Classical and quantum computation

This book is an introduction to a new and rapidly developing topic: the 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. Some of the results have not appeared elsewhere while others improve on existing works. 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 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. Introduction 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 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 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 A. Yu. Kitaev, A.h. Shen, M.n. Vyalyi ; [translated From The Russian By Lester J. Senechal]. Includes Bibliographical References (p. 251-254) And Index. An introduction to a rapidly developing topic: the theory of quantum computing. Following the basics of classical theory of computation, the book provides an exposition of quantum computation theory. In concluding sections, related topics, including parallel quantum computation, are discussed