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Anonymous
Posted October 2, 2005
In planning a course on quantum computing, an instructor would want to cover the significant highpoints in the subject: Shor¿s factoring algorithm, Grover¿s search algorithm, Deutsch¿s problem, the hidden subgroup problem. I for one found that this book does precisely that. Students will want an accessible and attractive presentation. This book is beautifully presented, nicely organized, and pedagogically presented with motivation, clear explanations, and well chosen exercises. While the subject has a variety of facets, physics, math, computer science, this book emphasizes the last two. In a highly interdisciplinary subject, each author (or team of authors) must make selections. In selecting what to cover, the authors had the classroom and students in mind. More precisely the subject here is presented in the form of quantum gates, channels, and circuits. Yet, quantum physics and the foundations are not neglected. The graphic presentation (figures and diagrams) is done in a way to aid learning, and I expect that this book will be the preferred text in courses in the subject for some time to come. Advanced undergraduates will be able to follow the logical progression of subjects. Several special features in the book help: Exercises, an extensive and instructive glossary, historical insight, motivation, appendices (including key math topics, e.g., modular arithmetic and Hadamard transforms which perhaps may not be widely known), and circuit diagrams illustrating at the same time matrix factorization and the complexity of circuits. Contents: 1. History and background, 2. Rudiments of quantum physics as it is needed, 3. Qubits and computer science rewritten in the form of quantum gates, 4. The key quantum algorithms (Shor, Grover, Simon) and the highpoints in the subject, 5. Entanglement, decoherence, error-correcting codes, Bell, dense coding, EPR, reversible computation, thermodynamic entropy, and more. Highly recommended!
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More About This Textbook
Overview
With a clear writing style and matter-of-fact approach, this rigorous yet accessible introduction to quantum computing is designed for readers with a solid mathematical background but limited knowledge of physics and quantum mechanics. Using a methodical approach and an abundance of worked examples, this handbook delivers a thorough introduction to the quantum circuit model, including the mathematical formalism required for quantum computing. Concentrates on the quantum circuit model to make complex subject matter more accessible. Provides a phenomenological introduction to quantum computing, encouraging readers to view the subject as a fundamentally new approach to computing. Detailed presentation of quantum algorithms demonstrates the logic behind the development of Deutsch’s problem, quantum Fourier transform, Shor’s factoring algorithm, Simon’s algorithm for phase estimation, and discrete logarithms evaluation problems. For anyone interested in learning more about quantum computing.
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Read an Excerpt
RATIONALE
Tremendous progress has been made in the areas of quantum computing and quantum information theory during the past decade. Thousands of research papers, a few solid reference books, and many popular science books have been published in recent years on this subject. The growing interest in quantum computing and quantum information theory is motivated by the incredible impact these disciplines are expected to have on how we store, process, and transmit data and knowledge in this information age.
Computer and communication systems using quantum effects have remarkable properties. Quantum computers enable efficient simulation of the most complex physical systems we can envision. Quantum algorithms allow efficient factoring of large integers with applications to cryptography. Quantum search algorithms considerably speed up the process of identifying patterns in apparently random data. We can guarantee the security of our quantum communication systems because eavesdropping on a quantum communication channel can be detected with high probability.
It is true that we are years, possibly decades, away from building a quantum computer requiring little, if any power at all, filling up the space of a grain of sand, and computing at speeds that are unattainable today even by covering tens of acres of floor space with clusters made from tens of thousands of the fastest processors built with current state-of-the-art solid-state technology. All we have at the time of this writing is a 7-qubit quantum computer capable of computing the prime factors of a small integer, 15 139. To break a code with a key size of 1024 bits requires more than 3000 qubits and 108 quantum gates 82. Building a quantum computer presents tremendous technological and theoretical challenges. At the same time, we are witnessing a faster rate of progress in quantum information theory than in quantum computing. Applications of quantum cryptography seem ready for commercialization. Recently, a successful quantum key distribution experiment over a distance of some 100 km has been announced.
It is difficult to predict how much time will elapse from the moment of a great discovery until it materializes into a device that profoundly changes our lives or drastically affects our understanding of natural phenomena. The first atomic bomb was detonated in 1945, less than 10 years after the discovery of nuclear fission by Lise Meitner and Otto Hahn 91. The first microprocessor was built in late 1970s, some 30 years after the creation of the transistor on December 23, 1947 by William Shockley, John Bardeen, and Walter Brattain. Francis Harry Compton Crick and James Dewey Watson discovered the double helix structure of the genetic material in 1957 and the full impact of their discovery will continue to reverberate for years to come.
