*482*
*41*
*24MB*

*English*
*Pages 292
[291]*
*Year 1999*

- Author / Uploaded
- Samuel L. Braunstein

*Table of contents : Content: Quantum Computation / S. L. Braunstein --The Los Alamos Trapped Ion Quantum Computer Experiment / R. J. Hughes, D. F. V. James and J. J. Gomez / [et al.] --Experimental Primer on the Trapped Ion Quantum Computer / D. H. Weineland, C. Monroe and W. M. Itano / [et al.] --Measurement and State Prparation via Ion Trap Quantum Computing / S. Schneider, H. M. Wiseman and W. J. Munro / [et al.] --Photon-Wavepackets as Flying Quantum Bits / K. M. Gheri, K. Ellinger and T. Pillizzari / [et al.] --Quantum Logic Gate Operating on Atomic Scattering by Standing Wave Field in Bragg Regime / A. A. Khan and M. S. Zubairy --Models of Quantum Turing Machines / P. Benioff --Space, Time, Parallelism and Noise Requirements for Reliable Quantum Computing / A. M. Steane --The Quantum Hamming and Hexacodes / Th. Beth and M. Grassl --Tight Bounds on Quantum Searching / M. Boyer, G. Brassard and P. Hoyer / [et al.] --Making an Empty Promise with a Quantum Computer / H. F. Chau and H.-K. Lo --Flocks of Quantum Clones: Multiple Copying of Qubits / V. Buzek, M. Hillery and P. L. Knight --Information Gain vs. State Disturbance in Quantum Theory / Ch. A. Fuchs --On Multi-Particle Entanglement / N. Linden and S. Popescu --Generalized Coherent States and Phase-Space-Interference in Multi-Mode Systems / M. J. Gagen.*

Samuel L. Braunstein (Ed.)

Quantum Computing Where do we want to go tomorrow?

WILEY-VCH Weinheim · New York · Chichester Brisbane · Singapore · Toronto

Editor: Samuel L. Braunstein School of Electronic Engineering & Computer Systems University of Wales, Bangor Gwynedd LL57 1UT United Kingdom schmuel @ sees.bangor.ac.uk

This book was carefully produced. Nevertheless, editor, authors and publisher do not warrant the information contained therein to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details, or other items may inadvertently be inaccurate.

Library of Congress Card No.: applied for A catalogue record for this book is available from the British Library. Deutsche Bibliothek Cataloguing-in-Publication Data: Quantum computing : where do we want to go tomorrow? / Samuel L. Braunstein (ed.) - Weinheim ; New York ; Chichester ; Brisbane ; Singapore ; Toronto : Wiley-VCH, 1999 ISBN 3-527-40284-5 © WILEY-VCH Verlag GmbH, D-69469 Weinheim (Federal Republic of Germany), 1999 Printed on acid-free and chlorine-free paper. All rights reserved (including those of translation in other languages). No part of this book may be reproduced in any form - by photoprinting, microfilm, or any other means - nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printing: betz-druck gmbH, D-64291 Darmstadt. Bookbinding: J. Schaffer GmbH & Co. KG, D-67269 Grunstadt. Printed in the Federal Republic of Germany.

Preface

In 1975, Gerald Moore, a founder of Fairchild Semiconductor and Intel, formulated Moore's law which essentially states that every 18 to 24 months the number of transistors on a computer chip will double. Amazingly this prediction has continued to work for the last 25 years and is used by the semiconductor industry as its vroadmap' for the future. By about 2015 to 2020 Moore's law predicts that we will be building computers whose components operate at the atomic scale where quantum effects become dominant. The field of quantum computation and information has been mapping out what we can do at the atomic scale. Many of us have been following with excitement the stride of developments in this field. Ever since Shor's 1994 result on factoring large numbers, there have been great hopes mingled with suspicion and skepticism about the prospects of building a machine that would manipulate qubits (quantum bits). While still far from realization, prospects for an exponential speed-up of computation (at least for a limited class of problems) were soon followed by the first quantum error correction codes (Shor, 1995; and Steane, 1996). More recently, protocols have been devised which show great promise for performing quantum computations in the presence of inevitable errors (Shor, 1996). By now a new class of algorithms have been devised (Grover, 1996) allowing a square-root speed up over conventional computers for an exceedingly broad range of interesting and difficult computational questions. On the experimental front researchers, primarily in quantum and atom optics, have joined forces and built the first quantum gates (Monroe et al., 1995; Turchette et al., 1995). Multi-qubit devices, already under construction, are expected to lead to new physics and, in particular, important insight into the nature of decoherence. The subjects discussed in this book include both experimental and theoretical aspects on ion-trap quantum computers and other proposals for quantum information processing. It includes work on quantum error correction codes and their potential implications for v large' quantum computers. The optimal efficiency of Grover's algorithm is discussed here as well as cryptographic problems that are unsolvable even with a quantum computer. Other questions in the processing and representation of quantum information are studied. This book consists of a collection of articles on quantum computation and information that appeared in Fortschritte der Physik in 1998. In addition, I have included a tutorial introducing quantum computation which I first wrote in 1995 and then updated in early 1998. The field grows so rapidly that I would recommend anyone interested in keeping up to date with the latest developments to refer to the Los Alamos preprint archive (http://xxx.lanl .gov/archi ve/quant-ph) where many of the articles in this field first appear. With the publication of this book I would

