Quantum Information Processing, Quantum Computing, and Quantum Error Correction: An Engineering Approach [2 ed.] 9780128219829

Citation preview

Quantum Information Processing, Quantum Computing, and Quantum Error Correction An Engineering Approach Second Edition

Ivan B. Djordjevic

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2021 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-821982-9 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Mara Conner Acquisitions Editor: Tim Pitts Editorial Project Manager: Rachel Pomery Production Project Manager: Prem Kumar Kaliamoorthi Cover Designer: Greg Harris Typeset by TNQ Technologies

To Milena

Preface Quantum information is related to the use of quantum mechanics concepts to perform information processing and transmission of information. Quantum information processing (QIP) is an exciting research area with numerous applications, including quantum key distribution (QKD), quantum teleportation, quantum computing, quantum lithography, and quantum memories. This area currently experiences rapid development, which can be judged from the number of published books on QIP and the number of conferences devoted solely to QIP concepts. Given the novelty of underlying QIP concepts, it is expected that this topic will be a subject of interest to a broad range of scientists, not just of those involved in QIP research. Moreover, based on Moore’s law, which claims that the number of chips that can be etched on a single chip doubles every 18 months, leading to the doubling of both memory and computational speed, the ever-increasing demands in miniaturization of electronics will eventually lead us to the point where quantum effects become important. Given this fact, it seems that a much broader range of scientists will be forced to turn to the study of QIP much sooner than expected. Note that, on the other hand, as multicore architecture is becoming a prevailing high-performance chip design approach, improvement in computational speed can be achieved even without reducing the feature size through parallelization. Therefore thanks to multicore processor architectures, the need for QIP might be prolonged for a certain amount of time. Another point might be that, despite intensive development of quantum algorithms, the number of available quantum algorithms is still small compared to that of classical algorithms. Furthermore, current quantum gates can operate only on several tens of quantum bits (also known as qubits), which is too low for any meaningful quantum computation operation. Until recently, it was widely believed that quantum computation would never become a reality. However, recent advances in various quantum circuits implementations, as well as the proof of accuracy threshold theorem, have given rise to the optimism that quantum computers might soon become a reality. Because of the interdisciplinary nature of the fields of QIP, quantum computing, and quantum error correction, this book aims to provide the right balance between quantum mechanics, quantum error correction, quantum computing, and quantum communication. This book has the following objectives: 1. It describes the trends in QIP, quantum error correction, and quantum computing. 2. It represents a self-contained introduction to QIP, quantum computing, and quantum error correction. 3. It targets a very wide range of readers: electrical engineers, optical engineers, applied mathematicians, computer scientists, and physicists. 4. It does not require prior knowledge of quantum mechanics. The basic concepts of quantum mechanics are provided in Chapter 2.

xvii

xviii

Preface

5. It does not require any background knowledge, except for understanding the basic concepts of vector algebra at an undergraduate level. An appendix is provided with basic descriptions of abstract algebra at a level sufficient to follow the book easily. 6. It offers in-depth exposition on the design and realization of QIP and quantum error correction circuits. 7. For readers not familiar with the concepts of information theory and channel coding, Chapter 6 provides basic concepts and definitions from coding theory. Only the concepts from coding theory of importance in quantum error correction are covered in this chapter. 8. Readers not interested in quantum error correction can approach the chapters related to QIP and quantum computing. 9. Readers interested only in quantum error correction do not need to read the chapters on quantum computing, except Chapters 3 and 4. An effort has been made to create self-independent chapters, while ensuring a proper flow between the chapters. 10. Various concepts are gradually introduced starting from intuitive and basic concepts, through medium difficulty topics, to highly mathematically involved topics. 11. At the end of each chapter, a set of problems is provided so that readers can have a deeper understanding of underlying concepts and be able to perform independent research in quantum computing and/or quantum error correction. 12. Several different courses can be offered by using this book: (1) quantum error correction, (2) quantum information processing, (3) quantum computing, and (4) integrating the concepts of quantum information processing, quantum computing, and quantum error correction. This book is a self-contained introduction to quantum information, quantum computation, and quantum error correction. After completion of the book readers will be ready for further study in this area and be prepared to perform independent research. They will also be able to design QIP circuits, stabilizer codes, CalderbankeShoreSteane (CSS) codes, subsystem codes, topological codes, surface codes, and entanglement-assisted quantum error correction codes, as well as propose corresponding physical implementation, and be proficient in fault-tolerant design. The book starts with basic principles of quantum mechanics, including state vectors, operators, density operators, measurements, and dynamics of a quantum system. It continues with fundamental principles of quantum computation, quantum gates, quantum algorithms, quantum teleportation, and quantum information theory fundamentals. Significant space is given to quantum error correction codes, in particular stabilizer codes, CSS codes, quantum low-density parity-check codes, subsystem codes (also known as operator-quantum error correction codes), topological codes, surface codes, and entanglement-assisted quantum error correction codes. Other topics covered are fault-tolerant quantum error correction coding,

Preface

fault-tolerant quantum computing, and the cluster states concept, which is applied to one-way quantum computation. Relevant space is devoted in investigating physical realizations of quantum computers, encoders, and decoders, including nuclear magnetic resonance, ion traps, photonic quantum realization, cavity quantum electrodynamics, and quantum dots. Focus is also given to quantum machine learning algorithms. The final chapter in the book is devoted to QKD. The following are some unique opportunities available to readers of the book: 1. They will be prepared for further study in these areas and will be qualified to perform independent research. 2. They will be able design the information processing circuits, stabilizer codes, CSS codes, topological codes, surface codes, subsystem codes, quantum LDPC codes, and entanglement-assisted quantum error correction codes, and will be proficient in fault-tolerant design. 3. They will be able to propose the physical implementation of QIP, quantum computing, and quantum error correction circuits. 4. They will be proficient in quantum machine learning. 5. They will be proficient in QKD too. 6. They will have access to extra material to support the book on an accompanying website. The author would like to acknowledge the support of the National Science Foundation. Finally, special thanks are extended to Rachel Pomery and Tim Pitts of Elsevier for their tremendous effort in organizing the logistics of the book, including editing and promotion, which has been indispensible in making this book a reality.

