An Introduction to Quantum Computing / Edition 1

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This concise, accessible text provides a thorough introduction to quantum computing - an exciting emergent field at the interface of the computer, engineering, mathematical and physical sciences. Aimed at advanced undergraduate and beginning graduate students in these disciplines, the text is technically detailed and is clearly illustrated throughout with diagrams and exercises. Some prior knowledge of linear algebra is assumed, including vector spaces and inner products. However, prior familiarity with topics such as tensor products and spectral decomposition is not required, as the necessary material is reviewed in the text.

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Editorial Reviews

From the Publisher
"The book is very accessible and the authors do an excellent job breaking up Shor's factoring algorithm into pieces that students can easily digest." Jonathan R. Friedman, Physics Today

"A reasonably brief and very accessible introductory graduate or senior undergraduate textbook."—Mathematical Reviews

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Product Details

  • ISBN-13: 9780198570493
  • Publisher: Oxford University Press, USA
  • Publication date: 1/28/2007
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 288
  • Product dimensions: 9.10 (w) x 6.10 (h) x 0.60 (d)

Meet the Author

Phillip Ronald Kaye was born in Toronto, and raised in Waterloo, Ontario, Canada. In 1995 Phil was accepted to the Faculty of Engineering at the University of Waterloo with an entrance scholarship. He completed his undergraduate degree in Systems Design Engineering in 2000 and was awarded the George Dufault Medal for Excellence in Communication at his convocation. During the Summer months following his undergraduate convocation, Phil worked as an encryption software developer at Research in Motion (RIM), where he continued to work on a part-time basis during his graduate studies. Phil did his Master's degree in the department of Combinatorics and Optimization at Waterloo. His Master's thesis was entitled 'Quantum Networks for Concentrating Entanglement, and a Logical Characterization of the Computational Complexity Class BPP.' Phil is currently a PhD student at the School of Computer Science at the University of Waterloo. Raymond Laflamme completed his undergraduate studies in Physics at Université Laval. He then moved to Cambridge, UK, where he took Part III of the Mathematical Tripos before doing a PhD in the Department of Applied Mathematics and Theoretical Physics (DAMTP) under the direction of Professor Stephen Hawking. Following posts at UBC, Cambridge and Los Alamos National Laboratory, Raymond moved to the University of Waterloo in 2001 as a Canada Research Chair in Quantum Information. Raymond is a recipient of Ontario's Premier Research Award and a Director of the Quantum Information program of the Canadian Institute for Advanced Research. He was named the Ivey Foundation Fellow of the Canadian Institute for Advanced Research (CIAR) in September of 2005. Michele Mosca obtained a DPhil in quantum computer algorithms in 1999 at the University of Oxford. Since then he has been a faculty member in Mathematics at St. Jerome's University and in the Combinatorics and Optimization department of the Faculty of Mathematics, University of Waterloo, and a member of the Centre for Applied Cryptographic Research. He holds a Premier's Research Excellence Award (2000-2005), is the Canada Research Chair in Quantum Computation (since January 2002), and is a CIAR scholar (since September 2003). He is a co-founder and the Deputy Director of the Institute for Quantum Computing, and a founding member of the Perimeter Institute for Theoretical Physics.

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Table of Contents

Preface     x
Acknowledgements     xi
Introduction and Background     1
Overview     1
Computers and the Strong Church-Turing Thesis     2
The Circuit Model of Computation     6
A Linear Algebra Formulation of the Circuit Model     8
Reversible Computation     12
A Preview of Quantum Physics     15
Quantum Physics and Computation     19
Linear Algebra and the Dirac Notation     21
The Dirac Notation and Hilbert Spaces     21
Dual Vectors     23
Operators     27
The Spectral Theorem     30
Functions of Operators     32
Tensor Products     33
The Schmidt Decomposition Theorem     35
Some Comments on the Dirac Notation     37
Qubits and the Framework of Quantum Mechanics     38
The State of a Quantum System     38
Time-Evolution of a Closed System     43
Composite Systems     45
Measurement     48
Mixed States and General Quantum Operations     53
Mixed States     53
Partial Trace     56
General Quantum Operations     59
A Quantum Model of Computation     61
The Quantum Circuit Model     61
Quantum Gates     63
1-Qubit Gates     63
Controlled-U Gates     66
Universal Sets of Quantum Gates     68
Efficiency of Approximating Unitary Transformations     71
Implementing Measurements with Quantum Circuits     73
Superdense Coding and Quantum Teleportation     78
Superdense Coding     79
Quantum Teleportation     80
An Application of Quantum Teleportation     82
Introductory Quantum Algorithms
Probabilistic Versus Quantum Algorithms     86
Phase Kick-Back     91
The Deutsch Algorithm     94
The Deutsch-Jozsa Algorithm     99
Simon's Algorithm     103
Algorithms with Superpolynomial Speed-Up     110
Quantum Phase Estimation and the Quantum Fourier Transform     110
Error Analysis for Estimating Arbitrary Phases     117
Periodic States     120
GCD, LCM, the Extended Euclidean Algorithm     124
Eigenvalue Estimation     125
Finding-Orders     130
The Order-Finding Problem     130
Some Mathematical Preliminaries      131
The Eigenvalue Estimation Approach to Order Finding     134
Shor's Approach to Order Finding     139
Finding Discrete Logarithms     142
Hidden Subgroups     146
More on Quantum Fourier Transforms     147
Algorithm for the Finite Abelian Hidden Subgroup Problem     149
Related Algorithms and Techniques     151
Algorithms Based on Amplitude Amplification     152
Grover's Quantum Search Algorithm     152
Amplitude Amplification     163
Quantum Amplitude Estimation and Quantum Counting     170
Searching Without Knowing the Success Probability     175
Related Algorithms and Techniques     178
Quantum Computational Complexity Theory and Lower Bounds     179
Computational Complexity     180
Language Recognition Problems and Complexity Classes     181
The Black-Box Model     185
State Distinguishability     187
Lower Bounds for Searching in the Black-Box Model: Hybrid Method     188
General Black-Box Lower Bounds     191
Polynomial Method     193
Applications to Lower Bounds     194
Examples of Polynomial Method Lower Bounds     196
Block Sensitivity     197
Examples of Block Sensitivity Lower Bounds     197
Adversary Methods     198
Examples of Adversary Lower Bounds     200
Generalizations     203
Quantum Error Correction     204
Classical Error Correction     204
The Error Model     205
Encoding     206
Error Recovery     207
The Classical Three-Bit Code     207
Fault Tolerance     211
Quantum Error Correction     212
Error Models for Quantum Computing     213
Encoding     216
Error Recovery     217
Three- and Nine-Qubit Quantum Codes     223
The Three-Qubit Code for Bit-Flip Errors     223
The Three-Qubit Code for Phase-Flip Errors     225
Quantum Error Correction Without Decoding     226
The Nine-Qubit Shor Code     230
Fault-Tolerant Quantum Computation     234
Concatenation of Codes and the Threshold Theorem     237
Appendix A     241
Tools for Analysing Probabilistic Algorithms     241
Solving the Discrete Logarithm Problem When the Order of a Is Composite     243
How Many Random Samples Are Needed to Generate a Group?     245
Finding r Given k/r for Random k     247
Adversary Method Lemma     248
Black-Boxes for Group Computations     250
Computing Schmidt Decompositions     253
General Measurements     255
Optimal Distinguishing of Two States     258
A Simple Procedure     258
Optimality of This Simple Procedure     258
Bibliography     260
Index     270
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