural network (QNN) model. The QNN description was adapted in this work in order to resemble classical artificial neural networks (ANNs). Three stock market series were predicted. The results are discussed over several statistics and are compared with ANN experiments with equivalent degrees of freedom; see [24, 25]. Quantum computation, quantum information, and artificial intelligence have all contributed for the new non-standard learning scheme named qubit neural network (QNN). Some QNN based on the qubit neuron model is used for real-world time series forecasting problem, where one chaotic series and one stock market series were predicted. Experimental results show evidences that the simulated system is able to preserve the relative phase information of neurons’ quantum states and, thus, automatically adjust the forecast’s time shift; see [26]. The most important problem in finance is forecasting of time series. With quantum technology, one can give some good prediction. We intend to explain some aspects of this problem in this section.

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5.7 Portfolio Creation When creating a portfolio, the security problem is the most important one. In this section we concern with some aspects of this events. “Quantum Portfolio Management is a provider of an investment management service specifically designed for recipients of large awards of damages for personal injury and clinical negligence. One develops and maintains close relationships with clients over the long term, offering the level of service many organisations find it difficult to sustain. In order to provide the best possible level of service and advice, our services are focused on the needs of personal injury clients and are founded on years of experience with special needs clients who require bespoke advice.” One presents a quantum algorithm for portfolio optimization and discusses the market data input, the processing of such data via quantum operations, and the output of financially relevant results. Given quantum access to the historical record of returns, the algorithm determines the optimal risk–return tradeoff curve and allows one to sample from the optimal portfolio. The algorithm can in principle attain a run time of poly(log(N)), where N is the size of the historical return data set. Direct classical algorithms for determining the risk–return curve and other properties of the optimal portfolio take time poly(N), and we discuss potential quantum speedups in light of the recent works on efficient classical sampling approaches; cf. [27]. The modern portfolio theory (MPT) [28] can be formalized as follows. Let N be the number of assets, μi be the expected return of asset i, σij be the covariance between returns for assets i and j , R be the target portfolio return, and wi ∈ R are the weights of decision variables, i.e., the investment associated with asset i. The standard Markowitz mean–variance approach is the quadratic optimization problem: N N  

wi wj σij → min,

i=1 j =1

subject to the constraints N  i=1

wi = 1 and

N 

wi μi = R.

i=1

This quadratic optimization problem can be solved by the quadratic programming with linear constraints. If the covariance matrix Σ = (σij ) is positive definite, the problem can be transformed, namely by using the Lagrangian cofactor method, into a unconstrained quadratic optimization problem (QUBO), which is a suitable formulation for quantum annealers. In the last few years some proof-of-principle papers, performing runs on the first generation D-Wave machines solving the discrete portfolio optimization problem, which is very hard and known as NP-

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complete. The paper [28] is devoted to an approach to portfolio optimization by extending the mean–variance model to a general dependence structure and allows portfolio managers to express discretionary views on relative attractiveness of different assets and their combinations. The choice of M funds without replacement from the universe of N funds is done by putting qi = 1 if the asset i is selected and qi = 0 if the asset i is not selected. Then the task of encoding the relationship among M selected funds can be formulated as a quadratic form: QU BO :

N 

O(q) =

ai qi +

N N  

bij qi qj → min .

i=1 j =i+1

i=1

The optimal solutions optimize the risk-adjusted returns by the use of metrics of Sharpe ratio (r − r0 )/σ where σ is asset’s volatility (annualized standard deviation of the asset’s long-returns). The QUBO problem is translated into a corresponding Ising problem of N variables si , i = 1, . . . , N with si ∈ {+1, −1} by the Hamiltonian operator OIsing (s) =

N 

hi si +

N N  

Jij si sj ,

i=1 j =i+1

i=1

 Ji = 14 bij and hi = ( 12 ai + j bij ). The problem is to find the ground state and evaluate the minimum energy level. One uses therefore the adiabatic programming to solve the problem of finding the annealing schedule for the time-dependent Hamiltonian HQA (t) = A[t]

&N N c   i=1

' σicx + B[t]Hχ −Ising

c=1

with the Chimera graph where the qubits are arranged in ordered 1D chains interlaced ⎡ ⎤ 1N 2 N N N c 1 −1    h i z z z |JF | ⎣ σic σi(c+1 ⎦ + σic Hχ −Ising = − Nc i=1

+

N  i,j =1

i=1

Jij

Nc  ci ,cj =1

i=1

c=1

χ

δij (ci , cj )σicz i σjzcj ,

where σicz are Ising spins ferromagnetically coupled directly by string JF forming an ordered 1D chain subgraph of χ , the value of JF should be strong enough to correlate the value of the magnetization of each individual spin if measured in the

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computation basis σicz = σicz  , and each coupler Jij is active only between one specific pair of qubits σicz ∗ , σjzc∗ , which is specified by the adjacency check function i

j

δij (ci , cj ), which is 1 only for the case ci = ci∗ , cj = cj∗ and is 0 otherwise. The authors introduce also the following algorithms to compute the optimal solutions. χ

Algorithm 1 Greedy search heuristic 1. for i from 0 to (number of qubits) N − 1 do # Initialize energy tuples with magnetic field h energy[i] = {−|h(i)|, h(i), i} end for 2. Energies = heap(energy) # Reorder energy tuples into a heap according to absolute magnitude 3. While Energies do # Initialize solutions {x, e, i} = heappop(Energies) # Energy tuple elements (largest magnitude energy tuple first) if e > 0 then Solution[i] = −1 else Solution[i] = +1 end if for z in Energies do # update the rest of heap n = z[2] z[1] = z[1] + Solutioni] ∗ (J (i, n) + J (n, i)) end for end while

In the case of portfolio optimization problem the solution (chromosome) is a binary vector q = (a1 , . . . , qN ) of N = 2n genes ai ∈ {0, 1}, and we do find a combination of genes that minimizes the objective fitness function O(q). The following algorithm solves the problem. Algorithm 2 Genetic algorithm: unconstrained portfolio optimization 1. Generation of L initial solutions by populating the chromosomes through the random draw from the pool of possible gene values {0, 1}. 2. Evaluation of the objective fitness function for each solution. 3. Ranking of solutions from ‘best’ to ‘worst’ according the objective function evaluation results. 4. for i from 0 to (number of iterations) N − 1 do - Selection of K best solutions from the previous generation and production of L new solutions by randomly changing the values of one or more genes. - With L = mK every one of the ‘best’ solutions will be used to produce m new solutions. - Evaluation of the objective fitness function for each solution. - Ranking of solutions from ‘best to ‘worst’ according to the objective function evaluation results. end for

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We finish this chapter by illustrating some quantum algorithms for quadratic speedup of the portfolio problem in risk analysis (see [23]): a quantum algorithm to estimate risk, e.g., for portfolios of financial assets, resulting in a quadratic speedup compared to classical Monte Carlo methods. The algorithm has been demonstrated on real hardware for a small model and the scalability and impact of noise has been studied using a more complex model and simulation. The approach is very flexible, and simulations of the two-asset portfolio show that circuit depth is limited for current hardware. In order to perform the calculation of VaR for the two-asset portfolio on real quantum hardware it is likely that qubit coherence times will have to be increased by several orders of magnitude and that cross-talk will have to be further suppressed. The quantum amplitude estimation algorithm AE. The formally introduced algorithm in Supplementary Information allows us to estimate the amplitude a in the state A|0 n+1 =