VI

Preface

like to take the opportunity to encourage new authors to submit their articles to Fortschritte der Physik. Finally, I would like to thank Dr. Michael Bar at the VCH Verlag and the editors of Fortschritte der Physik for their generous support for this project. February 26, 1999

Samuel L. Braunstein

Contents

Quantum Computation S. L. Braunstein Introduction Computing at the Atomic Scale Reversible Computation Classical Universal Machines and Logic Gates 3.1 FANOUT and ERASE 3.2 Computation without ERASE 4 Elementary Quantum Notation 5 Logic Gates for Quantum Bits 6 Logic Gates in the Laboratory 7 Model Quantum Computer and Quantum Code 8 Quantum Parallelism: Period of a Sequence 9 The Complexity of Factoring 10 Security and RSA 11 Shor's Result: Factoring Numbers 12 Quantum Error Correction 13 Prospects Glossary Appendix Works Cited 1 2 3

1 2 2 3 3 4 6 6 8 9 10 12 13 14 16 17 18 19 19

The Los Alamos Trapped Ion Quantum Computer Experiment R. J. Hughes, D. F. V. James, J. J. Gomez, M. S. Gulley, Μ Η. Holzscheiter, P. G. Kwiat, S. K. Lamoreaux, C. G. Peterson, V. D. Sandberg, M. M. Schauer, C. M. Simmons, C. E, Thorburn, D. Tupa, P. Z. Wang, A. G. White 1 2 3

Abstract Introduction The Principles of quantum Computation Quantum Factoring

23 23 24 27

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4 5

6

7

Contents

Quantum computer Technologies Theory of Quantum Computation with Ions in an Linear trap 5.1 Phonon Modes 5.2 Laser-ion Interactions 5.2.1 "V" Type Operations: Single Qubit Interactions 5.2.2 "U" Type Operations: Interactions with the Quantum bus Channel 5.3 Readout 5.4 Tolerances and Laser Requirements 5.4.1 Puls Durations and Standing Waves 5.4.2 Laser Power Requirements 5.4.3 Error Rates and Fault Tolerant Quantum Computing Experimental Considerations 6.1 Choice of Ion 6.2 The radio Frequency Ion trap 6.2.1 Radial Confinement 6.2.2 Axial Confinement 6.2.3 Thermalization of trapped Ions and Noise Driven Decoherence 6.3 Laser Systems 6.4 Quibit Addressing Optics 6.5 Imaging System Summary and Conclusions Acknowledgements References

29 30 31 33 33 . . . 36 38 38 38 40 40 41 41 42 43 45 47 48 49 52 53 53 53

Experimental Primer on the Trapped Ion Quantum Computer D. H. Weineland, C. Monroe, W. M. Itano, B. E. King, D. Leibfried, D. M. Meekhof, C. Myatt, C. Wood Abstract Introduction Background A Internal states and detection Β Ion traps and motional states C Coupling between internal and motional states D Laser cooling to the motional ground state III Quantum Logic with Trapped Ions IV Packing Ions into a Trap A Individual ion addressing Β Multimode interference 1 Effects of motion in spectator modes on logic gates (Debye-Waller factors) 2 Mode cross-coupling from static electric field imperfections 3 Mode cross coupling induced by logic operations V Decoherence A Internal state decoherence from spontaneous emission Β Motional decoherence 1 Thermal or blackbody noise 2 Noise on trap voltages C Induced decoherence from applied field amplitude noise VI Conclusion Acknowledgements

I II

57 57 58 58 59 61 62 63 66 66 70 70 71 73 74 74 75 77 78 80 81 82

Contents

References

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82

Measurement and State Prparation via Ion Trap Quantum Computing S. Schneider, H. M. Wiseman, W. J. Munro, G. J. Milburn Abstract I Introduction II The Model III Superposition of Coherent States on a Circle IV Searching for a Fock State V Discussion Acknowledgments References

85 85 86 88 91 92 92 93

Photon-Wavepackets as Flying Quantum Bits K. M. Gheri, K. Ellinger, T. Pillizzari, P. Zoller Abstract I Introduction II One Photon Wave Packets III Rederivation of the Master Equation IV Example: Driven 2-Level Atom V Wave Function Simulations VI Making the Connection VII Tailoring Wave-Packets VIII Correlation Functions IX Two Photon Wavepackets A the general case Β Narrow band two-photon source C Independent sources X Conclusions Acknowledgements References

95 95 96 97 100 101 102 103 104 105 106 107 108 108 108 108

Quantum Logic Gate Operating on Atomic Scattering by Standing Wave Field in Bragg Regime A. A. Khan, M. S. Zubairy Abstract References

