This textbook presents the elementary aspects of quantum computing in a mathematical form. It is intended as core or supplementary reading for physicists, mathematicians, and computer scientists taking a first course on quantum computing. It starts by introducing the basic mathematics required for quantum mechanics, and then goes on to present, in detail, the notions of quantum mechanics, entanglement, quantum gates, and quantum algorithms, of which Shor's factorisation and Grover's search algorithm are discussed extensively. In addition, the algorithms for the Abelian Hidden Subgroup and Discrete Logarithm problems are presented and the latter is used to show how the Bitcoin digital signature may be compromised. It also addresses the problem of error correction as well as giving a detailed exposition of adiabatic quantum computing. The book contains around 140 exercises for the student, covering all of the topics treated, together with an appendix of solutions.

Autorentext

Wolfgang Scherer was active in research and teaching in the USA and Germany before settling in London where until recently he worked in risk management for a financial institution. His scientific interests include geometric methods in mathematical physics and fundamental problems in quantum mechanics. He takes pleasure in communicating the joy of mathematics to the younger generation, and is an aficionado of two-wheeled vehicles, with and without motor.

Inhalt

Introduction.- Basic Notions of Quantum Mechanics.- Tensor Products and Composite Systems.- Entanglement.- Quantum Gates and Circuits for Elementary Calculations.- On the Use of Entanglement.- Error Correction.- Adiabatic Quantum Computing.- Epilogue Appendices: A Elementary Probability Theory.- B Elementary Arithmetic Operations.- C LANDAU Symbols.- D Modular Arithmetic.- E Continued Fractions.- F Some Group Theory.- G Proof of a Quantum Adiabatic Theorem.- Solutions to Exercises.