√ √ 1 − a|0 n + a|1 n

for the operator A acting n + 1 qubits through the states of n qubits. One uses also additional m sampling qubits to produce an estimator a¯ = sin2 ( yπ M ) of a, satisfying the inequality |a − a| ¯ ≤



π2 π + 2 = O M −1 M M

with probability of at least 8/π 2 . Therefore this AE algorithm has a quadratic speedup in comparison with the classical Monte Carlo method O(M −1/2 ). Let X be a random variable. By using n qubits, the states of X can be presented in the form: |ψ n =

n −1 2

2 −1 √ pi = 1, pi |i n , with n

i=0

i=0

where pi = P(X = i) is the probability of measuring the state |i of i ∈ {0, . . . , 2n − 1}. For a function f : {0, . . . , 2n − 1} → [0, 1], we may produce the operator Ff : |i n |0 $→ |i n

$  $ 1 − f (i)|0 + f (i)|1 .

Therefore, Ff |ψ |0 ancilla =

n −1 2 $

i=0

√

1 − f (i) pi |i n |0 +

n −1 2 $

√ f (i) pi |1 .

i=0

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Using a bisection search over I , one can find Ia such that P[X ≤ Ia ] ≥ 1 − a in at most n steps. The smallest Ia by definition is VaRn (X). This estimation of VaRn (X) has a quadratic speed-up accuracy O(M −1 ) compared with the classical Monte Carlo method accuracy O(M −1/2 ). The conditional expectation CVaRn (X) = E[X|Ia = VaRn (X)] can be estimated by using the operator Ff with f (i) = Iia fIa (i) on |ψ n |0 , ⎛ Ff |ψ n = ⎝

n −1 2

√ pi |i n +

i=Ia +1

Ia 

*

i=0

⎞

a  i √ 1− pi |i n ⎠ |0 + Ia

I

*

i=0

i √ pi |i n |1 . Ia

Using AE approximation, the probability of measuring |1 for the ancilla is Ia i i=0 Ia pi . Normalizing the probability of measuring |1 , we have a  i Ia pi . CVaRn (X) = P[X ≤ Ia ] Ia

I

i=0

With the same success, we can estimate CVaRn ( IXa ). However, we are usually interested in E[f (X)] for a more general function f : {0, . . . , 2n − 1} →  {0, . . . , 2n − 1}, representing some payoff or loss depending on X. In some cases, we can adjust Ff accordingly. The AE algorithm approximates the probability of measuring |1 in the last qubit, and keeping the n previous qubits unchanged, we have n −1 2

f (i)pi = E(f (X)).

i=0

Choosing f (i) =

i 2n −1 ,

one estimates the expectation E( 2nX−1 ) and therefore the

expectation E[X]. Choosing f (i) =

i2 , (2n −1)2

one can estimate E[X2 ] and then

Var(X) = E[X2 ] − E[X]2 . The technique can be extended to calculate more. Let us remind that for a given confidence level a ∈ [0, 1], VaRn (X) is the smallest value x ∈ {0, . . . , 2n − 1} such that the probability P[X ≤ x] ≥ 1 − a. For any fixed I ∈ {0, . . . , 2n − 1}, one chooses a function fI (i) = 1 if i ≤ I and 0 otherwise. Therefore, FfI |ψ n |0 =

n −1 2

i=I +1

√

pi |i n |0 +

I  √ i=0

pi |i n |1 .

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T-Bill. The model consists of a zero coupon bond discounted at an interest rate r. To find the bond value today given that in the next time step there might be a δr rise in r. The face value VF , which the holder receives when the bond matures is V =

ρVF (1 − ρ)VF + = (1 − ρ)Vlows + ρVhigh , 1 + r + δr 1+r

where ρ and 1 − ρ denote the probability of a constant interest rate and a rise. It is possible to count the value of the investor’s T-bill by running AE algorithm. Two-asset portfolio example. One of the most important tasks is to count what is the current total asset, i.e., the portfolio state, if one produces some bills, some amount of transactions. Let us illustrate how to use the AE algorithm to calculate the daily risk in a portfolio made up of 1-year US Treasury bills and 2-year US Treasury notes with face values VF1 and VF2 , respectively. Choose a simple portfolio in order to put the focus on the AE algorithm applied to VaR. The portfolio is worth  rc VF VF1 VF2 2 + + , 1 + r1 (1 + r2 /2)i (1 + r2 /2)4 4

V (r1 , r2 ) =

i=1

where rc is the annual coupon rate, which is equal to the annual payment for bondholder divided by the face value of the bond, paid every 6 month by the 2year treasury note and r1 and r2 are the rates yield to maturity of the 1-year bill and 2-year bill notes, respectively. Again, it is possible to count the current asset value of the investor by running the above amplitude estimation AE algorithm.

References 1. Wiesner, S. (1983). Conjugate coding. SIGACT News, 15(1), 78–88. https://doi.org/10.1145/ 1008908.1008920. ISSN 0163-5700 2. Horodecki, K., & Stankiewicz, M. (2020). Semi-device-independent quantum money. New Journal of Physics, 22, 023007. https://doi.org/10.1088/1367-2630/ab6872. 3. Bozzio, M., et al. (2018). Experimental investigation of practical unforgeable quantum money. NPJ Quantum Information, 4. ˇ 4. Jiráková, K., Bartkiewicz, K., Cernoch, A., et al. (2019). Experimentally attacking quantum money schemes based on quantum retrieval games. Science Reports, 9, 16318. https://doi.org/ 10.1038/s41598-019-51953-9. 5. Nakamoto, S. (2008). Bitcoin: A peer-to-peer electronic cash system. Consulted, 1(2012), 28. 6. Shatoshi Nakamoto-Wikipedia.html 7. Myers, E. C. (2012). Quantum coin. Kindle Edition. 8. Phoenix, S. J. D. (1993). Quantum cryptography without conjugate coding. Physical Review A, 48, 96. https://doi.org/10.1103/PhysRevA.48.96. 9. Orrell, D. (2020). Quantum economics and finance: The quantum coin trick. CQF Institute, Speaker. 10. Behera, A., & Sattath, O. (2020). Almost public quantum coins. arXiv: 2002.12438v2 [quantph].