xix

About the Author Ivan B. Djordjevic is a professor of electrical and computer engineering and optical sciences at the University of Arizona; director of the Optical Communications Systems Laboratory and Quantum Communications Lab; and codirector of the Signal Processing and Coding Lab. He is both an IEEE Fellow and OSA Fellow. He received his PhD degree from the University of Nis, Yugoslavia, in 1999. Professor Djordjevic has authored or coauthored eight books, more than 540 journal and conference publications, and 54 US patents. He presently serves as an Area Editor/Associate Editor/Member of Editorial Board for the following journals: OSA/IEEE Journal of Optical Communications and Networking, IEEE Communications Letters, Optical and Quantum Electronics, and Frequenz. He served as an editorial board member/associate editor for IOP Journal of Optics and Elsevier Physical Communication Journal from 2016 to 2021. Prior to joining the University of Arizona, Dr. Djordjevic held appointments at the University of Bristol and University of the West of England in the United Kingdom, Tyco Telecommunications in the United States, National Technical University of Athens in Greece, and State Telecommunication Company in Yugoslavia.

xxi

CHAPTER

Introduction

1

CHAPTER OUTLINE 1.1 Photon Polarization .............................................................................................. 4 1.2 The Concept of Qubit .......................................................................................... 11 1.3 The Spin-1/2 Systems......................................................................................... 12 1.4 Quantum Gates and QIP....................................................................................... 13 1.5 Quantum Teleportation........................................................................................ 16 1.6 Quantum Error Correction Concepts ..................................................................... 17 1.7 Quantum Key Distribution.................................................................................... 19 1.8 Organization of the Book..................................................................................... 21 References ............................................................................................................... 28 Further reading ......................................................................................................... 29

This chapter is devoted to the introduction to quantum information processing (QIP), quantum computing, and quantum error correction coding (QECC) [1e12]. Quantum information is related to the use of quantum mechanics concepts to perform information processing and transmission of information. QIP is an exciting research area with numerous applications, including quantum key distribution (QKD), quantum teleportation, quantum computing, quantum lithography, and quantum memories. This area currently experiences intensive development, and given the novelty of underlying concepts, it might become of interest to a broad range of scientists, not just those involved in research. Moreover, it is based on Moore’s law, which claims that the number of chips that can be etched on a single chip doubles every 18 months, leading to the doubling of both memory and computational speed. Therefore the ever-increasing demands in miniaturization of electronics will eventually lead to the point where quantum effects become important. Given this fact, it seems that a much broader range of scientists will be forced to study QIP much sooner than it appears right now. Note that because multicore architecture is becoming a prevailing high-performance chip design approach, improving the computational speed can be achieved even without reducing the feature size through parallelization. Therefore thanks to multicore processor architectures, the need for QIP might be prolonged for a certain amount of time. Another point might be that, despite intensive development of quantum algorithms, the number of available quantum algorithms is still small compared to that of classical algorithms. Until Quantum Information Processing, Quantum Computing, and Quantum Error Correction https://doi.org/10.1016/B978-0-12-821982-9.00016-2 Copyright © 2021 Elsevier Inc. All rights reserved.

1

2

CHAPTER 1 Introduction

recently, it was widely believed that quantum computation would never become a reality. However, recent advances in various quantum gate implementations, as well as proof of the accuracy threshold theorem, have given rise to the optimism that quantum computers will soon become a reality. Fundamental features of QIP are different from that of classical computing and can be summarized as: (1) linear superposition, (2) entanglement, and (3) quantum parallelism. The following are basic details of these features: 1. Linear superposition. Contrary to the classical bit, a quantum bit or qubit can take not only two discrete values 0 and 1 but also all possible linear combinations of them. This is a consequence of a fundamental property of quantum states: it is possible to construct a linear superposition of quantum state j0i and quantum state j1i. 2. Entanglement. At a quantum level it appears that two quantum objects can form a single entity, even when they are well separated from each other. Any attempt to consider this entity as a combination of two independent quantum objects, given by a tensor product of quantum states, fails unless the possibility of signal propagation at superluminal speed is allowed. These quantum objects, which cannot be decomposed into a tensor product of individual independent quantum objects, are called entangled quantum objects. Given the fact that arbitrary quantum states cannot be copied, which is the consequence of no-cloning theorem, communication at superluminal speed is not possible, and as a consequence the entangled quantum states cannot be written as the tensor product of independent quantum states. Moreover, it can be shown that the amount of information contained in an entangled state of N qubits grows exponentially instead of linearly, which is the case for classical bits. 3. Quantum parallelism. Quantum parallelism is the possibility of performing a large number of operations in parallel, which represents the key difference from classical computing. Namely, in classical computing it is possible to know what is the internal status of a computer. On the other hand, because of no-cloning theorem, it is not possible to know the current state of a quantum computer. This property has led to the development of Shor’s factorization algorithm, which can be used to crack the RivesteShamireAdleman (RSA) encryption protocol. Some other important quantum algorithms include Grover’s search algorithm, which is used to perform a search for an entry in an unstructured database; quantum Fourier transform, which is a basis for a number of different algorithms; and Simon’s algorithm. These algorithms are the subject of Chapter 5. The quantum computer is able to encode all input strings of length N simultaneously into a single computation step. In other words, the quantum computer is able simultaneously to pursue 2N classical paths, indicating that the quantum computer is significantly more powerful than the classical one. Although QIP has opened up some fascinating perspectives, as indicated earlier, there are certain limitations that should be overcome before QIP becomes a