The authors believe that the time to spread the knowledge about quantum computing and quantum information outside the circle of quantum computing researchers and students majoring in physics is ripe. Students and professionals interested in information sciences need to adopt a different way of thinking than the one used to construct today's algorithms. This certainly presents tremendous challenges, since, for many years, computer science students have been led to believe that they can get by with some knowledge of discrete mathematics and little understanding of physics at all. We are going back to the age when a strong relationship between physics and computer science existed.
TOPICS, PREREQUISITES, AND CHAPTER DESCRIPTIONS
This text is devoted to quantum computing. We treat the quantum computer as a mathematical abstraction. Yet, we discuss in some depth the fundamental properties of a quantum system necessary to understand the subtleties of counterintuitive quantum phenomena such as entanglement. Chapter 1 introduces the reader to the quantum world by way of several experiments. Chapter 2 provides the most basic concepts of quantum mechanics and of the supporting mathematical apparatus. Chapter 3 introduces the qubit and hints at simple physical realizations of a qubit. Chapter 4 is devoted to quantum gates and quantum circuits. Chapter 5 presents quantum algorithms. The last chapter, Chapter 6, introduces the reader to quantum teleportation, quantum key distribution, and dense coding, and then presents reversible computations. The text is intended to be self-contained; concepts, definitions, and theorems from linear algebra necessary to develop the mathematical apparatus of quantum mechanics are introduced in Chapter 2. Appendix A introduces basic algebraic structures. Appendix B presents modular arithmetic necessary for understanding the factoring algorithms. Appendix C is devoted to the Walsh-Hadamard transform, and Appendix D summarizes basic concepts related to the Fourier transform. Approaching Quantum Computing is intended as a textbook for a one-semester first course in quantum computing. The time table we suggest for covering the material is: five weeks for Chapters 1 and 2, five weeks for Chapters 3 and 4, four weeks for Chapter 5 and Appendices A, B, C and D, and two weeks for Chapter 6. Any graduate or undergraduate student with a solid background in linear algebra, calculus, and physics should be able to do well in the class.
Features
This volume combines a qualitative presentation with a more rigorous, quantitative analysis. Whenever possible, we attempt to avoid the sometimes difficult mathematical apparatus, the trademark of quantum mechanics. In his marvelous book A Brief History of Time 71, Stephen Hawking, the astrophysicist, who is now the Lucasian professor, shares with his readers the warning he got from his editor: "Expect the sales to be cut in half for every equation in your book." There are k° 102 equations in this series of lectures and 2100 ~ 100010 is a very large number. Detailed presentation and step-by-step analysis to illustrate the behavior of quantum circuits are given, along with numerous examples that will guide the reader in solving the problems at the end of each chapter. A solutions manual for instructors who adopt the book is available through the publisher.