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11. Jogenfos, J. (2016). Quantum bitcoin: an anonymous and distributed currency secured by the no-cloning theorem of quantum mechanics. ArXiv: 1604.01383v1[quant-ph]. 12. Fernández-Caramès, T. M., & Fraga-Lamas, P. (2020). Towards post-quantum blockchain: a review on blockchain cryptography resistant to quantum computing attacks. In IEEE access (vol. 8, pp. 21091–21116). https://doi.org/10.1109/ACCESS.2020.2968985. 13. Tessler, L., & Byrnes, T. (2017). Bitcoin and quantum computing. arXiv:1711.04235[quantph]. 14. Rajan, D., & Visser, M. (2019). Quantum blockchain using entanglement in time. Quantum Reports, 1, 2. https://doi.org/10.3390/quantum1010002. www.mdpi.com/journal/quantumrep. 15. Fedorov, A. K., Kiktenko, E. O., & Lvovsky, A. I. (2018). Quantum computers put blockchain security at risk. Nature, 563, 465. 16. Moulick, S. R., & Panigrahi, P. K. (2016). Quantum cheques. Quantum Information Processing, 15, 2475–2486. 17. Diep, D. N. Multiparty quantum telecommunication using quantum Fourier transforms. arXiv: 1705.02608[quant-ph]. 18. Cabello, A. (2000). Multiparty key distribution and secret sharing based on entanglement swapping. arXiv:quant-ph/0009025v1. 19. Jaffe, A., Liu, Z.-W., & Wozniakowsk, A. (2016). Holographic software for quantum networks. Science China Mathematics, 61(4). https://doi.org/10.1007/s11425-017-9207-3 20. Diep, D. N. Quantum e-cheques in nonsecured channels. arXiv: 1705.10083 [quant-ph]. 21. von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior. Princeton: Princeton University. 22. Savage, L. (1954). The foundations of statistics. New York: Wiley. 23. Woerner, S., & Egge, D. J. (2019). Quantum risk analysis. NPJ Quantum Information, 5, 15. https://doi.org/10.1038/s41534-019-0130-6. 24. Azevedo, C. R. B., & Ferreira, T. A. E. (2007). Time series forecasting with qubit neural networks. In Conference: ASC ’07 Proceedings of The Eleventh IASTED International. 25. Azevedo, C. R. B., & Ferreira, T. A. E. (2007). The application of qubit neural networks for time series forecasting with automatic phase adjustment mechanism. In Proceedings of XXVII Congress of the Brazilian Computer Science Society (VI National Meeting of Artificial Intelligence). 26. Azevedo, C. R. B., & Ferreira, T. A. E. (2007). The Application of Qubit Neural Networks for Time Series Forecasting with Automatic Phase Adjustment Mechanism. Conference: Encontro Nacional de Inteligência Artificial (ENIA). 27. Rebentrost, P., & Lloyd, S. (2018). Quantum computational finance: quantum algorithm for portfolio optimization, arXiv:1811.03975v1[quant-ph]. 28. Venturelli, D., & Kondratyev, A. (2018). Reverse quantum annealing approach to portfolio optimization problems, arXiv: 1810.08584v2[quant-ph].

Chapter 6

Future Perspectives of Quantum Applications Using AI H. U. Leena and R. Lawrance

6.1 Introduction Quantum computing is trending with prospective of future computing for communication and security. The future applications of quantum computing is connected to cybersecurity, financial modeling, cleaner fertilization, artificial intelligence, drug development, better batteries, weather forecasting, climate change, and traffic optimization. The quantum internet and quantum security are the two prime features that revolutionize the computing world. The current quantum communication and continued research work on it pave the pathway for highly secured quantum key distribution that powers the security feature. Quantum key distribution is the new trend for secured communication that involves QKD protocols for sending information securely; this novel communication is entirely different from the classical public key exchange method. The big contenders, research organization, and government bodies are pitching heavily on QKD networks to secure the data communication. In this chapter, we discuss the prime quantum inventions such as quantum internet, quantum security, quantum materials, and quantum sensing using the machine learning and artificial intelligence models.

H. U. Leena () XIPHIAS Software Technologies Pvt. Ltd., Bangalore, Karnataka, India e-mail: [email protected] R. Lawrance Department of Computer Applications, ANJA College, Sivakasi, Tamil Nadu, India © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Iyengar et al. (eds.), Quantum Computing Environments, https://doi.org/10.1007/978-3-030-89746-8_6

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6.2 Quantum Internet and Satellite and Underwater Communication Using Teleportation Techniques The future prospective of the internet depends on a world-shattering new technology, consenting a strong and secured network to transmit information faster ever before just matching the speed of light. This ground-breaking quantum internet will surpass all the current hurdles and interfaces all global network and will be rapid enough to send and receive information through the quantum bits known as qubits by adopting the governance of quantum mechanics. The prime aspect of this futuristic research is to power the quantum communication across any sender and receiver on the earth with the speed of light. Recently, to achieve this, a deep-seated new technology is introduced, that is, a potential of game-changing in quantum applications and services. Specifically, one such is on principally protecting the communication path naturally through laws of physics which is powerful at the moment. In addition, the quantum processors are interfaced to build quantum grid network in order to build quantum clusters for large computing. This method is known as quantum cluster computing that proposes reliability, secured, and scalability. The amalgamation of quantum clustered network with quantum internet enables users to access the services remotely just like the way as classical cloud known as “quantum cloud.” A research by Mario Mastriani et al. [1–3] presents a satellite alternative to quantum repeaters based on the terrestrial laid of optical fiber, where the latter have the following disadvantages: a propagation speed (v) equal to 2/3 of the speed of light (c), losses and an attenuation in the material that requires the installation of a repeater every 50 km, while satellite repeaters can cover greater distances at a speed v = c, with less attenuation and losses than in the case of optical fiber except for relative environmental aspects to the ground–sky link, that is, clouds that can disrupt the distribution of entangled photons. Two configurations are presented, the first one of a satellite and the second one of two satellites in the event that both the points on the ground cannot access the same satellite. Finally, a series of implementations for evaluating the performance and robustness of both the configurations are implemented on a 5 qubits IBM Q processor. The research work [2] presents a protocol for a relay configuration of one quantum CubeSat and two quantum drones positioned at distant places over the earth, where: (a) an entangled pair is generated and distributed by the CubeSat between both the drones located over the clouds, (b) each drone descends through the clouds with its respective entangled photon, and (c) each drone generates a new entangled photon pair, conserves one, and distributes the other one to mobile ground stations (MGS) (Fig. 6.1). These latter photons allow to teleport both the CubeSat’s entangled photons to the MGSs. Once on earth, the CubeSat’s entangled photons constitute a bridge for the teleportation of an arbitrary qubit among the MGSs. In this way, we solve the main problem of all quantum communication between a satellite and the earth: the weather as well as unfavorable environmental conditions. To conclude, this chapter compares the protocol performance by teleporting the Cubesat’s entangled photons

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Fig. 6.1 Quantum teleportation for an arbitrary satellite

on IBM Q experience using a 16-qubit Melbourne Processor. This comparison is considered to be the first stage and is specifically performed on communication between two distant points on the earth at any time regardless of weather.