Ill 115

Models of Quantum Turing Machines P. Benioff Abstract

117

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I Introduction II The Physical Model III The Step Operator IV Γ as a Sum of elementary Step Operators V Distinct Path Generation A Basis Independent Description of Distinct Path Generation Β Eigenfunctions, Spectrum of Η C Sum Over Path Representation D Effective Determination of Distinct Path Generation . VI Examples A The Erasure Operator Β General Product Qubit Transformation and Add 1 C QTMS with Interferometer Graph Structures Acknowledgements References

117 119 119 121 122 123 124 125 126 126 127 128 132 135 135

Space, Time, Parallelism and Noise Requirements for Reliable Quantum Computing A. M. Steane I II III IV V VI

Abstract Choice of Method Assumptions Analysis Code Comparison Discussion Ancilla Factory References

137 138 139 141 145 148 149 151

The Quantum Hamming and Hexacodes Th. Beth , M. Grassl Abstract Introduction Binary Codes and Spin l/i Quantum Systems Modelling the quantum Channel From Pauli Matrices to GF(4) From Codes over Finite Fields back to Quantum Codes Encoding/Decoding the Quantum Hamming Code A Encoding Β Computation of the Syndrome C Error Correction VII The Quantum Hexacode A Encoding Β Computation of the Syndrome C Error Correction VIII Conclusion Acknowledgements References

I II III IV V VI

153 153 155 159 162 168 177 178 179 179 181 182 182 182 184 184 184

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Tight Bounds on Quantum Searching M. Boyer, G. Brassard, P. H0yer, A. Tapp 1 2 3 4 5 6 7 8

Abstract Introduction Overview of Grover's Algorithm Finding a Unique Solution The Case of Multiple Solutions The Case t = N/4 Unknown Number of Solutions An Improved Lower Bound Conclusions and Future Directions Acknowledgements References

187 187 188 189 190 191 192 194 198 199 199

Making an Empty Promise with a Quantum Computer H. F. Chau, H.-K. Lo Abstract Introduction Bit Commitment - from the Ancient to the Post-Modern World A Bit Commitment in the Ancient World Β Bit Commitment in the Modern World C Bit Commitment in the Post-Modern World III Insecurity of Quantum Bit Commitment A General Form of a Quantum Bit Commitment Scheme Β Unitary Description C Generality of the above Description D Schmidt Decomposition Ε Alice's Cheating Strategy IV Concluding Remarks A Secure Computations Β Security Analysis of composite Quantum Protocols C Lessons We Learn Acknowledgements References

I II

201 201 202 202 202 203 204 204 205 205 206 207 210 210 211 211 212 212

Flocks of Quantum Clones: Multiple Copying of Qubits V. Bub'k, M. Hillery, P. L Knight Abstract I Introduction II Universal Quantum copying Machine III Copying Network A Preparation of quantum copier Β Quantum copying IV Multiple copying A Preparation of the quantum copier Β Copying of information V Properties of copied Qubits

215 215 216 218 219 220 221 222 223 224

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Conclusions Acknowledgements References

227 227 227

Information Gain vs. State Disturbance in Quantum Theory Ch. A. Fuchs Abstract Introduction The Model A Evolutions Β Measurements C Information and Distinguishability D Disturbance Measures Ε Tradeoff III Historical Context IV Pure States A The Probe and Interaction Β After the Interaction C The Tradeoff Curve V Mixed States VI Foundations VII Appendix: Error Probability Acknowledgements References

I II

229 229 230 231 232 233 236 239 240 242 243 248 250 252 253 256 257 257

On Multi-Particle Entanglement N. Linden, S. Popescu 1 2

3. 4. 5.

Abstract Introduction The Number of Parameters Needed to Describe Inequivalent States 2.1 Dimension of a General Orbit 2.2 A single spin 2.3 Two spins 2.4 Three spins Invariants 3.1 Examples 3.2 General case Orbit Types conclusion Acknowledgements References

261 261 263 264 264 265 266 267 267 268 269 271 271 272

Generalized Coherent States and Phase-Space-Interference in Multi-Mode Systems M. J. Gag en I

Abstract Introduction

273 273

Contents II III IV V VI VII

Single Mode Quasi-Probability Distributions Multi-Mode Quasi-Probability Distributions Two Photon Coherent States Visualizing Multi-Mode Phase Volumes PSI Approaches in Multi-Mode Systems Conclusion Acknowledgements References

Index

XIII 274 275 276 277 278 282 282 282 283

Quantum Computation SAMUEL L. BRAUNSTEIN SEECS, University of Wales, Bangor, United Kingdom

Introduction A quantum computer is a device that can arbitrarily manipulate the quantum state of a part of itself. The field of quantum computation is largely a body of theoretical promises for some impressively fast algorithms that could be executed on quantum computers. However, since the first significant algorithm was proposed in 1994 (Shor, 1994) experimental progress has been rapid with several schemes yielding two- (Turchette et