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Book DetailsGraduate Studies in MathematicsVolume: 47; 2002; 257 ppMSC: Primary 68; 81
This book is an introduction to a new rapidly developing theory of quantum computing. It begins with the basics of classical theory of computation: Turing machines, Boolean circuits, parallel algorithms, probabilistic computation, NPcomplete problems, and the idea of complexity of an algorithm. The second part of the book provides an exposition of quantum computation theory. It starts with the introduction of general quantum formalism (pure states, density matrices, and superoperators), universal gate sets and approximation theorems. Then the authors study various quantum computation algorithms: Grover's algorithm, Shor's factoring algorithm, and the Abelian hidden subgroup problem. In concluding sections, several related topics are discussed (parallel quantum computation, a quantum analog of NPcompleteness, and quantum errorcorrecting codes).
Rapid development of quantum computing started in 1994 with a stunning suggestion by Peter Shor to use quantum computation for factoring large numbers—an extremely difficult and timeconsuming problem when using a conventional computer. Shor's result spawned a burst of activity in designing new algorithms and in attempting to actually build quantum computers. Currently, the progress is much more significant in the former: A sound theoretical basis of quantum computing is under development and many algorithms have been suggested.
In this concise text, the authors provide solid foundations to the theory—in particular, a careful analysis of the quantum circuit model—and cover selected topics in depth. Included are a complete proof of the SolovayKitaev theorem with accurate algorithm complexity bounds, approximation of unitary operators by circuits of doubly logarithmic depth. Among other interesting topics are toric codes and their relation to the anyon approach to quantum computing.
Prerequisites are very modest and include linear algebra, elements of group theory and probability, and the notion of a formal or an intuitive algorithm. This text is suitable for a course in quantum computation for graduate students in mathematics, physics, or computer science. More than 100 problems (most of them with complete solutions) and an appendix summarizing the necessary results are a very useful addition to the book. It is available in both hardcover and softcover editions.
ReadershipAdvanced undergraduates, graduate students, research mathematicians, physicists, and computer scientists interested in computer science and quantum theory.

Table of Contents

Chapters

Introduction

Classical computation

1. Turing machines

2. Boolean circuits

3. The class NP: Reducibility and completeness

4. Probabilistic algorithms and the class BPP

5. The hierarchy of complexity classes

Quantum computation

6. Definitions and notation

7. Correspondence between classical and quantum computation

8. Bases for quantum circuits

9. Definition of quantum computation. Examples.

10. Quantum probability

11. Physically realizable transformations of density matrices

12. Measuring operators

13. Quantum algorithms for Abelian groups

14. The quantum analogue of NP: the class BQNP

15. Classical and quantum codes

Part 3. Solutions

S1. Problems of Section 1

S2. Problems of Section 2

S3. Problems of Section 3

S5. Problems of Section 5

S6. Problems of Section 6

S7. Problems of Section 7

S8. Problems of Section 8

S9. Problems of Section 9

S10. Problems of Section 10

S11. Problems of Section 11

S12. Problems of Section 12

S13. Problems of Section 13

S15. Problems of Section 15

Appendix A. Elementary Number Theory


Reviews

Apart from its conciseness and rigor, one of the main strengths of this book is the attention it gives to Kitaev's contributions to quantum computing. ...A good understanding of the quantum part of this book will provide the researcher with invaluable insights and tools for new research. ...a lot can be learned from this book. ...a valuable resource for people who want to look up or learn the intricacies of things like the circuit model, quantum NPcompleteness, etc.
SIGACT News 
The first part of the book ... consists of a compact introduction to classical complexity theory ... provides an elegant summary of the definitions and some of the tools required for the rest of the book ... The book is concluded with the solutions to all (!) exercises ... I liked this book a lot and think that it provides an excellent complement to the existing books on quantum computation ... Big pluses are the rigorous treatment of complexity issues, the introduction of the density matrix formalism early on, and complete solutions to all exercises ... translation has been done remarkably well ... concise ... researchers in the area will like it.
Mathematical Reviews 
The aim of the book is to teach the wonders of the qubitalgorithms. While other books, such as NielsenChuang, serve as (more or less) comprehensive references, the present book is focused on complexity. Mathematical prerequisites are minimal, but a reader with some understanding of basic ideas from CS, and quantum theory will get more out of Kitaev, et al ... Really well done, and nicely updated; a handy appendix was added, covering elementary math terms that are used ... The book does a great job in explaining the fundamentals ... The big question is why some qubitalgorithms are a lot better than classical counterparts ... a reader comes away with a good understanding of this in the end.
Palle Jorgensen 
Definitions and theorems are stated precisely ... proofs are written with an eye towards rigor ... most mathematicians will feel at home with the presentation of the material ... main points are explained carefully and precisely ... contains a number of exercises, with solutions to all ... well suited to mathematicians interested in quantum algorithms.
MAA Monthly