6.3 Countering Cyberattacks on Quantum Key Distribution Using ML Models on Security, Privacy, and Cryptography Quantum key distribution (QKD) is one of the most promising quantum information technology applications presently [4–6]. QKD allows two parties, Alice and Bob, to exchange a key for encrypting their messages in an information in a theoretically

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safe manner. In theory, if any eavesdropping on their key exchange protocol creates mistakes in the measured data, then key communication is discarded. Otherwise, they abandon the QKD protocol in favor of a different quantum channel (or communicate later). As a result, the privacy of their messages is never jeopardized. In practice, however, the theoretical “security” model may not be properly implemented, or the security proof, which is at the heart of this security model may be based on incorrect or insufficient assumptions. In such instances, the eavesdropper could launch an assault to obtain (partial or full) knowledge of the key without generating too many errors, thus compromising the QKD system’s security. Such theory–practice discrepancies are frequently caused by technical flaws and operational flaws in the system’s hardware or firmware. Many proof-of-principle quantum hacking attacks have been created and tested on practical QKD systems in the previous decade [7–21], and the area of quantum hacking studies such deviations [22–24]. Fiberoptic connectors connect distinct system components in the vast majority of contemporary QKD implementations. All of the input light could be perfectly transferred to the output with ideal components and interfaces. However, a nonzero percentage of light will be reflected or scattered back while going through an interface or inside a component. Fresnel reflection is caused by a change in refractive index during propagation, whereas Rayleigh or Brillouin scattering is caused by density variations in the optical fiber substance. One year later, the core concepts and applicable criteria for executing a genuine big pulse attack were thoroughly examined [25]. This paper presented the findings of a modest experimental investigation of Alice and Bob devices’ “static” settings, as well as several plausible countermeasures. Trojan-horse attacks [26] were coined in 2006 to describe these concepts. These attacks were once again examined experimentally, albeit in a static sense, with a focus on plug-and-play architecture [27–29]. Recent attention has focused on both realistic and real-time Trojan-horse attacks on practical QKD implementations [19–21]. Commercial QKD systems such as Clavis2 from ID Quantique [30] and Cygnus from SeQureNet [31] are among the implementations. In all of these works, the wavelength used for the assault is (in the area of) 1550 nm, which is the standard telecom wavelength used by Alice and Bob for their connection. The possibility and restrictions of launching Trojan-horse attacks at various wavelengths are still to be investigated, particularly from an experimental standpoint. This research is necessary for a thorough examination of the threat posed by these attacks, as well as the (re)design of appropriate prevention and countermeasure methods. Researchers examine the spectral behavior of a number of optical devices pertinent to QKD, ranging from single passive components to a whole QKD subsystem, in these works. In this work, all investigations are thoroughly performed to provide highest level of security to the proposed system. Specifically, from Trojan-horse attacks and the experimental results obtained on the spectral behaviour of two common components in QKD systems are a placed in circulator and an isolator modes.

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6.4 Quantum Sensing by Applying AI Learned Techniques to Detect Early Cancer and Improved Treatment Early cancer identification is critical for successful therapy. As a result, researchers are attempting to create a model that can predict if a malignant tumor will form in a tissue. One of the most difficult aspects of making a successful prediction is determining the most essential components that influence cancer development. Supervised learning, a branch of AI, has a lot of potential for developing such a model. A computer is given a big dataset with known labels, and it creates a model based on the patterns observed in the data. Because of its low Raman crosssection, graphene quantum dots are being used as a carrier for plasmonic materials rather than for cancer detection. As a result, a sensor with an effective Raman cross-section and the ability to preserve cellular homeostasis is required for reliable detection. We describe a SERS-activated graphene oxide (GO) quantum Cytosensor [32] for whole-cell cancer detection that can detect cancer at the cellular level. By introducing the functionality of quantum confinement at the quantum scale, we were able to achieve SERS activation of GO quantum dots. We facilitated rapid self-cellular absorption and boosted SERS sensitivity by changing the number of functional groups on the surface. We use three cell lines to demonstrate cancer diagnosis using two methods: external biomarker detection and intracellular detection. The quantum Cytosensor detects and differentiates malignant features of a cell by sensing complex intracellular biomolecular activities. The augmentation of DNA, RNA, and protein, respectively, increased by 3000-, 2500-, and 3500-folds. Machine learning approaches were used to detect SERS spectral signatures, which primarily detected the distinctions between cancer cells and normal cells. The diagnostic sensitivity and specificity of the classification and clustering approaches were 84.83% and 92.3%, respectively. SERS-based early cancer diagnosis down to a single cellular level utilizing a quantum Cytosensor and machine learning presents a new stepping stone toward adoption of SERS-based early cancer detection.

6.5 Designing Quantum Materials Using AI and Machine Learning The discovery, synthesis, and characterization of quantum materials have all been accelerated up thanks to artificial intelligence and machine learning (AI/ML). The collaboration of materials scientists, chemists, physicists, and others is driving the expansion of this interdisciplinary discipline. This project will progress from the development of algorithms to the study of quantum material interactions and the application of that knowledge to new technologies. Then talk about how AI/ML is interdisciplinary, as well as cooperation with researchers, governmental labs, and businesses.

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Quantum material breakthroughs in fundamental physics and chemistry are critical for new technologies in computation, information processing, and communication. Interdisciplinary teams must take into account the nature of quantum states and materials, as well as engineering and scalability concerns. This project will focus on their groundbreaking work in the realm of quantum information processing, such as the synthesis of novel materials and the characterization of the physical processes that underpin them. When generating model inputs, traditional machine learning approaches for predicting material qualities from elemental compositions have stressed the need of utilizing the domain knowledge. We show how, even with only a few thousand training examples, we can use a deep learning strategy to avoid such manual feature building that requires domain knowledge and produce considerably superior outcomes. We demonstrate the design and implementation of ElemNet [33], a deep neural network model that uses artificial intelligence to capture the physical and chemical interactions and similarities between distinct elements, allowing it to forecast material properties more accurately and quickly. ElemNet’s speed and best-in-class accuracy allow us to screen hundreds of thousands of chemical systems for new material candidates in a vast combinatorial space.

6.6 Quantum Dots Abstract This article examines at the current research on semiconductor quantum dots, as well as a quick look at the theory behind their unique properties and a summary of the importance of quantum dots in multidisciplinary research areas. Because of their size-tunable optical properties, quantum dots, colloidal semiconductor nanocrystals, have a wide range of uses. Since their origin, quantum dots have been a central idea in nanoscience and nanotechnology, and they had a significant impact on the field’s development.

6.6.1 Introduction Quantum dots concept was first proposed in the 1980s after being postulated in the 1970s. The continuum band structure in the mass is changed into specific energy levels in quantum dots, resulting in a wide range of physicochemical properties that differ drastically from their bulk counterparts. Quantum effects come into play when semiconductor particles are made tiny enough, limiting electrons and holes have different energies that can exist in the particles [34]. QDs have been the key notion of nanoscience from their inception and have guided the development of physics, chemistry, and materials science in related domains. They continue to play an essential role in nanoscience for nearly four decades [35, 36].

