Classical and Quantum Computation (Graduate Studies in Mathematics) [Paperback]

A. Yu. Kitaev (Author), A. H. Shen (Author), M. N. Vyalyi (Author)

Book Description

Publication Date: May 31, 2002 | Series: Graduate Studies in Mathematics

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). <P>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. <P>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.

Editorial Reviews

Review

"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

Language Notes

Text: English (translation)
Original Language: Russian --This text refers to an out of print or unavailable edition of this title.

Product Details

• Paperback: 272 pages
• Publisher: Amer Mathematical Society (May 31, 2002)
• Language: English
• ISBN-10: 0821832298
• ISBN-13: 978-0821832295
• Product Dimensions: 9.8 x 6.9 x 0.5 inches
• Shipping Weight: 1 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #1,040,482 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

28 of 28 people found the following review helpful:

5.0 out of 5 stars Complexity of algorithms., August 30, 2002

By 

Palle E T Jorgensen "Palle Jorgensen" (Iowa City, Iowa United States) - See all my reviews
(VINE VOICE)    (REAL NAME)   

This review is from: Classical and Quantum Computation (Graduate Studies in Mathematics) (Hardcover)

The book covers classical and quantum algorithms;-- of the 250 or so, pages of text, roughly the first 50 pages are "classical", the rest quantum;-- and indeed 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 (by Kitaev et al) is focussed on complexity. The mathematical prerequisits are minimal, but a reader with some understanding of basic ideas from CS, and from quantum theory (at the level of ch 1 in Nielsen-Chuang), will get more out of Kitaev et al. The book is a translation of a Russian 1999 original, but it is really well done, and nicely updated;-- for example, a handy appendix was added, covering elementary math terms that are used.
The book does a great job in explaining the fundamentals, both at the level of the *intuitive ideas*, as well as the mathematical proofs. The big question is why some qubit-algorithms (such as P Shor's factoring algorithm), are a lot better than classical counterparts(for example polynomial vs exponential), and a reader comes away with a good understanding of this in the end.

6 of 7 people found the following review helpful:

5.0 out of 5 stars A clear, concise exposition, April 8, 2007

By 

Si Hui Tan (Singapore) - See all my reviews
(REAL NAME)   

This review is from: Classical and Quantum Computation (Graduate Studies in Mathematics) (Paperback)

I started off learning Quantum Computation and Quantum Information by reading Nielsen and Chuang's book in order to do research in my junior year on quantum cyptography. Despite the completeness and popularity of that book, it did not exhibit enough explanation and insights for me to be truly satisfied that quantum computation will truly take flight one day to be implementable in place of classical computation.

Recently, in my preparation for my qualifying exam in Quantum information at MIT, I commenced reading this book. The feeling was like drinking a long cool sip of water after a 10 mile run. In particular, I really like the mathematical rigor of the writers. I have known Kitaev as a clear and careful presentator while I was at CalTech as an undergrad, and this is clearly reflected in his book. I definitely would recommend this book to anyone interested in Quantum computing and quantum information, professionally or amateurishly to buy this book (and no, I was not bribed to give this review in order to pass my quals!).