Mathematical Foundations of Quantum Computing
Why This Book
Mathematical Foundations of Quantum Computing: A Scaffolding Approach is intended as a streamlined, accessible guide for learners. Our goal is to bridge the gap between traditional mathematical education and the specialized demands of quantum computing, equipping readers with a solid foundation to support their future studies in quantum mechanics and quantum algorithms. The book is organized into four main parts:
- Part I: Preliminaries. A concise review of complex numbers, trigonometry, summation rules, and sets, groups, and functions, laying the groundwork for the following sections.
- Part II: Vectors, Matrices, and Linear Spaces. Covers topics traditionally studied in a first-year linear algebra course, but with an early introduction to complex numbers and Dirac notation, which are used throughout.
- Part III: Matrix Methods for Quantum Systems. Focuses on advanced linear algebra concepts essential for quantum computing, including tensor products, matrix functions, Pauli matrices and strings, and singular value decomposition.
- Part IV: Probability Foundations for Computation. Explores topics such as probability, stochastic processes, Markov chains, and Monte Carlo methods, with an emphasis on their relevance to quantum computing.
More Information
Please reach us at qci401@polarisqci.com if you need additional information.
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.
This book provides the foundational mathematics necessary for further study in quantum computing and quantum algorithms. It serves both as a stepping stone to the second and third books in this series and as a standalone reference.
To fully engage with the material, students are encouraged to complete the exercises and problems at the end of each chapter. For a two-semester course, this approach allows for a comprehensive exploration of the content. However, students already familiar with basic linear algebra may complete the book in a single semester by focusing primarily on Parts III and IV.
Furthermore, while not all topics covered here are essential for the second book, Quantum Computing and Information, they will be important for the third book, Quantum Algorithms and Applications. These topics can be deferred until studying the third book or specific quantum algorithms that require them. They include:
- Discrete Fourier Transform (§ 11.2.5)
- Pauli String Basis (§ 12.4) and Pauli Groups (§ 12.5)
- Jordan, Singular Value, and Schmidt Decompositions (Chapter 13)
- Markov Chains (Chapter 16) and Monte Carlo Methods (Chapter 17)
- 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: Holds a Ph.D. in Electrical Engineering from Princeton University. His research at Princeton focused on quantum nanostructures, the fractional quantum Hall effect, and Wigner crystals. Following his academic tenure, he joined Bell Labs, making significant contributions to the fields of photonics and optical communications and securing over 20 patents. Dr. Lee's multifaceted expertise extends to educational settings; he has a rich history of teaching, academic program oversight, and computer programming.
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.
Ying Nian Wu, Professor, Department of Statistics and Data Science, University of California in Los Angeles
The QCI book (Quantum Computing and Information: A Scaffolding Approach) presents quantum computing in a wonderfully friendly manner, making this complex field accessible to anyone with basic undergraduate math preparation. The companion text (Mathematical Foundations of Quantum Computing: A Scaffolding Approach), with its comprehensive coverage of mathematical foundations, provides all the essential tools needed to dive into quantum concepts with confidence. I found the chapters on probability to be expertly written, offering a clear, engaging, and quantum-relevant introduction. Together, these books form an inviting and masterful gateway for learners eager to explore quantum computing.
Andrew Kent, Professor of Physics, The Center for Quantum Phenomena, New York University
This comprehensive and accessible text presents, in a single volume, the mathematical foundation of quantum information. Beginning with the essentials—linear algebra, probability, and matrix analysis—and advancing to topics like tensor products, spectral decompositions, and Markov Chain Monte Carlo simulations, the authors guide the reader with clarity and rigor. Rarely is so much mathematical depth presented in such a student-friendly way. This volume will serve both newcomers and experts alike, providing a strong foundation for gaining facility with the mathematics required to understand quantum systems.
Steven Frankel, Rosenblatt Professor, Faculty of Mechanical Engineering, Technion - Israel Institute of Technology
A beautiful, colorfully crystal clear, and veritable one-stop-shop, this resource offers everything mathematical essential to quantum computing. Covering vector spaces, matrix methods including tensor products, and probability theory, it is a must-read for quantum computing researchers and practitioners alike.
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, 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.
Yamamoto Fujio, Professor Emeritus, Kanagawa Institute of Technology, Japan
This book provides a thorough explanation of the mathematics underlying quantum computing. Dirac (bra–ket) notation is introduced right at the beginning of Part II. Part III then offers a detailed treatment of matrix operations fundamental to quantum computing, with particular emphasis on tensor products. The text also gives careful attention to change of basis—crucial in applications such as quantum key distribution—and to the Kronecker product, which is central to describing composite quantum systems. Equally significant, Part IV presents an in-depth discussion of probability, an essential tool for understanding quantum computing in contrast to classical computing.