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Blue light (450 nm wavelength) Quantum dot size, nanometers (nm)

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Fig. 6.2 Quantum dots transform a spectrum of light in different color depending on its size (in nanoscale)

Fig. 6.3 A cadmium-based quantum dot vials produce vibrant hues displaying a pure and precise green color response. (Source: NASA photo)

Nanotechnology is the study of the development, manipulation, and implementation of structures that are 1–100 nm in size. The dimensional similarities between semiconductor and metal nanoparticles in the range of 2–6 nm in size and biological macromolecules like nucleic acids and proteins are striking as shown in Fig. 6.2. Many important milestones in nanoscience and nanotechnology can be traced back to QD-related research including the invention of quantum confinement effect, biological and device applications, size-controlled synthesis, and so on, all of which have aided in the development of fundamental principles and conceptions in the field of nanotechnology [37]. The particles based on their optical features can be thinly regulated depending on the size because energy is proportional to wavelength or color [37] as shown in Fig. 6.3. By manipulating the size of particles, they are programmable to release or absorb explicit light colors or wavelengths. A quantum dot’s attributes are dictated not only by its size but also by its structure, composition, and form, which includes whether it is solid or hollow. A dependable manufacturing process that uses quantum dots’ capabilities for a wide range of applications in the field

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of electronics, catalysis, information storage, photonics, grapheme, bioimaging, biomedicine, or sensors that are able to create vast quantities of nano-crystals with the same parameters in each batch. Nano-crystals act as primary building blocks for self-assembled functional nano-devices since molecular recognition and selfassembly are capabilities of biological molecules. The energy states of atoms in quantum dots contribute to unique optical features like size-dependent particles of fluorescence wavelength that can be used in medical and biological imaging to make optical probes [38]. Colloidal quantum dots have found the largest range of applications in bio-analytics and bio-labeling so far. The early invention of quantum dots demonstrated their potential and hard work to enhance fundamental features, particularly in salt-containing solutions to identify colloidal stability. Quantum dots were first used in highly controlled conditions, and in the real-world scenario, particles would have precipitated, for example, blood. These issues have been resolved, and quantum dots are now widely used in realworld applications [39]. Quantum dots are used in composite materials, solar cells [40, 41], and fluorescent-based biological labels to track a biological molecule right from small particle size to tunable energy levels [40, 42]. The advancement in the field of chemistry has enabled the creation and protection of monolayers, mono-dispersed, highly calibrated, and crystallized quantum dots that are in diameters as tiny as 2 nm that can be handled and processed in the same way as any other chemical reagent [43].

6.6.2 Applications of Quantum Dots (Fig. 6.4) 6.6.2.1

Biomedicine

Quantum dots are excellent device in the field of imaging, diagnostics, and biosensors, bio-imaging applications, environmental and life sciences because of their bright color, narrow emission, and low toxicity and biocompatibility [42]. Researchers are investigating the process involved in biomedicine down to the level of a single molecule with quantum dots in enhancing the current diagnosis and treatment in cancer diseases [43, 44]. Quantum dots are used in high-resolution cellular imaging by injecting active sensor-based elements because of its luminous properties. The characteristics of each quantum dots vary from sample to sample by labeling each receptor molecules (antibodies) at each surface of the dots. Quantum dots can bring change in biomedicine potentially, but unfortunately, the majority of them are poisonous. Ironically, dots contain heavy metals called cadmium which is toxic to human and also the carcinogen present is at high risks when injected to human bodies. Environmental contamination and toxicity must be addressed in using nanomaterials for the growth of biomedical applications that are non-toxic in nature, and biocompatible nanomaterials are the forthcoming research areas.

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Fig. 6.4 Applications of quantum dots

6.6.2.2

Photovoltaic

The appeal of employing quantum dots to make solar-based stem cells is advantageous when compared with conventional methods [45, 46]. These cells can be created using low energy at room temperature technique and developed using lowcost materials that do not require much purification process involved in silicon. Dots are applied to a range of low-cost, flexible substrate materials as lightweight plastics [47]. Photovoltaic devices are challenging in using quantum dots to develop solarbased stem cells by converting solar light into electricity [48, 49]. Semiconductor inks are used in coating the surface of solar cell substrates in a single deposition step, avoiding tens of deposition steps required with the prior layer-by-layer process for routing the quantum dot solar-based stem cells [45, 50].

6.6.2.3

Graphene Quantum Dots

Graphene is a planar form of a carbon nanotube and has emerged as a promising potential material for nanoscale electronics. Researchers have demonstrated that nanoscale transistors can be carved out of a single graphene quantum dots [51]. Graphene material remains extremely stable and conductive even when chopped into nanometer-sized devices. Due to its unique photoluminescence [52] qualities, such as exceptional low toxicity, biocompatibility, and high stability besides photo-bleaching and photo-

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blinking, graphene quantum dots (GQDs) have a lot of potential in the domains of photovoltaics [47], optoelectronics [53], bioimaging, and biosensing [43]. Scientists are still trying to come up with efficient and universal ways for making GQDs with surface properties that are controlled and have a high level of stability with the option of changing photoluminescence emission wavelengths.

6.6.2.4

Display Devices

The most well-known application of quantum dots today is display devices [37]. Companies like LG and Samsung launched their quantum-based LED TVs in the year 2015, followed by many other companies [54, 55]. Quantum dots will be at the heart of next-generation displays due to its photo-luminescent and electroluminescent properties [48, 50]. Quantum-based materials feature lower manufacturing costs, bright colors, and lower power consumption compared to organic luminescent materials used in organic light-emitting diodes (OLEDs) [37, 56]. Another significant benefit is that, quantum dots are deposited on any substrate, irrespective of any size are printed and flexible to be rolled back.

6.6.2.5

Agriculture

Quantum dots may hold the answer to allowing indoor farms to generate much more food or tiny farms to produce significantly more food. Indoor farms can also be found in regions that are not traditionally conducive to farming, such as the cities where the majority of the world’s population now resides. Lighting is one of the most expensive aspects of indoor farming, yet some visible light wavelengths (colors) [57] are more effective than others. Green plants, for example, love the color magenta. Quantum dots can be tweaked to efficiently create magenta light [58–60]. Because the wavelengths that the plants want are employed, less overall light and electricity are required [59, 61]. There is no energy wasted in synthesizing green wavelengths that are not required for a certain plant species, for example. Quantum dots can help plants develop quicker, not just on a per-plant basis, but also based on where they are in their growth cycle. For a young plant, certain wavelengths can be employed, and for a more mature plant, somewhat different wavelengths can be used. Researchers have also been able to accelerate the growth of plants [59–61]. Nanoco, the company that makes the lights you see in the image to the right and above, claims that plants can grow twice as fast with the lights as they can with ordinary LED lights in some instances.

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Data Storage

Quantum dots or memory produced from tiny islands of semiconductors fill a gap left of today’s computer memory by allowing for both quick and long-term storage. Many researchers have demonstrated that within nanoseconds they can write data into quantum dot memories [62]. There are two types of solid-state memory chips such as short-term memory where computers employ rapid dynamic access memory, or DRAM. However, the data exists for limited period and requires being refreshed and repeating this process over 100 times each second in order to maintain its memory [63, 64]. The other type is flash memory that is used in memory cards and is the equivalent of solid memory. The data can be stored for years without refreshing by writing the data 1000 times slower compared to DRAM. According to recent study, memory-based quantum dots are able to deliver the best for long-term storage along with write rates comparable to DRAM. The researchers claim that a dense array of the tiny islands around 15 nm each could hold one terabyte (1000 gigabytes) of data per square inch. Dieter Bimberg [64] and his colleagues from the Technical University of Berlin, Germany, demonstrated that information can be written to quantum dots in about 6 ns with researchers from Istanbul University, Turkey. This is considered to be the initial prototype of new quantum dot-based memory which is faster than DRAM. The researchers used quantum dots constructed by mixing two semiconductors called gallium arsenide and indium arsenide to achieve the fastest writing times. At 14 ns, similar dots manufactured with other materials were still exceedingly rapid. The concept of electron shunting was used for isolated silicon electrode in and out electrically by changing the bits 0 s and 1 s of data in flash memory. However, those processes must cross a “energy barrier,” a laborious process that gradually destroys the quality of a memory chip and drastically reduces its longevity. In similar passion, the quantum dots work in storing the data by shunting the electrons of a dot into a special energy band. However, quantum dot memory is quicker and further reliable when compared to an electric field when uses temporarily lower than the energy barrier. In conclusion, quantum dots are still a hot topic in nanomaterial-applied research. We anticipate that the outstanding publication lists will provide a good representation of recent contributions in the quantum dots field to the reader, as well as a glimpse into future technological advancements based on the findings of these studies.

