Cover -- Title page -- Copyright -- Contents -- Preface -- The advent of quantum computing -- The goal of the book -- The structure of the book -- What is not covered -- For instructors -- Acknowledgements -- Chapter 1. Classical Computation -- 1.1. Deterministic algorithms -- 1.2. Probabilistic algorithms -- 1.3. Analysis of probabilistic algorithms -- 1.4. Complexity theory -- 1.5. The circuit model -- 1.6. Circuit families and circuit complexity -- 1.7. Reversible circuits -- Chapter 2. Hilbert Spaces -- 2.1. Kets and state spaces -- 2.2. Inner products -- 2.3. Linear maps -- 2.4. Endomorphisms -- 2.5. Tensor products -- Chapter 3. Quantum Mechanics -- 3.1. State spaces -- 3.2. State spaces of composite systems -- 3.3. Time evolution -- 3.4. Measurements -- 3.5. Density operators -- 3.6. The quantum postulates for mixed states -- 3.7. Partial trace and reduced density operators -- Chapter 4. The Theory of Quantum Algorithms -- 4.1. Simple single-qubit operators -- 4.2. More geometry in R3 -- 4.3. Rotation operators -- 4.4. Controlled operators -- 4.5. Swap and permutation operators -- 4.6. Ancillary and erasure gates -- 4.7. Quantum circuits revisited -- 4.8. Universal sets of quantum gates -- 4.9. Implementation of controlled operators -- 4.10. Perfectly universal sets of quantum gates -- 4.11. A universal set of quantum gates -- 4.12. Quantum algorithms and quantum complexity -- Chapter 5. The Algorithms of Deutsch and Simon -- 5.1. The Deutsch algorithm -- 5.2. Oracle complexity -- 5.3. The Deutsch-Jozsa algorithm -- 5.4. Simon's algorithm -- 5.5. Generalization of Simon's algorithm -- Chapter 6. The Algorithms of Shor -- 6.1. Idea of Shor's factoring algorithm -- 6.2. The Quantum Fourier Transform -- 6.3. Quantum phase estimation -- 6.4. Order finding -- 6.5. Integer factorization -- 6.6. Discrete logarithms.

Includes bibliographical references and index.

Classical computation -- Hilbert spaces -- Quantum mechanics -- The theory of quantum algorithms -- The algorithms of Deutsch and Simon -- The algorithms of Shor -- Quantum search and quantum counting -- The HHL algorithm.

Quantum algorithms are among the most important, interesting, and promising innovations in information and communication technology. They pose a major threat to today's cybersecurity and at the same time promise great benefits by potentially solving previously intractable computational problems with reasonable effort. The theory of quantum algorithms is based on advanced concepts from computer science, mathematics, and physics. Introduction to Quantum Algorithms offers a mathematically precise exploration of these concepts, accessible to those with a basic mathematical university education, while also catering to more experienced readers. This comprehensive book is suitable for self-study or as a textbook for one- or two-semester introductory courses on quantum computing algorithms. Instructors can tailor their approach to emphasize theoretical understanding and proofs or practical applications of quantum algorithms, depending on the course's goals and timeframe.