Introduction

commercial reality. The first is related to existing quantum algorithms, whose number is significantly lower than that of classical algorithms. The second problem is related to physical implementation issues. There are many potential technologies such as nuclear magnetic resonance (NMR), ion traps, cavity quantum electrodynamics, photonics, quantum dots, and superconducting technologies, just to mention a few. Nevertheless, it is not clear which technology will prevail. Regarding quantum teleportation, most probably the photonic implementation will prevail. On the other hand, for quantum computing applications there are many potential technologies that compete with each other. Moreover, presently, the number of qubits that can be efficiently manipulated is in the order of tens, which is well below that needed for meaningful quantum computation, which is in the order of thousands. Another problem, which can be considered as problem number one, is related to decoherence. Decoherence is related to the interaction of qubits with the environment that blurs the fragile superposition states. Decoherence also introduces errors, indicating that the quantum register should be sufficiently isolated from the environment so that only a few random errors occur occasionally, which can be corrected by QECC techniques. One of the most powerful applications of quantum error correction is the protection of quantum information as it dynamically undergoes quantum computation. Imperfect quantum gates affect the quantum computation by introducing errors in computed data. Moreover, the imperfect control gates introduce the errors in a processed sequence since wrong operations can be applied. The QECC scheme now needs to deal not only with errors introduced by the quantum channel by also with errors introduced by imperfect quantum gates during the encoding/decoding process. Because of this, the reliability of data processed by a quantum computer is not a priori guaranteed by QECC. The reason is threefold: (1) the gates used for encoders and decoders are composed of imperfect gates, including controlled imperfect gates; (2) syndrome extraction applies unitary operators to entangle ancilla qubits with code blocks; and (3) error recovery action requires the use of a controlled operation to correct for the errors. Nevertheless, it can be shown that arbitrary good quantum error protection can be achieved even with imperfect gates, providing that the error probability per gate is below a certain threshold; this claim is known as accuracy threshold theorem and will be discussed later in Chapter 11, together with various fault-tolerant concepts. This introductory chapter is organized as follows. In Section 1.1, photon polarization is described as it represents a simple and most natural connection to QIP. In the same section, we introduce some basic concepts of quantum mechanics, such as the concept of state. We also introduce Dirac notation, which will be used throughout the book. In Section 1.2, we formally introduce the concept of qubit and provide its geometric interpretation. In Section 1.3, another interesting example of the representation of qubits, the spin-1/2 system, is provided. Section 1.4 is devoted to basic quantum gates and QIP fundamentals. The basic concepts of quantum teleportation are introduced in Section 1.5. Section 1.6 is devoted to basic QECC concepts. Section 1.7 is devoted to QKD, also known as quantum cryptography. The organization of the book is described in Section 1.8.

3

4

CHAPTER 1 Introduction

1.1 PHOTON POLARIZATION The electric/magnetic field of plane linearly polarized waves is described as follows [13]: Aðr; tÞ ¼ pA0 exp½jðut  k $ rÞ;

A˛fE; Hg

(1.1)

where E(H) denotes the electric (magnetic) field, p denotes the polarization orientation, r ¼ xex þ yey þ zez is the position vector, and k ¼ kxex þ kyey þ kzez denotes the wave propagation vector whose magnitude is k ¼ 2p/l (l is the operating wavelength). For the x-polarization waves (p ¼ ex, k ¼ kez), Eq. (1.1) becomes: Ex ðz; tÞ¼ ex E0x cosðut  kzÞ;

(1.2)

while for y-polarization (p ¼ ey, k ¼ kez) it becomes: Ey ðz; tÞ ¼ ey E0y cosðut  kz þ dÞ;

(1.3)

where d is the relative phase difference between the two orthogonal waves. The resultant wave can be obtained by combining Eqs. (1.2) and (1.3) as follows: Eðz; tÞ ¼ Ex ðz; tÞ þ Ey ðz; tÞ ¼ ex E0x cosðut  kzÞ þ ey E0y cosðut  kz þ dÞ. (1.4) The linearly polarized wave is obtained by setting the phase difference to an integer multiple of 2p, namely d ¼ m$2p: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eðz; tÞ ¼ ðex E cos q þ ey E sin qÞcosðut  kzÞ; E ¼ E20x þ E20y ; (1.5) E0y q ¼ tan1 . E0x By ignoring the time-dependent term, we can represent the linear polarization as shown in Fig. 1.1A. On the other hand, if dsm$2p; the elliptical polarization is obtained. From Eqs. (1.2) and (1.3), by eliminating the time-dependent term we obtain the following equation of ellipse:  2  2 2E0x E0y cos d Ey Ex Ex E y þ 2 cos d ¼ sin2 d; tan 2 f ¼ ; (1.6) 2  E2 E0x E0y E0x E0y E0x 0y

E0y

Ey

f

q Ex

(A)

Ey

E

E

(B)

E0x

E

Ex

Ex

(C)

FIGURE 1.1 Various forms of polarizations: (A) linear polarization, (B) elliptic polarization, and (C) circular polarization.

1.1 Photon Polarization

which is shown in Fig. 1.1B. By setting d ¼ p/2,  3p/2, ... the equation of ellipse becomes:  2  2 Ey Ex þ ¼ 1: (1.7) E0x E0y By setting further E0x ¼ E0y ¼ E0 ; d ¼ p/2,  3p/2, ... the equation of ellipse becomes the circle: Ex2 þ Ey2 ¼ 1;

(1.8)

and corresponding polarization is known as circular polarization (Fig. 1.1C). The right circularly polarized wave is obtained for d ¼ p/2 þ 2mp: E ¼ E0 ½ex cosðut  kzÞ  ey sinðut  kzÞ:

(1.9)

Otherwise, for d ¼ p/2 þ 2mp, the polarization is known as left circularly polarized. Very often, the Jones vector representation of the polarization wave is used: " pffiffiffiffiffiffiffiffiffiffiffi #   Ex ðtÞ 1k (1.10) EðtÞ ¼ ¼ E pffiffiffi jd ejðutkzÞ ; Ey ðtÞ ke where k is the power-splitting ratio between states of polarizations (SOPs), with the complex phasor term being typically omitted in practice. Another interesting representation is Stokes vector representation: 3 2 S0 ðtÞ 7 6 6 S1 ðtÞ 7 7; 6 (1.11) SðtÞ ¼ 6 7 4 S2 ðtÞ 5 S3 ðtÞ where the parameter S0 is related to the optical intensity by: S0 ðtÞ ¼ jEx ðtÞj2 þ jEy ðtÞj2 .

