Classical and Quantum Computation (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). 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 This Book Presents A Concise Introduction To An Emerging And Increasingly Important Topic, The Theory Of Quantum Computing. The Development Of Quantum Computing Exploded In 1994 With The Discovery Of Its Use In Factoring Large Numbers--an Extremely Difficult And Time-consuming Problem When Using A Conventional Computer. In Less Than 300 Pages, The Authors Set Forth A Solid Foundation To The Theory, Including Results That Have Not Appeared Elsewhere And Improvements On Existing Works. The Book Starts With The Basics Of Classical Theory Of Computation, Including Np-complete Problems And The Idea Of Complexity Of An Algorithm. Then The Authors Introduce General Principles Of Quantum Computing And Pass To The Study Of Main 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). This Is A Suitable Textbook For A Graduate Course In Quantum Computing. Prerequisites Are Very Modest And Include Linear Algebra, Elements Of Group Theory And Probability, And The Notion Of An Algorithm (on A Formal Or An Intuitive Level). The Book Is Complete With Problems, Solutions, And An Appendix Summarizing The Necessary Results From Number Theory. 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