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This book is an introduction to a new rapidly developing theory of quantum computing. It begins with the basics of classical theory of computation: Turing machines, Boolean circuits, parallel algorithms, probabilistic computation, NPcomplete problems, and the idea of complexity of an algorithm. The second part of the book provides an exposition of quantum computation theory. It starts with the introduction of general quantum formalism (pure states, density matrices, and superoperators), universal gate sets and approximation theorems. Then the authors study various quantum computation algorithms: Grover's algorithm, Shor's factoring algorithm, and the Abelian hidden subgroup problem. In concluding sections, several related topics are discussed (parallel quantum computation, a quantum analog of NPcompleteness, and quantum errorcorrecting codes).
Rapid development of quantum computing started in 1994 with a stunning suggestion by Peter Shor to use quantum computation for factoring large numbers—an extremely difficult and timeconsuming problem when using a conventional computer. Shor's result spawned a burst of activity in designing new algorithms and in attempting to actually build quantum computers. Currently, the progress is much more significant in the former: A sound theoretical basis of quantum computing is under development and many algorithms have been suggested.
In this concise text, the authors provide solid foundations to the theory—in particular, a careful analysis of the quantum circuit model—and cover selected topics in depth. Included are a complete proof of the SolovayKitaev theorem with accurate algorithm complexity bounds, approximation of unitary operators by circuits of doubly logarithmic depth. Among other interesting topics are toric codes and their relation to the anyon approach to quantum computing.
Prerequisites are very modest and include linear algebra, elements of group theory and probability, and the notion of a formal or an intuitive algorithm. This text is suitable for a course in quantum computation for graduate students in mathematics, physics, or computer science. More than 100 problems (most of them with complete solutions) and an appendix summarizing the necessary results are a very useful addition to the book. It is available in both hardcover and softcover editions.
Advanced undergraduates, graduate students, research mathematicians, physicists, and computer scientists interested in computer science and quantum theory.

Chapters

Introduction

Classical computation

1. Turing machines

2. Boolean circuits

3. The class NP: Reducibility and completeness

4. Probabilistic algorithms and the class BPP

5. The hierarchy of complexity classes

Quantum computation

6. Definitions and notation

7. Correspondence between classical and quantum computation

8. Bases for quantum circuits

9. Definition of quantum computation. Examples.

10. Quantum probability

11. Physically realizable transformations of density matrices

12. Measuring operators

13. Quantum algorithms for Abelian groups

14. The quantum analogue of NP: the class BQNP

15. Classical and quantum codes

Part 3. Solutions

S1. Problems of Section 1

S2. Problems of Section 2

S3. Problems of Section 3

S5. Problems of Section 5

S6. Problems of Section 6

S7. Problems of Section 7

S8. Problems of Section 8

S9. Problems of Section 9

S10. Problems of Section 10

S11. Problems of Section 11

S12. Problems of Section 12

S13. Problems of Section 13

S15. Problems of Section 15

Appendix A. Elementary Number Theory

Apart from its conciseness and rigor, one of the main strengths of this book is the attention it gives to Kitaev's contributions to quantum computing. ...A good understanding of the quantum part of this book will provide the researcher with invaluable insights and tools for new research. ...a lot can be learned from this book. ...a valuable resource for people who want to look up or learn the intricacies of things like the circuit model, quantum NPcompleteness, etc.
SIGACT News 
The first part of the book ... consists of a compact introduction to classical complexity theory ... provides an elegant summary of the definitions and some of the tools required for the rest of the book ... The book is concluded with the solutions to all (!) exercises ... I liked this book a lot and think that it provides an excellent complement to the existing books on quantum computation ... Big pluses are the rigorous treatment of complexity issues, the introduction of the density matrix formalism early on, and complete solutions to all exercises ... translation has been done remarkably well ... concise ... researchers in the area will like it.
Mathematical Reviews 
The aim of the book is to teach the wonders of the qubitalgorithms. While other books, such as NielsenChuang, serve as (more or less) comprehensive references, the present book is focused on complexity. Mathematical prerequisites are minimal, but a reader with some understanding of basic ideas from CS, and quantum theory will get more out of Kitaev, et al ... Really well done, and nicely updated; a handy appendix was added, covering elementary math terms that are used ... The book does a great job in explaining the fundamentals ... The big question is why some qubitalgorithms are a lot better than classical counterparts ... a reader comes away with a good understanding of this in the end.
Palle Jorgensen 
Definitions and theorems are stated precisely ... proofs are written with an eye towards rigor ... most mathematicians will feel at home with the presentation of the material ... main points are explained carefully and precisely ... contains a number of exercises, with solutions to all ... well suited to mathematicians interested in quantum algorithms.
MAA Monthly