Mathematical Foundations of Quantum Computing

• ISBN 978-1-961880-08-5 (ebook): Amazon | Google Play
• 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

In an era where quantum computing is poised to revolutionize technology and science, Mathematical Foundations of Quantum Computing: A Scaffolding Approach provides the essential mathematical tools needed to navigate this rapidly evolving field. This book is part of an educational series designed to systematically guide beginning graduate students, senior undergraduates, and all those intrigued by the mathematical underpinnings of quantum computation.

The book adopts a ``scaffolding approach,'' offering a step-by-step progression from foundational mathematics to more advanced concepts. Drawing inspiration from educational theories like Vygotsky's Zone of Proximal Development, this approach ensures that readers build confidence and understanding as they move through increasingly sophisticated topics. Key mathematical concepts—such as linear algebra, matrix theory, and probability—are introduced gradually, with frequent revisiting and reinforcement of core ideas. The use of diagrams, tables, and boxed highlights enhances learning while avoiding cognitive overload.

Structured into four parts, the book begins with a review of essential mathematical preliminaries, followed by an in-depth study of vectors, matrices, and linear spaces tailored for quantum computing. From there, it delves into advanced matrix analysis, introducing the indispensable tools of tensor products, matrix decompositions, and quantum operators. The final section focuses on the probability foundations necessary for quantum algorithms, exploring stochastic processes, Markov chains, and Monte Carlo methods.

Whether you are new to quantum computing or wish to solidify your understanding of the mathematical framework, this book offers a clear, structured pathway to mastering the subject. Prepare to engage deeply with the mathematical principles that form the backbone of quantum computing, and embark on your journey into this exciting and complex field.

• Pedagogically sound approach
• Up-to-date information
• Navigational aids
• Clean and clear layout
• Engaging exercises
• Suitable for senior undergraduates and early graduates
• 530 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 N`otations

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