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Classical and Quantum Computation
A. Yu. Kitaev, California Institute of Technology, Pasadena, CA, and A. H. Shen and M. N. Vyalyi, Independent University of Moscow, Russia

Graduate Studies in Mathematics
2002; 257 pp; softcover
Volume: 47
ISBN-10: 0-8218-3229-8
ISBN-13: 978-0-8218-3229-5
List Price: US$45
Member Price: US$36
Order Code: GSM/47.S
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See also:

Topological Quantum Computation - Zhenghan Wang

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, NP-complete 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 NP-completeness, and quantum error-correcting 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 time-consuming 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 Solovay-Kitaev 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.


"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 NP-completeness, 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 qubit-algorithms. While other books, such as Nielsen-Chuang, 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 qubit-algorithms 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

Table of Contents

  • Introduction
  • Classical computation
  • Quantum computation
  • Solutions
  • Elementary number theory
  • Bibliography
  • Index
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