(1.12a)

The parameter S1 > 0 is related to the preference for horizontal polarization and is defined by: S1 ðtÞ ¼ jEx ðtÞj2  jEy ðtÞj2 .

(1.12b)

The parameter S2 > 0 is related to the preference for p/4 SOP: S2 ðtÞ ¼ Ex ðtÞEy ðtÞ þ Ex ðtÞEy ðtÞ.

(1.12c)

Finally, the parameter S3 > 0 is related to the preference for right circular polarization and is defined by: h i (1.12d) S3 ðtÞ ¼ j Ex ðtÞEy ðtÞ  Ex ðtÞEy ðtÞ .

5

6

CHAPTER 1 Introduction

The parameter S0 is related to other Stokes parameters by: S20 ðtÞ ¼ S21 ðtÞ þ S22 ðtÞ þ S23 ðtÞ. The degree of polarization is defined by:  2 1=2 S þ S22 þ S23 ; p¼ 1 S0

(1.13)

0p1

(1.14)

For p ¼ 1 the polarization does not change in time. The Stokes vector can be represented in terms of Jones vector parameters as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s1 ¼ 1  2k; s2 ¼ 2 kð1  kÞcos d; s3 ¼ 2 kð1  kÞsin d. (1.15) After normalization with respect to S0, the normalized Stokes parameters are given by: si ¼

Si ; i ¼ 0; 1; 2; 3: S0

(1.16)

If the normalized Stokes parameters are used, the polarization state can be represented as a point on a Poincare´ sphere, as shown in Fig. 1.2. The points located at the opposite sides of the line crossing the center represent the orthogonal polarizations. The polarization ellipse is very often represented in terms of ellipticity and azimuth, which are illustrated in Fig. 1.3. The ellipticity is defined by the ratio of half-axes lengths. The corresponding angle is called the ellipticity angle and denoted by ε. The small ellipticity means that the polarization ellipse is highly elongated, while for zero ellipticity the polarization is linear. For ε ¼ p/4, the polarization is circular. For ε > 0, the polarization is right elliptical. On the other hand, the azimuth angle h defines the orientation of the main axis of ellipse with respect to Ex.

The right-elliptical polarization points are located in the northern hemisphere

s3

The radius is determined by S0

P 2ε 2η

s2 The poles represent rightand left-circular polarizations

The equator represents various forms of linear polarization

s1

FIGURE 1.2 Representation of polarization state as a point on a Poincare´ sphere.

1.1 Photon Polarization

The ellipticity (tanε) defines the length ratio of half-axes

Ey

ε

η

Ex

The azimuth defines the orientation of the main axis with respect to Ex

FIGURE 1.3 The ellipticity and azimuth of the polarization ellipse.

The parameters of the polarization ellipse can be related to the Jones vector parameters by: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 kð1  kÞsin d . (1.17) sin 2 ε ¼ 2 kð1  kÞsin d; tan 2 h ¼ 1  2k Finally, the parameters of the polarization ellipse can be related to the Stokes vector parameters by: s1 ¼ cos 2h cos 2ε; s2 ¼ sin 2h cos 2ε; s3 ¼ sin 2ε;

(1.18)

and corresponding geometrical interpretation is provided in Fig. 1.2. Let us now observe the polarizereanalyzer ensemble, shown in Fig. 1.4. When an electromagnetic wave passes through the polarizer, it can be represented as a vector in the xey plane, transversal to the propagation direction, as given by Eq. (1.5), where the angle q depends on the filter orientation. By introducing the unit vector pb ¼ ðcos q; sin qÞ; Eq. (1.5) can be rewritten as: E ¼ E0 pbcosðut  kzÞ.

(1.19)

If q ¼ 0 rad, the light is polarized along the x-axis, while for q ¼ p/2 rad it is polarized along the y-axis. The natural light is unpolarized as it represents x

pˆ

 cos T ,sin T

Ex

T

x

nˆ y

 cos I ,sin I

Ey

I

Polarizer

Ey

E0 cos Zt  kz

E0 cos Zt  kz

E

Analyzer

z y

Ex

Ex

E0 cos Zt  kz pˆ

Ey

E ' =  E × nˆ nˆ = E0 cos Zt - kz  pˆ × nˆ nˆ

FIGURE 1.4 The polarizereanalyzer ensemble for the study of photon polarization.

7

8

CHAPTER 1 Introduction

incoherent superposition of 50% of light polarized along the x-axis and 50% of light polarized along the y-axis. After the analyzer, whose axis makes an angle f with respect to the x-axis, which can be represented by the unit vector nb ¼ ðcos f; sin fÞ, the output electric field is given by: E0 ¼ ðE$b n Þb n ¼ E0 cosðut  kzÞðb p $b n Þb n ¼ E0 cosðut  kzÞ½ðcos q; sin qÞ$ðcos f; sin fÞb n ¼ E0 cosðut  kzÞ½cos q cos f þ sin q sin fb n ¼ E0 cosðut  kzÞ cosðq  fÞb n.

