Quantum computing : where do we want to go tomorrow? PDF
Preview Quantum computing : where do we want to go tomorrow?
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 1 1 Computing at the Atomic Scale 2 2 Reversible Computation 2 3 Classical Universal Machines and Logic Gates 3 3.1 FANOUT and ERASE 3 3.2 Computation without ERASE 4 4 Elementary Quantum Notation 6 5 Logic Gates for Quantum Bits 6 6 Logic Gates in the Laboratory 8 7 Model Quantum Computer and Quantum Code 9 8 Quantum Parallelism: Period of a Sequence 10 9 The Complexity of Factoring 12 10 Security and RSA 13 11 Shor's Result: Factoring Numbers 14 12 Quantum Error Correction 16 13 Prospects 17 Glossary 18 Appendix 19 Works Cited 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 Abstract 23 1 Introduction 23 2 The Principles of quantum Computation 24 3 Quantum Factoring 27 VIII Contents 4 Quantum computer Technologies 29 5 Theory of Quantum Computation with Ions in an Linear trap 30 5.1 Phonon Modes 31 5.2 Laser-ion Interactions 33 5.2.1 "V" Type Operations: Single Qubit Interactions 33 5.2.2 "U" Type Operations: Interactions with the Quantum bus Channel . . . 36 5.3 Readout 38 5.4 Tolerances and Laser Requirements 38 5.4.1 Puls Durations and Standing Waves 38 5.4.2 Laser Power Requirements 40 5.4.3 Error Rates and Fault Tolerant Quantum Computing 40 6 Experimental Considerations 41 6.1 Choice of Ion 41 6.2 The radio Frequency Ion trap 42 6.2.1 Radial Confinement 43 6.2.2 Axial Confinement 45 6.2.3 Thermalization of trapped Ions and Noise Driven Decoherence 47 6.3 Laser Systems 48 6.4 Quibit Addressing Optics 49 6.5 Imaging System 52 7 Summary and Conclusions 53 Acknowledgements 53 References 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 57 I Introduction 57 II Background 58 A Internal states and detection 58 Β Ion traps and motional states 59 C Coupling between internal and motional states 61 D Laser cooling to the motional ground state 62 III Quantum Logic with Trapped Ions 63 IV Packing Ions into a Trap 66 A Individual ion addressing 66 Β Multimode interference 70 1 Effects of motion in spectator modes on logic gates (Debye-Waller factors) 70 2 Mode cross-coupling from static electric field imperfections 71 3 Mode cross coupling induced by logic operations 73 V Decoherence 74 A Internal state decoherence from spontaneous emission 74 Β Motional decoherence 75 1 Thermal or blackbody noise 77 2 Noise on trap voltages 78 C Induced decoherence from applied field amplitude noise 80 VI Conclusion 81 Acknowledgements 82 Contents IX References 82 Measurement and State Prparation via Ion Trap Quantum Computing S. Schneider, H. M. Wiseman, W. J. Munro, G. J. Milburn Abstract 85 I Introduction 85 II The Model 86 III Superposition of Coherent States on a Circle 88 IV Searching for a Fock State 91 V Discussion 92 Acknowledgments 92 References 93 Photon-Wavepackets as Flying Quantum Bits K. M. Gheri, K. Ellinger, T. Pillizzari, P. Zoller Abstract 95 I Introduction 95 II One Photon Wave Packets 96 III Rederivation of the Master Equation 97 IV Example: Driven 2-Level Atom 100 V Wave Function Simulations 101 VI Making the Connection 102 VII Tailoring Wave-Packets 103 VIII Correlation Functions 104 IX Two Photon Wavepackets 105 A the general case 106 Β Narrow band two-photon source 107 C Independent sources 108 X Conclusions 108 Acknowledgements 108 References 108 Quantum Logic Gate Operating on Atomic Scattering by Standing Wave Field in Bragg Regime A. A. Khan, M. S. Zubairy Abstract Ill References 115 Models of Quantum Turing Machines P. Benioff Abstract 117 X Contents I Introduction 117 II The Physical Model 119 III The Step Operator 119 IV Γ as a Sum of elementary Step Operators 121 V Distinct Path Generation 122 A Basis Independent Description of Distinct Path Generation 123 Β Eigenfunctions, Spectrum of Η 124 C Sum Over Path Representation 125 D Effective Determination of Distinct Path Generation . 