Review
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.
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.
Ying Nian Wu, Professor, Department of Statistics and Data Science, University of California in Los Angeles
The QCI book (Quantum Computing and Information) 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), 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.
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.
From the Author
Quantum Computing and Information (QCI) represents a paradigm shift not only in computation but also in the mathematical framework necessary for advancing in the field. While linear algebra is central to QCI, the applications here extend beyond its traditional role. In quantum computing, matrices operate as dynamic tools—taking on the roles of operators and transformations—and matrix algebra, including tensor products, trace operations, matrix decompositions, and matrix functions, becomes indispensable. These sophisticated operations are essential for the mathematical precision and versatility required in quantum mechanics.
In this context, Dirac notation serves as the primary language for expressing vectors, operators, and their interactions, helping students transition smoothly into quantum mechanics. Special matrices such as Hermitian, unitary, and Pauli matrices are introduced not merely as abstract constructs but as essential building blocks that encode quantum states and govern quantum transformations, directly supporting the mathematical requirements of quantum algorithms and quantum error correction.
The probabilistic nature of quantum mechanics differs from classical probability, and this book equips students with foundational tools in probability, sampling theory, and key stochastic methods like Markov chains and MCMC. This preparation supports their understanding of probabilistic quantum algorithms and paves the way for more advanced quantum probability concepts.
Recognizing the broad mathematical prerequisites for quantum computing, we begin the book with a focused review of complex numbers, trigonometry, and summation rules, tailored specifically to quantum applications. By omitting areas such as differential equations and complex functional analysis, which, while valuable, are not essential to QC studies, this book emphasizes efficiency and accessibility, allowing students to concentrate on mastering topics directly aligned with QC.
Rather than serving as a comprehensive mathematical reference, Mathematical Foundations of Quantum Computing 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.
From the Inside Flap
A Journey Well Traveled
Congratulations to all who have completed this book, especially those who engaged with the exercises and problems throughout. You've successfully established a strong foundation in the mathematical concepts underpinning quantum computing, equipping yourselves for further exploration in this rapidly advancing field.
The Mathematical Horizon
As quantum computing continues to evolve, so too does its mathematical framework. New discoveries in areas such as quantum algorithms, cryptography, and error correction challenge us to deepen our mathematical understanding and extend it into new realms. The foundational topics you have mastered—linear algebra, Dirac notation, matrix analysis, and probability—will remain central as quantum technology progresses.
But there are still many exciting mathematical challenges ahead. As the quest for quantum algorithms and scalable, fault-tolerant quantum computing continues, new theoretical tools will be required. Quantum computing presents unique mathematical demands, and the journey ahead will offer opportunities to expand upon the knowledge you have gained here.
Looking Forward
As you continue your journey, consider the following next steps:
- Study the fundamentals of quantum computing and information, by exploring the second book in this series, Quantum Computing and Information: A Scaffolding Approach.
- Delve into quantum algorithms after establishing a solid understanding of quantum computing, as these offer solutions across a wide range of applications.
- Explore other fields within quantum science, such as quantum sensing, where new frontiers in technology and mathematics are rapidly emerging.
- Stay updated with the latest advancements in quantum computing, including hardware innovations, error correction techniques, and emerging algorithms.
With the knowledge gained from this book, you are well-positioned to further explore the fascinating realm of quantum computing and related fields. Keep learning, stay curious, and continue to expand your understanding of this ever-evolving subject. From the Back Cover
Essential Mathematics for Quantum Computing
This focused guide connects key mathematical principles with their specialized applications in quantum computing, equipping students with the essential tools to succeed in this transformative field. It is ideal for educators, students, and self-learners seeking a strong mathematical foundation to master quantum mechanics and quantum algorithms.
Features
- Covers key mathematical concepts, including matrix algebra, probability, and Dirac notation, tailored for quantum computing.
- Explains essential topics like tensor products, matrix decompositions, Hermitian and unitary matrices, and their roles in quantum transformations.
- Offers a streamlined introduction to foundational math topics for quantum computing, with an emphasis on accessibility and application.
Authors- Dr. Peter Y. Lee (Ph.D., Princeton University) - Expert in quantum nanostructures with extensive experience in teaching and academic program leadership.
- James M. Yu (Ph.D., Rutgers University) - Expert in mathematical modeling, applied mathematics, and quantum computing, with extensive teaching experience.
- Dr. Ran Cheng (Ph.D., University of Texas at Austin) - Specialist in condensed matter theory and an award-winning physicist.
About the Author
Dr. 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 his time 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.
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