(1.20)

The intensity of the analyzer output field is given by: 2

I 0 ¼ jE0 j ¼ I cos2 ðq  fÞ;

(1.21)

which is commonly referred to as Malus’ law. The polarization decomposition by a birefringent plate is now studied (Fig. 1.5). Experiment shows that photodetectors PDx and PDy are never triggered simultaneously, which indicates that an entire photon reaches either PDx or PDy (a photon never splits). Therefore the corresponding probabilities that a photon is detected by photodetector PDx and PDx can be determined by: px ¼ PrðPDx Þ ¼ cos2 q; py ¼ PrðPDy Þ ¼ sin2 q.

(1.22)

If the total number of photons is N, the number of detected photons in x-polarization will be Nx y Ncos2 q, and the number of detected photons in y-polarization will be Ny y Nsin2 q. In the limit, as N / ∞ we would expect Malus’ law to be obtained. Let us now study polarization decomposition and recombination by means of birefringent plates, as illustrated in Fig. 1.6. Classical physics prediction of total probability of a photon passing the polarizereanalyzer ensemble is given by: ptot ¼ cos2 q cos2 f þ sin2 q sin2 f s cos2 ðq  fÞ;

x

Extraordinary ray

q

Ix=I cos2q PDx PDy

Ordinary ray

y

Iy=I sin2q

Birefringent plate

FIGURE 1.5 Polarization decomposition by a birefringent plate. PD, Photodetector.

(1.23)

1.1 Photon Polarization

x

x Extraordinary ray

f

q Ordinary ray

y

Polarizer

y

Analyzer

FIGURE 1.6 Polarization decomposition and recombination by a birefringent plate.

which is inconsistent with Malus’ law, given by Eq. (1.21). To reconstruct the results from wave optics, it is necessary to introduce into quantum mechanics a concept of probability amplitude that a is detected as b, which is denoted as aða /bÞ; and is a complex number. The probability is obtained as the squared magnitude of probability amplitude: pða / bÞ ¼ jaða/bÞj2 .

(1.24)

The relevant probability amplitudes related to Fig. 1.6 are: aðq/xÞ ¼ cos q aðq/yÞ ¼ sin q

aðx/fÞ ¼ cos f aðx/fÞ ¼ sin f

(1.25)

The basic principle of quantum mechanics is to sum up the probability amplitudes for indistinguishable paths: atot ¼ cos q cos f þ sin q sin f ¼ cosðq  fÞ.

(1.26)

The corresponding total probability is: ptot ¼ jatot j2 ¼ cos2 ðq  fÞ;

(1.27)

and this result is consistent with Malus’ law! Based on the previous discussion, the state vector of photon polarization is given by:   jx ; (1.28) jji ¼ jy where jx is related to x-polarization and jy is related to y-polarization, with the normalization condition as follows: 2 jjx j2 þ jy ¼ 1: (1.29)

9

10

CHAPTER 1 Introduction

In this representation, the x- and y-polarization photons can be represented by:     1 0 ; jyi ¼ ; (1.30) jxi ¼ 0 1 and the right and left circular polarization photons are represented by:     1 1 1 1 ; jLi ¼ pffiffiffi . jRi ¼ pffiffiffi 2 j 2 j

(1.31)

In Eqs. (1.28)e(1.31) we used Dirac notation to denote the column vectors (kets). In Dirac notation, with each column vector (“ket”) jji, we associate a row vector (“bra”) hjj as follows:

 (1.32) hjj ¼ jx jy . The scalar (dot) product of ket jfi bra hjj is defined by “brackets” as follows: hfjji ¼ fx jx þ fy jy ¼ hjjfi .

(1.33)

The normalization condition can be expressed in terms of scalar product by: hjjji ¼ 1:

(1.34)

Based on Eqs. (1.30) and (1.31) it is evident that: hxjxi ¼ hyjyi ¼ 1;

hRjRi ¼ hLjLi ¼ 1:

Because the vectors jxi and jyi are orthogonal, their dot product is zero: hxjyi ¼ 0;

(1.35)

and they form the basis. Any state vector jji can be written as a linear superposition of basis kets as follows:   jx jji ¼ (1.36) ¼ jx jxi þ jy jyi. jy We can now use Eqs. (1.34) and (1.36) to derive an important relation in quantum mechanics, known as completeness relation. The projections of state vector jji along basis vectors jxi and jyi are given by: hxjji ¼ jx hxjxi þ jy hxjyi ¼ jx ; |ffl{zffl} |ffl{zffl} 1

hyjji ¼ jx hyjxi þ jy hyjyi ¼ jy .

(1.37)

0

By substituting Eq. (1.37) into Eq. (1.36) we obtain:

jji ¼ jxihxjji þ jyihyjji ¼ ðjxihxj þ jyihyjÞ ji; |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} I

(1.38)

1.2 The Concept of Qubit

and from the right side of Eq. (1.38) we derive the completeness relation: jxihxj þ jyihyj ¼ I.

(1.39)

The probability that the photon in state jji will pass the x-polaroid is given by: px ¼

jjx j2 2 2 2 ¼ jjx j ¼ jhxjjij ; 2 jjx j þ jy

(1.40)

the probability amplitude of the photon in state jji to pass the x-polaroid is: ax ¼ hxjji.

(1.41)

Let jfi and jji be two physical states. The probability amplitude of finding f in j, denoted as a(f / j), is given by: aðf / jÞ ¼ hfjji;

(1.42)

and the probability for f to pass the j test is given by: pðf / jÞ ¼ jhfjjij2 .