References 1. Mastriani, M., Iyengar, S. S., & Latesh Kumar, K. J. (2021). Satellite quantum communication protocol regardless, of the weather. Optical and Quantum Electronics, 53, 181. https://doi.org/ 10.1007/s11082-021-02829-8

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2. Mastriani, M., Iyengar, S. S., & Latesh Kumar, K. J. (2021). Bidirectional teleportation for underwater quantum communications. Quantum Information Processing, 20, 22. https://doi.org/10.1007/s11128-020-02970-5. Springer Science+Business Media, LLC part of Springer Nature. 3. Mastriani, M., Iyengar, S. S., & Latesh Kumar, K. J. (2020). Satellite quantum repeaters for a quantum internet. Willey. https://doi.org/10.1002/que2.55. Springer Science+Business Media, LLC part of Springer Nature. 4. Jain, N., Stiller, B., Khan, I., Makarov, V., Marquardt, C., & Leuchs, G. (2015). Risk analysis of Trojan-horse attacks on practical quantum key distribution systems. IEEE Journal of Selected Topics in Quantum Electronics, 21, 168–177. https://arxiv.org/ftp/arxiv/papers/1408/ 1408.0492.pdf 5. Bennett, C. H., & Brassard, G. (1984). Quantum cryptography: Public key distribution and coin tossing. In Proceedings of the IEEE international conference on computers systems and signal processing, Bangalore (p 175). 6. Gisin, N., Ribordy, G., Tittel, W., & Zbinden, H. (2002). Quantum cryptography. Reviews of Modern Physics, 74(1), 145–195. http://journals.aps.org/rmp/abstract/10.1103/ RevModPhys.74.145 7. Lydersen, L. (2011). Practical security of quantum cryptography. Ph.D. dissertation, Norwegian University of Science and Technology, Trondheim. 8. Bugge, N., Sauge, S., Ghazali, A. M. M., Skaar, J., Lydersen, L., & Makarov, V. (2014). Laser damage helps the eavesdropper in quantum cryptography. Physical Review Letters, 112(7), 070503. 9. Gerhardt, J., Liu, Q., Lamas-Linares, A., Skaar, J., Kurtsiefer, C., & Makarov, V. (2011). Fullfield implementation of a perfect eavesdropper on a quantum cryptography system. Nature Communications, 2(2027), 349. 10. Jain, N., Wittmann, C., Lydersen, L., Wiechers, C., Elser, D., Marquardt, C., Makarov, V., & Leuchs, G. (2011). Device calibration impacts security of quantum key distribution. Physical Review Letters, 107(11), 110501. Available: http://journals.aps.org/prl/abstract/10.1103/ PhysRevLett.107.1105 01 11. Lamas-Linares, A., & Kurtsiefer, C. (2007). Breaking a quantum key distribution system through a timing side channel. Optics Express, 15(15), 9388. 12. Li, H.-W., Wang, S., Huang, J.-Z., Chen, W., Yin, Z.-Q., Li, F.-Y., Zhou, Z., Liu, D., Zhang, Y., Gou, G.-C., Bao, W.-S., & Han, Z.-F. (2011). Attacking a practical quantum-keydistribution system with wavelength-dependent beam-splitter and multiwavelength sources. Physical Review A, 84(6), 062308. 13. Lydersen, L., Jain, N., Wittmann, C., Marøy, Ø., Skaar, J., Marquardt, C., Makarov, V., & Leuchs, G. (2011). Superlinear threshold detectors in quantum cryptography. Physical Review A, 84(3), 032320. 14. Lydersen, L., Wiechers, C., Wittmann, C., Elser, D., Skaar, J., & Makarov, V. (2010). Hacking commercial quantum cryptography systems by tailored bright illumination. Nature Photonics, 4(10), 686–689. 15. Nauerth, S., Fürst, M., Schmitt-Manderbach, T., Weier, H., & Weinfurter, H. (2009). Information leakage via side channels in freespace BB84 quantum cryptography. New Journal of Physics, 11(6), 065001. 16. Sun, S.-H., Jiang, M.-S., & Liang, L.-M. (2011). Passive Faraday-mirror attack in a practical two-way quantum-keydistribution system. Physical Review A, 83(6), 062331. 17. Wiechers, C., Lydersen, L., Wittmann, C., Elser, D., Skaar, J., Marquardt, C., Makarov, V., & Leuchs, G. (2011). After-gate attack on a quantum cryptosystem. New Journal of Physics, 13(1), 013043. 18. Xu, F., Qi, B., & Lo, H.-K. (2010). Experimental demonstration of phase-remapping attack in a practical quantum key distribution system. New Journal of Physics, 12(11), 113026. 19. Zhao, Y., Fung, C.-H., Qi, B., Chen, C., & Lo, H.-K. (2008). Quantum hacking: Experimental demonstration of time-shift attack against practical quantum-key-distribution systems. Physical Review A, 78(4), 042333.

