Mathematical Foundations of Quantum Computing

• ISBN 978-1-961880-08-5 (ebook, color): Amazon
• ISBN 978-1-961880-09-2 (paperback, b/w): Amazon
• ISBN 978-1-961880-10-8 (hardcover, b/w): Amazon
• Library of Congress Control Number (LCCN) 2024947285

Quantum Computing and Information (QCI) requires a shift in mathematical thinking, going beyond the traditional applications of linear algebra and probability. This book focuses on building the specialized mathematical foundation needed for QCI, explaining the unique roles of matrices, outer products, tensor products, and the Dirac notation. Special matrices crucial to quantum operations are explored, and the connection between quantum mechanics and probability theory is made clear.

Recognizing that diving straight into advanced concepts can be overwhelming, this book starts with a focused review of essential preliminaries like complex numbers, trigonometry, and summation rules. It serves as a bridge between traditional math education and the specific requirements of quantum computing, empowering learners to confidently navigate this fascinating and rapidly evolving field.

• Pedagogically sound approach
• Up-to-date information
• Navigational aids
• Clean and clear layout
• Engaging exercises
• Suitable for senior undergraduates and early graduates
• 570 pages, 100+ illustrations

Peter Y. Lee: Earned a Ph.D. in E.E. from Princeton, specializing in quantum nanostructures and the fractional quantum Hall effect. Post-academia, he joined Bell Labs, contributing to photonics and securing 20+ patents. He brings extensive teaching experience and is now a faculty member at Fei Tian College, NY.

James M. Yu: Earned his Ph.D. in Mechanical Engineering from Rutgers University at New Brunswick, specialized in mathematical modeling and simulation of biophysical phenomena. Following his doctorate studies, He continued to conduct research as a postdoctoral associate at Rutgers University. Currently, he is a faculty member at Fei Tian College, Middletown where he dedicates to teaching mathematics, statistics, and computer science.
Ran Cheng: Earned his Ph.D. in Physics from the University of Texas at Austin, with a specialization in condensed matter theory, particularly in spintronics and magnetism. Following a postdoctoral position at Carnegie Mellon University, he joined the faculty at the University of California, Riverside, where he was honored with the NSF CAREER and DoD MURI awards.

Preface
Reviews

I. Preliminaries

1. Summation and Product Notations

2. Trigonometry

3. Complex Numbers

4. Sets, Groups, and Functions

II. Vectors, Matrices, and Linear Spaces
5. Vectors and Vector Spaces
6. Inner Product Spaces
7. Fundamentals of Matrix Algebra
8. Matrices as Linear Operators
9. Spectral Decomposition of Matrices

III. Matrix Methods for Quantum Computing
10. Tensor Products of Vector Spaces
11. Functions of Vectors and Matrices
12. Pauli Matrices, Strings, and Groups
13. Advanced Matrix Decompositions

IV. A Probability Primer for Quantum Computing
14. Fundamentals of Probability
15. Stochastic Processes
16. Markov Chains
17. Monte Carlo Methods

V. Supporting Materials
Key Formulas and Concepts
Bibliography
List of Figures
List of Tables
Index
Journey Forward

Leonard Kahn, Professor and Chair, Department of Physics, University of Rhode Island

With the move toward introducing quantum computing as a first-year course, the structure of Mathematical Foundations of Quantum Computing makes it a strong contender as a text that can be used throughout an academic career. The authors have successfully designed a text that can be used at multiple stages of development, from introductory, through intermediate and graduate levels, as well as a useful reference work. From the introduction of vectors and matrices, each topic is revisited with increasing complexity, an ideal implementation of the scaffolding approach. The layout of the text, accompanied by a variety of exercises, examples, and clear graphics, advances the authors' goal of creating a valuable learning and teaching aid. The text, along with its companion Quantum Computing and Information, deserves serious consideration by those who are designing a full-range quantum computing curriculum.

Tony Holdroyd, Retired Senior Lecturer in Computer Science and Mathematics

This book is a learned and thorough exposition of the mathematics that supports quantum computing.
The authors have gone to great lengths to make it both learner-friendly and detailed while maintaining rigor. It covers topics ranging from the fundamentals of quantum mathematics to the complexities of vector and matrix algebra, as well as the probabilities central to quantum computing. The text is complemented by numerous supporting figures that effectively illustrate key concepts. Applications of quantum computing are introduced and seamlessly integrated throughout the book. This volume, along with its companion, \textit{Quantum Computing and Information - a Scaffolding Approach}, is an essential addition to the bookshelf of anyone seeking a deeper understanding of quantum computing and its mathematical foundations.