126 VI Examples 126 A The Erasure Operator 127 Β General Product Qubit Transformation and Add 1 128 C QTMS with Interferometer Graph Structures 132 Acknowledgements 135 References 135 Space, Time, Parallelism and Noise Requirements for Reliable Quantum Computing A. M. Steane Abstract 137 I Choice of Method 138 II Assumptions 139 III Analysis 141 IV Code Comparison 145 V Discussion 148 VI Ancilla Factory 149 References 151 The Quantum Hamming and Hexacodes Th. Beth , M. Grassl Abstract 153 I Introduction 153 II Binary Codes and Spin l/i Quantum Systems 155 III Modelling the quantum Channel 159 IV From Pauli Matrices to GF(4) 162 V From Codes over Finite Fields back to Quantum Codes 168 VI Encoding/Decoding the Quantum Hamming Code 177 A Encoding 178 Β Computation of the Syndrome 179 C Error Correction 179 VII The Quantum Hexacode 181 A Encoding 182 Β Computation of the Syndrome 182 C Error Correction 182 VIII Conclusion 184 Acknowledgements 184 References 184 Contents XI Tight Bounds on Quantum Searching M. Boyer, G. Brassard, P. H0yer, A. Tapp Abstract 187 1 Introduction 187 2 Overview of Grover's Algorithm 188 3 Finding a Unique Solution 189 4 The Case of Multiple Solutions 190 5 The Case t = N/4 191 6 Unknown Number of Solutions 192 7 An Improved Lower Bound 194 8 Conclusions and Future Directions 198 Acknowledgements 199 References 199 Making an Empty Promise with a Quantum Computer H. F. Chau, H.-K. Lo Abstract 201 I Introduction 201 II Bit Commitment - from the Ancient to the Post-Modern World 202 A Bit Commitment in the Ancient World 202 Β Bit Commitment in the Modern World 202 C Bit Commitment in the Post-Modern World 203 III Insecurity of Quantum Bit Commitment 204 A General Form of a Quantum Bit Commitment Scheme 204 Β Unitary Description 205 C Generality of the above Description 205 D Schmidt Decomposition 206 Ε Alice's Cheating Strategy 207 IV Concluding Remarks 210 A Secure Computations 210 Β Security Analysis of composite Quantum Protocols 211 C Lessons We Learn 211 Acknowledgements 212 References 212 Flocks of Quantum Clones: Multiple Copying of Qubits V. Bub'k, M. Hillery, P. L Knight Abstract 215 I Introduction 215 II Universal Quantum copying Machine 216 III Copying Network 218 A Preparation of quantum copier 219 Β Quantum copying 220 IV Multiple copying 221 A Preparation of the quantum copier 222 Β Copying of information 223 V Properties of copied Qubits 224 XII Contents VI Conclusions 227 Acknowledgements 227 References 227 Information Gain vs. State Disturbance in Quantum Theory Ch. A. Fuchs Abstract 229 I Introduction 229 II The Model 230 A Evolutions 231 Β Measurements 232 C Information and Distinguishability 233 D Disturbance Measures 236 Ε Tradeoff 239 III Historical Context 240 IV Pure States 242 A The Probe and Interaction 243 Β After the Interaction 248 C The Tradeoff Curve 250 V Mixed States 252 VI Foundations 253 VII Appendix: Error Probability 256 Acknowledgements 257 References 257 On Multi-Particle Entanglement N. Linden, S. Popescu Abstract 261 1 Introduction 261 2 The Number of Parameters Needed to Describe Inequivalent States 263 2.1 Dimension of a General Orbit 264 2.2 A single spin 264 2.3 Two spins 265 2.4 Three spins 266 3. Invariants 267 3.1 Examples 267 3.2 General case 268 4. Orbit Types 269 5. conclusion 271 Acknowledgements 271 References 272 Generalized Coherent States and Phase-Space-Interference in Multi-Mode Systems M. J. Gag en Abstract 273 I Introduction 273
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