6 Future Perspectives of Quantum Applications Using AI

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20. Khan, I., Jain, N., Stiller, B., Jouguet, P., Kunz-Jacques, S., Diamanti, E., Marquardt, C., & Leuchs, G. (2014). Trojan-horse attacks on practical continuous-variable quantum key distribution systems. QCRYPT. 21. Sajeed, S., Radchenko, I., Kaiser, S., Bourgoin, J.-P., Monat, L., Legré, M., & Makarov, V. (2014). Securing two-way quantum communication: The monitoring detector and its flaws. QCRYPT. 22. Scarani, V., Bechmann-Pasquinucci, H., Cerf, N., Dušek, M., Lütkenhaus, N., & Peev, M. (2009). The security of practical quantum key distribution. Reviews of Modern Physics, 81(3), 1301–1350. 23. Jain, N. (2014). Security of practical quantum key distribution systems. Ph.D. dissertation (in progress), Max Planck Institute for the Science of Light, Erlangen. 24. Makarov, V. (2007). Quantum cryptography and quantum cryptanalysis. Ph.D. dissertation, Norwegian University of Science and Technology, Trondheim. 25. Bethune, D. S., & Risk, W. P. (2000). An autocompensating fiber-optic quantum cryptography system based on polarization splitting of light. IEEE Journal of Quantum Electronics, 36(3), 340–347. 26. Vakhitov, A., Makarov, V., & Hjelme, D. R. (2001). Large pulse attack as a method of conventional optical eavesdropping in quantum cryptography. Journal of Modern Optics, 48(13), 2023–2038. 27. Jain, N., Anisimova, E., Khan, I., Makarov, V., Marquardt, C., & Leuchs, G.. (2014). Trojanhorse attacks threaten the security of practical quantum cryptography. New Journal of Physics. arXiv: 1406.5813. https://doi.org/10.1088/1367-2630/16/12/123030. 28. Gisin, N., Fasel, S., Kraus, B., Zbinden, H., & Ribordy, G. (2006). Trojan-horse attacks on quantum-key-distribution systems. Physical Review A, 73(2), 022320. Available: http:// journals.aps.org/pra/abstract/10.1103/PhysRevA.73.022320 29. Bourennane, M., Gibson, F., Karlsson, A., Hening, A., Jonsson, P., Tsegaye, T., Ljunggren, D., & Sundberg, E. (1999). Experiments on long wavelength (1550 nm) “plug and play” quantum cryptography systems. Optics Express, 4(10), 383. Available: http://www.opticsinfobase.org/ oe/abstract.cfm?uri=oe-4-10-383 30. Stucki, D., Gisin, N., Guinnard, O., Ribordy, G., & Zbinden, H. (2002). Quantum key distribution over 67 km with a plug&play system. New Journal of Physics, 4(1), 41. Available: http://iopscience.iop.org/1367-2630/4/1/341/. ID Quantique, http://www.idquantique.com/ 31. SeQureNet. http://www.sequrenet.com/ 32. Ganesh, S., Venkatakrishnan, K., & Tan, B. (2019). Quantum cytosensor for early detection of cancer. https://doi.org/10.1002/mds3.10058A. 33. Jha, D., Ward, L., Paul, A., Liao, W., Choudhary, A., Wolverton, C., & Agrawal, A. (2018). ElemNet: Deep learning the chemistry of materials from only elemental composition. Sci Rep, 8, 17593. https://doi.org/10.1038/s41598-018-35934-y 34. Alferov, Z. I. (1998). The history and future of semiconductor heterostructures. Semiconductors, 32, 1–14. 35. Quantum Dots. Nanosys – Quantum dot pioneers. Retrieved December 4, 2015. 36. Brus, L. E. (2007). Chemistry and physics of semiconductor nanocrystals. Retrieved July 7, 2009. 37. Morozova, S., Alikina, M., Vinogradov, A., & Pagliaro, M. (2020). Silicon quantum dots: Synthesis, encapsulation, and application in light-emitting diodes. Frontiers in Chemistry. https://doi.org/10.3389/fchem.2020.00191 38. Bukowski, T. J., & Simmons, J. H. (2002). Quantum dot research: Current state and future prospects. Critical Reviews in Solid State and Materials Sciences, 27(3–4), 119–142. https:// doi.org/10.1080/10408430208500496 39. Shirasaki, Y., Supran, G. J., Bawendi, M. G., & Bulovic, V. (2013). Emergence of colloidal quantum-dot light-emitting technologies. Nature Photonics, 7, 13–23.

206

H. U. Leena and R. Lawrance

40. Bhandari, S., Pramanik, S., Biswas, N. K., Roy, S., & Pan, U. N. (2019). Enhanced luminescence of a quantum dot complex following interaction with protein for applications in cellular imaging, sensing, and white-light generation. ACS Applied Nano Materials, 2, 2358– 2366. 41. Kokal, R. K., Bredar, A. R. C., Farnum, B. H., & Deepa, M. (2019). Solid-state succinonitrile/sulfide hole transport layer and carbon fabric counter electrode for a quantum dot solar cell. ACS Applied Nano Materials, 2, 7880–7887. 42. Wolfbeisa, O. S. (2015). An overview of nanoparticles commonly used in fluorescent bioimaging. Chemical Society Reviews, 44, 4743–4768. 43. Caglayan, M. O., Mindivan, F., & Sahin, ¸ S. (2020). Sensor and bioimaging studies based on carbon quantum dots: The green chemistry approach. Critical Reviews in Analytical Chemistry. https://doi.org/10.1080/10408347.2020.1828029 44. Biju, V., Mundayoor, S., Omkumar, R. V., Anas, A., & Ishikawa, M. (2010). Bioconjugated quantum dots for cancer research: Present status, prospects and remaining issues. Biotechnology Advances, 28, 199–213. 45. Reference solar spectral irradiance: Air mass 1.5. National Renewable Energy Laboratory. https://www.nrel.gov/grid/solar-resource/spectra.html, Retrieved December 2020. 46. Dong, Q., Liu, H., Hara, Y., Starr, H. E., Dempsey, J. L., & Lopez, R. (2019). Impact of background oxygen pressure on the pulsed-laser deposition of ZnO nanolayers and on their corresponding performance as electron acceptors in PbS quantum-dot solar cells. ACS Applied Nano Materials, 2, 767–777. 47. Chen, G., Seo, J., Yang, C., & Prasad, P. N. (2013). Nanochemistry and nanomaterials for photovoltaics. Chemical Society Reviews, 42, 8304–8338. 48. Ahumada-Lazo, R., Fairclough, S. M., Hardman, S. J. O., Taylor, P. N., Green, M., Haigh, S. J., Saran, R., Curry, R. J., & Binks, D. J. (2019). Confinement effects and charge dynamics in Zn3N2 colloidal quantum dots: Implications for QD-LED displays. ACS Applied Nano Materials, 2, 7214–7219. 49. McDaniel, H., et al. (2014). Simple yet versatile synthesis of CuInSexS2-x quantum dots for sunlight harvesting. Journal of Physical Chemistry C, 118, 16987–16994. 50. Choi, J., Choi, W., & Jeon, D. Y. (2019). Ligand-exchange-ready CuInS2/ZnS quantum dots via surface-ligand composition control for film-type display devices. ACS Applied Nano Materials, 2, 5504–5511. 51. Li, W., Li, M., Liu, Y., Pan, D., Li, Z., Wang, L., & Wu, M. (2018). Three minute ultrarapid microwave-assisted synthesis of bright fluorescent graphene quantum dots for live cell staining and white LEDs. ACS Applied Nano Materials, 1, 1623–1630. 52. Santiago, S. R. M. S., Chang, C.-H., Lin, T.-N., Yuan, C.-T., & Shen, J.-L. (2019). Diethylenetriamine-doped graphene oxide quantum dots with tunable photoluminescence for optoelectronic applications. ACS Applied Nano Materials, 2, 3925–3933. 53. Miao, S., & Cho, Y. (2021). Toward green optoelectronics: Environmental-friendly colloidal quantum dots photodetectors. Frontiers in Energy Research, 9, 666534. https://doi.org/ 10.3389/fenrg.2021.666534 54. Liu, L., Bisri, S. Z., Ishida, Y., Hashizume, D., Aida, T., & Iwasa, Y. (2018). Ligand and solvent effects on hole transport in colloidal quantum dot assemblies for electronic devices. ACS Applied Nano Materials, 1, 5217–5225. 55. Cho, K.-S., et al. (2009). High-performance cross linked colloidal quantum-dot light emitting diodes. Nature Photonics, 3, 341–345. 56. Singh, V. K., Yadav, S. M., Mishra, H., Kumar, R., Tiwari, R. S., Pandey, A., & Srivastava, A. (2019). WS2 quantum dot graphene nanocomposite film for UV photo detection. ACS Applied Nano Materials, 2, 3934–3942. 57. Mallick, S., Kumar, P., & Koner, A. L. (2019). Freeze-resistant cadmium-free quantum dots for live-cell imaging. ACS Applied Nano Materials, 2, 661–666. 58. McCree, K. J. (1972). The action spectrum, absorptance and quantum yield of photosynthesis in crop plants. Agricultural and Forest Meteorology, 9, 191–216.