(1.43)

1.2 THE CONCEPT OF QUBIT Based on the previous section, it can be concluded that the quantum bit, also known as qubit, lies in a 2D Hilbert space H, isomorphic to C2-space, where C is the complex number space, and can be represented as:   a (1.44) ; a; b˛C; jaj2 þ jbj2 ¼ 1; jji ¼ aj0i þ bj1i ¼ b where j0i and j1i states are computational basis (CB) states, and jji is a superposition state. If we perform the measurement of a qubit, we will get j0i with probability jaj2 and j1i with probability jbj2. Measurement changes the state of a qubit from a superposition of j0i and j1i to the specific state consistent with the measurement result. If we parametrize the probability amplitudes a and b as follows:     q q a ¼ cos ; b ¼ ejf sin ; (1.45) 2 2 where q is a polar angle and f is an azimuthal angle, we can geometrically represent the qubit by the Bloch sphere (or the Poincare´ sphere for the photon) as illustrated in Fig. 1.7. (Note that the Bloch sphere from Fig. 1.7 is a little bit different from that of the Poincare´ sphere from Fig. 1.2.) Bloch vector coordinates are given by (cos f sin q, sin f sin q, cos q). This Bloch vector representation is related to CB by:   cosðq=2Þ ; (1.46) jjðq; fÞi ¼ cosðq = 2Þj0i þ ejf sinðq = 2Þj1i¼_ ejf sinðq=2Þ

11

12

CHAPTER 1 Introduction

z

|xñ®|0ñ

q |-ñ

|Lñ: q =p /2, f =-p /2

|Yñ |Rñ: q =p /2, f =p /2

f

y

|+ñ

x |yñ®|1ñ

FIGURE 1.7 Bloch (Poincare´) sphere representation of a single qubit.

where 0  q  p and 0  f < 2p. The north and south poles correspond to computational j0i (jxi-polarization) and j1i (jyi-polarization) basis kets, respectively. Other important bases are the diagonal basis {jþi, jei}, very often denoted as fjbi; jaig; related to CB by: 1 j þ i ¼ jbi ¼ pffiffiffi ðj0i þ j1iÞ; 2

1 j  i ¼ jai ¼ pffiffiffi ðj0i  j1iÞ; 2

(1.47)

and the circular basis {jRi,jLi}, related to the CB as follows: 1 jRi ¼ pffiffiffi ðj0i þ jj1iÞ; 2

1 jLi ¼ pffiffiffi ðj0i  jj1iÞ. 2

(1.48)

1.3 THE SPIN-1/2 SYSTEMS In addition to the photon, an important realization of the qubit is a spin-1/2 system. NMR is based on the fact that the proton possesses a magnetic moment m that can take only two values along the direction of the magnetic field. Namely, the projection along the nb-axis obtained by m$b n takes only two values, and this property characterizes a spin-1/2 particle. Experimentally, this was confirmed by Sterne Gerlach experiment, shown in Fig. 1.8. A beam of protons is sent into a nonhomogenic magnetic field in a direction nb orthogonal to the beam direction. It can be observed that the beam splits into two beams: one deflected to the direction þb n and the other in the opposite direction b n . Therefore the proton never splits. The basis in spin-1/2 systems can be written as fj þi; j ig, where the corresponding

1.4 Quantum Gates and QIP

N

Oven

B

y z

S

x

Collimating slits

FIGURE 1.8 The SterneGerlach experiment.

basis kets represent the spin-up and spin-down states. The superposition state can be represented in terms of these bases as follows:   jþ (1.49) ¼ jþ j þ i þ j j  i; jj j2 þ jjþ j2 ¼ 1; jji ¼ j where jjþj2 (jjej2) denotes the probability of finding the system in the spin-up (spin-down) state. By using the trigonometric identity sin2(q/2) þ cos2(q/2) ¼ 1, the expansion coefficients jþ and jþ can be expressed as follows: jþ ¼ ejf=2 cosðq = 2Þ

j ¼ ejf=2 sinðq = 2Þ;

(1.50)

so that: jji ¼ ejf=2 cosðq = 2Þj þ i þ ejf=2 sinðq = 2Þj  i:

(1.51)

Therefore the single superposition state of the spin-1/2 system can also be visualized as the point (q,4) on a unit sphere (Bloch sphere).

1.4 QUANTUM GATES AND QIP In quantum mechanics, the primitive undefined concepts are physical system, observable, and state. The concept of state has been introduced in previous sections. An observable, such as momentum and spin, can be represented by an operator, denoted by A, in the vector space of question. An operator, or gate, acts on a ket from the left: (A) , jai ¼ A jai, and results in another ket. A linear operator (gate) B can be expressed in terms of eigenkets {ja(n)i} of an Hermitian operator A. (An operator A is said to be Hermitian if Ay ¼ A; Ay ¼ ðAT Þ .) The operator X is associated with a square matrix (albeit infinite in extent), whose elements are: D E Xmn ¼ aðmÞ X aðnÞ ; (1.52)

13

14

CHAPTER 1 Introduction

and can be explicitly written as: 0D E að1Þ X að1Þ BD E B X ¼_ B að2Þ X að1Þ @ «

D E að1Þ X að2Þ D E að2Þ X að2Þ «

/

1

C C ; /C A

(1.53)

1

where we use the notation ¼_ to denote that operator X is represented by the foregoing matrix. Very important single-qubit gates are: Hadamard gate H, phase shift gate S, p/8 (or T) gate, controlled-NOT (or CNOT) gate, and Pauli operators X, Y, Z. The Hadamard gate H, phase shift gate, T gate, and CNOT gate have the following matrix representation in CB {j0i,j1i}: 3 2 1 0 0 0       60 1 0 07 1 0 1 0 1 1 1 7 6 H ^ pffiffiffi ; S^ ; T^ 7. jp=4 ; CNOT^6 5 4 1 1 0 j 0 e 0 0 0 1 2 0 0 1 0 (1.54) The Pauli operators, on the other hand, have the following matrix representation in CB:       0 1 0 j 1 0 X^ ; Y^ ; Z^ . (1.55) 1 0 j 0 0 1 The action of Pauli gates on an arbitrary qubit jji ¼ aj0i þ bj1i is given as follows: Xðaj0i þ bj1iÞ ¼ aj1i þ bj0i; Yðaj0i þ bj1iÞ ¼ jðaj1i  bj0iÞ; Zðaj0i þ bj1iÞ ¼ aj0i  bj1i.