Table of Contents
1 Preface
2 Introduction
2.1 Computing and the Laws of Physics
2.2 Quantum Information
2.3 Quantum Computers
2.4 The Wave and the Corpuscular Nature of Light
2.5 Deterministic versus Probabilistic Photon Behavior
2.6 State Description, Superposition, and Uncertainty
2.7 Measurements in Multiple Bases
2.8 Measurements of Superposition States
2.9 An Augmented Probabilistic Model. The Superposition Probability Rule.
2.10 A Photon Coincidence Experiment
2.11 A Three Beam Splitter Experiment
2.12 BB84, the Emergence of Quantum Cryptography
2.13 A Qubit of History
2.14 Summary and Further Readings
2.15Exercises and Problems
3 Quantum Mechanics, a Mathematical Model of the Physical World
3.1 Vector Spaces
3.2 n-Dimensional Real Euclidean Vector Space
3.3 Linear Operators and Matrices
3.4 Hermitian Operators in a Complex n -Dimensional Euclidean Vector Space
3.5 n -Dimensional Hilbert Spaces. Dirac Notations
3.6 The Inner Product in an n -Dimensional Hilbert Space
3.7 Tensor and Outer Products
3.8 Quantum States
3.9 Quantum Observables. Quantum Operators
3.10 Spectral Decomposition of a Quantum Operator
3.11 The Measurement of Observables
3.12 More about Measurements. The Density Operator
3.13 Double-Slit Experiments
3.14 Stern-Gerlach Type Experiments
3.15 The Spin as an Intrinsic Property
3.16 SchrodingerÕs Wave Equation
3.17 HeisenbergÕs Uncertainty Principle
3.18 A Brief History of Quantum Ideas
3.19 Summary and Further Readings
3.20 Exercises and Problems
4 Qubits and Their Physical Realization
4.1 One Qubit, a Very Small Bit
4.2 The Bloch Sphere Representation of One Qubit
4.3 Rotation Operations on the Bloch Sphere
4.4 The Measurement of a Single Qubit
4.5 Pure and Impure States of a Qubit
4.6 A Pair of Qubits. Entanglement
4.7 The Fragility of Quantum Information. SchrodingerÕs Cat
4.8 Qubits: from Hilbert Spaces to Physical Implementation
4.9 Qubits as Spin One-Half Particles
4.10 The Measurement of the Spin
4.11 The Qubit as a Polarized Photon
4.12 Entanglement
4.13 The Exchange of Information Using Entangled Particles
4.14 Summary and Further Readings
4.15 Exercises and Problems
5 Quantum Gates and Quantum Circuits
5.1 Classical Logic Gates and Circuits
5.2 One-Qubit Gates
5.3 The Hadamard Gate, Beam Splitters and Interferometers
5.4 Two-Qubit Gates. The CNOT Gate
5.5 Can We Build Quantum Copy Machines?
5.6 Three-Qubit Gates. The Fredkin Gate
5.7 The Toffoli Gate
5.8 Quantum Circuits
5.9 The No Cloning Theorem
5.10 Qubit Swapping and Full Adder Circuits
5.11 More about Unitary Operations and Rotation Matrices
5.12 Single-Qubit Controlled Operations
5.13 Multiple Qubit Controlled Operations
5.14 Universal Quantum Gates
5.15 A Quantum Circuit for the Walsh-Hadamard Transform
5.16 The State Transformation Performed by Quantum Circuits
5.17 Mathematical Models of a Quantum Computer
5.18 Errors, Uniformity Conditions, and Time Complexity
5.19 Summary and Further Readings
5.20 Exercises and Problems
6 Quantum Algorithms
6.1 From Classical to Quantum Turing Machines
6.2 Computational Complexity and Entanglement
6.3 Classes of Quantum Algorithms
6.4 Quantum Parallelism
6.5 DeutschÕs Problem
6.6 Quantum Fourier Transform
6.7 Tensor Product Factorization
6.8 A Circuit for Quantum Fourier Transform
6.9 A Case Study: A Three-Qubit QFT
6.10 ShorÕs Factoring Algorithm and Order Finding
6.11 A Quantum Circuit for Computing f(x)Modulo 2
6.12 SimonÕs Algorithm for Phase Estimation
6.13 The Fourier Transform on an Abelian Group
6.14 Periodicity and the Quantum Fourier Transform
6.15 The Discrete Logarithms Evaluation Problem
6.16 The Hidden Subgroup Problem
6.17 Quantum Search Algorithms
6.18 Historical Notes
6.19 Summary and Further Readings
6.20 Exercises and Problems
7 The "Entanglement" of Computing and Communication with Quantum Mechanics. Reversible Computations
7.1 Communication, Entropy, and Quantum Information
7.2 Information Encoding
7.3 Quantum Teleportation with Maximally Entangled Particles
7.4 Anti-Correlation and Teleportation
7.5 Dense Coding
7.6 Quantum Key Distribution
7.7 EPR Pairs and Bell States
7.8 Uncertainty and Locality
7.9 Possible Explanations of the EPR Experiment
7.10 The Bell Inequality. Local Realism
7.11 Reversibility and Entropy
7.12 Thermodynamics and Thermodynamic Entropy
7.13 The Maxwell Demon
7.14 Energy Consumption. Landauer Principle
7.15 Low Power Computing. Adiabatic Switching
7.16 Bennett Information Driven Engine
7.17 Logically Reversible Turing Machines and Physical Reversibility
7.18 Historical Notes
7.19 Summary and Further Readings 297
7.20 Exercises and Problems 299
8 Appendix I: Algebraic Structures
8.1 Rings, Commutative Rings, Integral Domains, Fields
8.2 Complex Numbers
8.3 Abstract Groups and Isomorphisms
8.4 Matrix Representation
8.5 Groups of Transformations
8.6 Symmetry in a Plane
8.7 Finite Fields
9 Appendix II: Modular Arithmetic
9.1 Elementary Number Theory Concepts
9.2 EuclidÕs Algorithm for Integers
9.3 The Chinese Remainder Theorem and its Applications
9.4 Computer Arithmetic for Large Integers
10 Appendix III: Welsh-Hadamard Transform
10.1 Hadamard Matrices
10.2 The Fast Hadamard Transform
11 Appendix IV: Fourier Transform and Fourier Series
12 Glossary
Preface
RATIONALE
Tremendous progress has been made in the areas of quantum computing and quantum information theory during the past decade. Thousands of research papers, a few solid reference books, and many popular science books have been published in recent years on this subject. The growing interest in quantum computing and quantum information theory is motivated by the incredible impact these disciplines are expected to have on how we store, process, and transmit data and knowledge in this information age.
