Quantum computing : where do we want to go tomorrow?

291 Pages·1999·23.805 MB·English

by  Samuel L. Braunstein

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
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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