6 Future Perspectives of Quantum Applications Using AI

207

59. Snowden, M. C., Cope, K. R., & Bugbee, B. (2016). Sensitivity of seven diverse species to blue and green light: Interactions with photon flux. PLoS One, 11, e0163121. 60. Cope, K. R., Snowden, M. C., & Bugbee, B. (2014). Photobiological interactions of blue light and photosynthetic photon flux: Effects of monochromatic and broadspectrum light sources. Photochemistry and Photobiology, 90, 574–584. 61. Nelson, J. A., & Bugbee, B. (2014). Economic analysis of greenhouse lighting: Light emitting diodes vs. high intensity discharge fixtures. PLoS One, 9, e99010. 62. Smullen, C. W., Mohan, V., Nigam, A., Gurumurthi, S., & Stan, M. R. (2011). Relaxing nonvolatility for fast and energy-efficient STT-RAM caches. In HPCA’11 proceedings of the 2011 IEEE 17th international symposium on high performance computer architecture (pp. 50–61). 63. Chen, A., Hutchby, J., Zhirnov, V., & Bourianoff, G. (2015). Emerging nanoelectronic devises (1st ed.). Wiley. 64. Geller, M., Marent, A., Nowozin, T., Bimberg, D., Akçay, N., & Öncan, N. (2008). A write time of 6ns6ns for quantum dot–based memory structures. Applied Physics Letters, 92, 092108. https://doi.org/10.1063/1.2890731

Index

A Adiabatic quantum computers (AQCs), 40–42, 44–46, 52–63, 67, 69 Agriculture, 202 Annealing functions, 44, 54, 59–61

B Bell’s Theorem, 10–12 Bernstein-Vazirani Algorithm, 29–30 Binary symmetric channel (BSC), 112 Birthday paradox, 27 Bob’s qubit, 10, 91, 92 Brewster angle, 94 Brickwork state, 83

C Classical circuit, 12–13 Clifford group, 19–20, 44 Composite system, 6, 7, 9, 15, 80

D Data storage, 203 Deutsch algorithm, 21–24 Deutsch Jozsa Algorithm, 24–26, 29 Dirac Ket notation, 78 Direct and distributed Quantum Computing, 116–117 Distributive property, 3

Double slit experiment, 78, 79 D-Wave 2X systems, 54 E Einstein-Podolsky-Rosen (EPR) pair, 84 Encryption key, 76 Entanglement, v, 1, 2, 9–12, 17, 27, 47, 67, 77, 80–89, 91–100, 103, 104, 109, 111, 113–115, 120, 126, 129–132, 139–142, 144, 145, 148, 149, 152, 155, 158, 159, 175 Entanglement swapping, 90–93, 97–99 F Fast Byzantine negotiation, 101, 106–108 Fault, Configuration, Accounting, Performance, Security (FCAPS), 102 G Gate-model quantum computers, 40, 41, 43, 44, 46–52, 67 Graphene based quantum dots, 197, 201–202 Grover’s Algorithm, 30–37, 174 H Hadamard, Hadamard Gate, 7, 8, 15, 16, 24, 25, 30, 85–87, 177 Hamiltonian, 2, 41, 43, 45, 53, 54, 57–60, 62, 138, 186

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Iyengar et al. (eds.), Quantum Computing Environments, https://doi.org/10.1007/978-3-030-89746-8

209

210 Higher-order Unconstrained Binary Optimization (HUBO), 46, 60–61 Hilbert Space, 4–7, 10, 15, 47, 78, 145, 146 Huang’s Law, 76 I Interferometric quantum radars, 140–142, 154, 155, 157 International Telecommunications Union (ITU-T), 77 Internet Research Task Force (IRTF), 77 J Josephson parametric amplifiers (JPA), 148, 151, 152, 158 L Lemma, 8–11, 16–17, 20–21, 24–25, 27 M Multi qubits, 58–59, 79–80, 82, 100 N Nobel Laureate Max Born, 78 P Parameterized gates, 18–19 Pauli gates, 15, 19, 44 Pauli group, 19 Phase kickback, 21–23, 25, 30 Photovoltaic devices, 201 Pirandola-Laurenza-Ottaviani-Banchi (PLOB), 97 Poissonian distribution, 133, 138 Polarization, 3, 79, 93, 142, 144, 145, 167 Positive operator valued measure (POVM), 8–9, 150, 171 Pumping scheduling algorithm, 95 Q Quantum Annealing (QA), 45, 53, 54, 56, 58–63 Quantum Annealing Correction (QAC), 59–60 Quantum blockchain, 174–175 Quantum compiling, v, 39–70 Quantum correlated noise (QCN) radar, 152–154 Quantum cryptography, 76, 82, 101, 105, 131, 167, 175

Index Quantum dots, 41, 43, 197–203 Quantum error correction (QEC), 16, 19, 43, 44, 46, 47, 59–60, 83–89, 131 Quantum firmware, 40, 44, 52–70 Quantum Fourier Transform, 37, 171 Quantum Internet architecture, 75–77, 101–104, 107, 115, 116, 119 Quantum Internet Research Group (QIRG), 101 Quantum Key Distribution (QKD), 76, 77, 82, 97, 101, 105, 108, 109, 114, 117, 120, 159, 167, 168, 170, 175, 177, 193, 195–196 Quantum mechanics, v, 1, 3, 5–8, 105, 111, 112, 131, 140, 194 Quantum money and quantum cheque, 167–170, 176–183 Quantum processing unit (QPU), 40, 41, 55, 56, 61, 62, 64, 67, 102 Quantum radar, 125–160 Quantum security, 105, 193 Quantum sensor types, 125, 129–130 Quantum teleportation, 109, 111, 131, 195 Quantum two-mode squeezing (QTMS) radar, 151, 152, 154, 155 Quantum verification protocols (1D-QPVè), 110–111 R Reversible computation, 6, 13–14, 18 S Schrodinger equation, 45 Shannon theory, 112 Simon’s algorithm, 26–29 Single qubit, 15, 17, 18, 47, 48, 58, 77–80, 83–88, 90, 100 Solovay-Kitaev theorem, 49–52 Spontaneous parametric down conversion (SPDC), 141, 147–150, 158 Superdense codes, 111–112, 131 SWAP gate, 17, 52, 66 T Teleportation, 84, 90–92, 109, 111, 131, 175, 194–195 The Quantum Information Software Kit (Qiskit), 52, 64–70 Toffoli Gate, 18 W Wavefunction, 2, 55, 130, 147