Computer and communication systems using quantum effects have remarkable properties. Quantum computers enable efficient simulation of the most complex physical systems we can envision. Quantum algorithms allow efficient factoring of large integers with applications to cryptography. Quantum search algorithms considerably speed up the process of identifying patterns in apparently random data. We can guarantee the security of our quantum communication systems because eavesdropping on a quantum communication channel can be detected with high probability.
It is true that we are years, possibly decades, away from building a quantum computer requiring little, if any power at all, filling up the space of a grain of sand, and computing at speeds that are unattainable today even by covering tens of acres of floor space with clusters made from tens of thousands of the fastest processors built with current state-of-the-art solid-state technology. All we have at the time of this writing is a 7-qubit quantum computer capable of computing the prime factors of a small integer, 15 139. To break a code with a key size of 1024 bits requires more than 3000 qubits and 108 quantum gates 82. Building a quantum computer presents tremendous technological and theoretical challenges. At the same time, we are witnessing a faster rate of progress in quantum information theory than in quantum computing. Applications of quantum cryptography seem ready for commercialization. Recently, a successful quantum key distribution experiment over a distance of some 100 km has been announced.
It is difficult to predict how much time will elapse from the moment of a great discovery until it materializes into a device that profoundly changes our lives or drastically affects our understanding of natural phenomena. The first atomic bomb was detonated in 1945, less than 10 years after the discovery of nuclear fission by Lise Meitner and Otto Hahn 91. The first microprocessor was built in late 1970s, some 30 years after the creation of the transistor on December 23, 1947 by William Shockley, John Bardeen, and Walter Brattain. Francis Harry Compton Crick and James Dewey Watson discovered the double helix structure of the genetic material in 1957 and the full impact of their discovery will continue to reverberate for years to come.
The authors believe that the time to spread the knowledge about quantum computing and quantum information outside the circle of quantum computing researchers and students majoring in physics is ripe. Students and professionals interested in information sciences need to adopt a different way of thinking than the one used to construct today's algorithms. This certainly presents tremendous challenges, since, for many years, computer science students have been led to believe that they can get by with some knowledge of discrete mathematics and little understanding of physics at all. We are going back to the age when a strong relationship between physics and computer science existed.
TOPICS, PREREQUISITES, AND CHAPTER DESCRIPTIONS
This text is devoted to quantum computing. We treat the quantum computer as a mathematical abstraction. Yet, we discuss in some depth the fundamental properties of a quantum system necessary to understand the subtleties of counterintuitive quantum phenomena such as entanglement. Chapter 1 introduces the reader to the quantum world by way of several experiments. Chapter 2 provides the most basic concepts of quantum mechanics and of the supporting mathematical apparatus. Chapter 3 introduces the qubit and hints at simple physical realizations of a qubit. Chapter 4 is devoted to quantum gates and quantum circuits. Chapter 5 presents quantum algorithms. The last chapter, Chapter 6, introduces the reader to quantum teleportation, quantum key distribution, and dense coding, and then presents reversible computations. The text is intended to be self-contained; concepts, definitions, and theorems from linear algebra necessary to develop the mathematical apparatus of quantum mechanics are introduced in Chapter 2. Appendix A introduces basic algebraic structures. Appendix B presents modular arithmetic necessary for understanding the factoring algorithms. Appendix C is devoted to the Walsh-Hadamard transform, and Appendix D summarizes basic concepts related to the Fourier transform. Approaching Quantum Computing is intended as a textbook for a one-semester first course in quantum computing. The time table we suggest for covering the material is: five weeks for Chapters 1 and 2, five weeks for Chapters 3 and 4, four weeks for Chapter 5 and Appendices A, B, C and D, and two weeks for Chapter 6. Any graduate or undergraduate student with a solid background in linear algebra, calculus, and physics should be able to do well in the class.
Features
This volume combines a qualitative presentation with a more rigorous, quantitative analysis. Whenever possible, we attempt to avoid the sometimes difficult mathematical apparatus, the trademark of quantum mechanics. In his marvelous book A Brief History of Time 71, Stephen Hawking, the astrophysicist, who is now the Lucasian professor, shares with his readers the warning he got from his editor: "Expect the sales to be cut in half for every equation in your book." There are k° 102 equations in this series of lectures and 2100 ~ 100010 is a very large number. Detailed presentation and step-by-step analysis to illustrate the behavior of quantum circuits are given, along with numerous examples that will guide the reader in solving the problems at the end of each chapter. A solutions manual for instructors who adopt the book is available through the publisher.