(1.56)

So, the action of the X gate is to introduce the bit flip, the action of the Z gate is to introduce the phase flip, and the action of the Y gate is to simultaneously introduce the bit and phase flips. Several important single-, double-, and triple-qubit gates are shown in Fig. 1.9. The action of a single-qubit gate is to apply the operator U on qubit jji, which results in another qubit. Controlled-U gate conditionally applies the operator U on target qubit jji, when the control qubit jci is in the j1i state. One particularly important controlled-U gate is the CNOT gate. This gate flips the content of the target qubit jti when the control qubit jci is in the j1i state. The purpose of SWAP gate is to interchange the positions of two qubits, and can be implemented by using three CNOT gates as shown in Fig. 1.9D. Finally, the Toffoli gate represents the

1.4 Quantum Gates and QIP

Control qubit

|cñ

|cñ

|y ñ

U

U|y ñ

U gate

(A)

U

Uc|y ñ

Controlled-U gate

2

|t Åcñ

|t ñ

|y ñ

|c1ñ |y 2ñ |c ñ

|y 1ñ

|cñ

CNOT-gate

(B)

|y 1ñ |t ñ

|y 2ñ SWAP gate

Target qubit

(C)

(D)

|t Å(c1c2)ñ Toffoli gate

(E)

FIGURE 1.9 Important quantum gates and their actions: (A) single-qubit gate, (B) controlled-U gate, (C) CNOT gate, (D) SWAP gate, and (E) Toffoli gate.

H

|y 1ñ

|EPR12ñ

|y 2ñ FIGURE 1.10 Bell states [Einstein-Podolsky-Rosen (EPR) pairs] preparation circuit.

generalization of the CNOT gate, where two control qubits are used. The minimum set of gates that can be used to perform an arbitrary quantum computation algorithm is known as universal set of gates. The most popular sets of universal quantum gates are: {H, S, CNOT, Toffoli} gates, {H, S, p/8 (T), CNOT} gates, Barenco gate [14], and Deutsch gate [15]. By using these universal quantum gates, more complicated operations can be performed. As an illustration, Fig. 1.10 shows the Bell states (EinsteinePodolskyeRosen [EPR] pairs) preparation circuit, which is of high importance in quantum teleportation and QKD applications. So far, single-, double-, and triple-qubit quantumPgates have been considered. An arbitrary quantum state of K qubits has the form s as jsi, where s runs over all binary strings of length K. Therefore there are 2K complex coefficients, all independent except for the normalization constraint: 11::11 X s¼00:::00

jas j2 ¼ 1:

(1.57)

For example, the state a00j00iþ a01j01iþ a10j10iþ a11j11i (with ja00j2 þ ja01j2 þ ja10j2 þ ja11j2 ¼ 1) is the general two-qubit state (we use j00i to denote the tensor product j0i5j0i). The multiple qubits can be entangled so that they cannot be decomposed into two separate states. For example, the Bell state or EPR pair (j00iþ j11i)/O2 cannot be written in terms of tensor product jj1i jj2i¼ (a1j0iþ b1j1i) 5 (a2j0iþ b2j1i) ¼ a1a2j00iþ a1b2j01iþ b1a2j10iþ b1b2j11i, because a1a2 ¼ b1b2 ¼ 1/O2, while a1b2 ¼ b1a2 ¼ 0, which a priori has no reason to be valid. This state can be obtained by using the circuit shown in

15

16

CHAPTER 1 Introduction

Fig. 1.10 for two-qubit input state j00i. For more details on quantum gates and algorithms, interested readers are referred to Chapters 3 and 5, respectively.

1.5 QUANTUM TELEPORTATION Quantum teleportation [16] is a technique to transfer quantum information from source to destination by employing the entangled states. Namely, in quantum teleportation, the entanglement in the Bell state (EPR pair) is used to transport arbitrary quantum state jji between two distant observers A and B (often called Alice and Bob), as illustrated in Fig. 1.11. The quantum teleportation system employs three qubits: qubit 1 is an arbitrary state to be teleported, while qubits 2 and 3 are in the Bell state jB00i¼(j00iþj11i)/O2. Let the state to be teleported be denoted by jji ¼ aj0iþbj1i. The input to the circuit shown in Fig. 1.11 is therefore jjijB00i, and can be rewritten as: pffiffiffi jjijB00 i ¼ ðaj0i þ bj1iÞðj00i þ j11iÞ= 2 (1.58) pffiffiffi ¼ ðaj000i þ aj011i þ bj100i þ bj111iÞ= 2. The CNOT gate is then applied with the first qubit serving as control and the second qubit as target, which transforms Eq. (1.58) into: pffiffiffi CNOTð12Þ jjijB00 i ¼ CNOTð12Þ ðaj000i þ aj011i þ bj100i þ bj111iÞ= 2 (1.59) pffiffiffi ¼ ðaj000i þ aj011i þ bj110i þ bj101iÞ= 2. In the next stage, the Hadamard gate is applied to the first qubit, which maps pffiffiffi pffiffiffi j0i to (j0iþj1i)/ 2 and j1i to (j0iej1i)/ 2, so that the overall transformation of Eq. (1.59) is as follows: 1 H ð1Þ CNOTð12Þ jjijB00 i ¼ ðaj000i þ aj100i þ aj011i þ aj111i 2 þ bj010i  bj110i þ bj001i  bj101iÞ.

|yñ

(1.60)

a

H b

| B00ñ=(|00ñ+|11ñ)/ Ö2

Xb FIGURE 1.11 Illustration of quantum teleportation principle.

Za

|yñ

1.6 Quantum Error Correction Concepts

The measurements are performed on qubits 1 and 2, and based on the results of measurements, denoted respectively as a and b, the controlled-X (CNOT) and controlled-Z gates are applied conditionally to lead to the following content on qubit 3: 1 ð2aj0i þ 2bj1iÞ ¼ aj0i þ bj1i ¼ jji; (1.61) 2 indicating that the arbitrary state jji is teleported to the remote destination and can be found at qubit 3 position.

1.6 QUANTUM ERROR CORRECTION CONCEPTS QIP relies on delicate superposition states, which are sensitive to interactions with the environment, resulting in decoherence. Moreover, the quantum gates are imperfect and the use of QECC is necessary to enable fault-tolerant computing and to deal with quantum errors [17e22]. QECC is also essential in quantum communication and quantum teleportation applications. The elements of quantum error correction codes are shown in Fig. 1.12A. The (N,K) QECC code performs encoding of the quantum state of K qubits, specified by 2K complex coefficients as, into a quantum state of N qubits, in such a way that errors can be detected and corrected, and all 2K complex coefficients can be perfectly restored up to the global phase shift. Namely, from quantum mechanics (see Chapter 2) we know that two states jji and ejqjji are equal up to a global phase shift as the results of measurement on both states are the same. A quantum error correction consists of four major steps: encoding, error detection, error recovery, and decoding. The sender (Alice) encodes quantum information in state jji with the help of local ancilla qubits j0i, and then sends the encoded qubits over a noisy quantum channel (say free-space optical channel or Quantum channel |y ñ

|y ñ

Error detection

Encoder Potential errors

Error recovery

Decoder

Error correction

(A) |y ñ

|Ψ ñ= a |0ñ + b |1ñ

1-p p

|Ψñ= α |0ñ + b |1ñ X |Ψñ= b |0ñ + a |1ñ

(B)

|0ñ |0ñ

(C)

FIGURE 1.12 (A) A quantum error correction principle. (B) Bit-flipping channel model. (C) Three-qubit flip-code encoder.

17

18

CHAPTER 1 Introduction

optical fiber). The receiver (Bob) performs multiqubit measurement on all qubits to diagnose the channel error and performs a recovery unitary operation R to reverse the action of the channel. The quantum error correction is essentially more complicated than classical error correction. Difficulties for quantum error correction can be summarized as follows: (1) the no-cloning theorem indicates that it is impossible to make a copy of an arbitrary quantum state, (2) quantum errors are continuous and a qubit can be in any superposition of the two bases states, and (3) the measurements destroy the quantum information. The quantum error correction principles will be more evident after the following simple example. Assume we want to send a single qubit jJi ¼ aj0i þ bj1i through the quantum channel in which during transmission the transmitted qubit can be flipped to X jJi ¼ b j0i þ a j1i with probability p. Such a quantum channel is called a bitflip channel and it can be described as shown in Fig. 1.12B. Three-qubit flip code sends the same qubit three times, and therefore represents the repetition

code

equivalent. The corresponding codewords in this code are 0 ¼ 000 ; 1 ¼ 111 . The three-qubit flip-code encoder is shown in Fig. 1.12C. One input qubit and two ancillas are used at the input of encoder, which can be represented by jj123 i ¼ aj000i þ bj100i. The first ancilla qubit (the second qubit at the encoder input) is controlled by the information qubit (the first qubit at encoder input) so that its output can be represented by CNOT12 ðaj000i þbj100iÞ ¼ aj000i þbj110i (if the control qubit is j1i the target qubit is flipped, otherwise it stays unchanged). The output of the first CNOT gate is used as input to the second CNOT gate in which the second ancilla qubit (the third qubit) is controlled by the information qubit (the first qubit) so that the corresponding encoder output is obtained as CNOT13 ðaj000i þbj110iÞ ¼ aj000i þbj111i, which indicates that basis codewords are indeed 0 and 1 . With this code, we are capable of correcting a single qubit flip error, which occurs with probability (1  p)3 þ 3p(1  p)2 ¼ 1  3p2 þ 2p3. Therefore the probability of an error remaining uncorrected or wrongly corrected with this code is 3p2  2p3. It is clear from Fig. 1.12C that a three-qubit bit-flip encoder is a systematic encoder in which the information qubit is unchanged, and the ancilla qubits are used to impose the encoding operation and create the parity qubits (the output qubits 2 and 3). Let us assume that a qubit flip occurred on the first qubit leading to the received quantum word jJri ¼ aj100i þ bj011i. To identify the error measurements need to be taken of the following observables Z1Z2 and Z2Z3, where the subscript denotes the index of qubit on which a given Pauli gate is applied. The result of measurement is the eigenvalue 1, and corresponding eigenvectors are two valid codewords, namely j000i and j111i. The observables can be represented as follows: Z1 Z2 ¼ ðj00ih11j þ j11ih11jÞ5I  ðj01ih01j þ j10ih10jÞ5I Z2 Z3 ¼ I5ðj00ih11j þ j11ih11jÞ  I5ðj01ih01j þ j10ih10jÞ

(1.62)

It can be shown that hjr jZ1 Z2 jjr i ¼ 1; hjr jZ2 Z3 jjr i ¼ þ1, indicating that an error occurred on either the first or second qubit, but not on the second or third

1.7 Quantum Key Distribution

Table 1.1 The three-qubit flip-code look-up table. Z1Z2

Z3Z4

Error

þ1 þ1 1 1

þ1 1 þ1 1

I X3 X1 X2

Z1

Z2

X1

Z2

X2

Z3

X3

|y ñ |0ñ

H

H

|0ñ

H

H

Control logic

(A)

(B)

FIGURE 1.13 (A) Three-qubit flip-code error detection and error correction circuit. (B) Decoder circuit configuration.

qubit. The intersection reveals that the first qubit was in error. By using this approach we can create the three-qubit look-up table (LUT), given as Table 1.1. Three-qubit flip-code error detection and error correction circuits are shown in Fig. 1.13. The results of measurements on ancillas (Fig. 1.13A) will determine the error syndrome [1 1], and based on the LUT gi