Editorial Reviews

A carefully structured guide to the core ideas of quantum algorithms, connecting foundational primitives to real application domains and helping readers build lasting intuition for quantum computational design.

This is an excellent resource for a student or professional coming from a classical STEM background. It manages to be technically rigorous without being impenetrable. If you find standard texts like Nielsen & Chuang too dense for a first pass, this Scaffolding Approach provides the necessary rungs to climb that ladder of complexity.

The one-stop resource for everything quantum computing. Whether you are developing new algorithms or exploring practical applications, this book has it all. True to the clear, signature style of the author’s earlier titles, this latest installment brings complex concepts into sharp focus through masterful presentation.

The field of quantum algorithms is advancing at a very rapid pace, and it is not easy to learn enough to reach the current research frontier. However, with this textbook, readers can efficiently study a wide range of topics, from the fundamentals to state-of-the-art algorithms. I would recommend it as an excellent first introduction for anyone who wishes to pursue research in this field.

Professor Peter Y. Lee and his coauthors, who have been building the Quantum Information Science series through a carefully scaffolded approach, have now published the long-awaited third volume, Quantum Algorithms and Applications, following Quantum Computing and Information (Vol. 1) and Mathematical Foundations of Quantum Computing (Vol. 2). I have used the first volume in teaching quantum information science to a broad range of undergraduate students and have seen an overwhelmingly positive response. This new volume is exceptionally well designed, enabling students to acquire a broad and up-to-date understanding of quantum algorithms and their applications in a clear, systematic, and accessible manner. I expect that many future quantum computer programmers will learn the foundations of using quantum computers from this book.

This book begins with a review of the fundamental concepts of quantum algorithms, followed by detailed explanations of key techniques such as the Quantum Fourier Transform (QFT) and Quantum Phase Estimation (QPE). It then bridges these foundations to Shor’s factoring algorithm. After demonstrating Shor’s algorithm through concrete examples, the discussion expands into the more general framework of Hidden Subgroup Problems.

The book also highlights the importance of Hamiltonian simulation, explaining time evolution as governed by the Schrödinger equation. And variational algorithms based on ansatz are treated with a rigor and depth that is particularly commendable.

Unlike many CS-oriented books, this book devotes substantial space to simulations in physics and chemistry. The Hamiltonian introduced earlier plays a central role here as well. In doing so, the book provides a concrete and efficient approach to simulating nature, staying true to the vision originally envisioned by Feynman.

In addition, readers can explore modern applications such as quantum optimization and quantum machine learning. Together with the other two volumes in the Scaffolding series, this book is likely to become a definitive reference in quantum computing for researchers, engineers, and students alike.

I recently had the opportunity to review the manuscript Quantum Algorithms and Applications: A Scaffolding Approach by Peter Y. Lee, Ran Cheng, and Huiwen Ji.

At more than 600 pages, the manuscript represents a substantial and technically serious contribution to the growing literature on quantum computing. What makes this work stand out is its architectural clarity. Rather than presenting quantum algorithms as isolated results, the book uses a scaffolding methodology that systematically builds the reader’s understanding from foundational principles to modern algorithmic frameworks.

The early chapters establish the conceptual groundwork of quantum computation and computational complexity before moving into the algorithmic core of the field. Topics such as block-encoding techniques, spectral transformations, and quantum linear system algorithms are presented within a structured framework that connects theory to emerging computational use cases.

Equally important is the manuscript’s balanced perspective on the current state of quantum computing. The text recognizes the genuine engineering challenges facing the field today, including noise, qubit scalability, error correction overhead, and the verification of quantum computations. In a field often surrounded by speculation, that level of realism is both refreshing and necessary.

The application discussions are particularly compelling. Examples involving optimization, financial modeling, and infrastructure systems illustrate how quantum algorithmic primitives could eventually intersect with economically and strategically important problems.

Quantum computing remains one of the most intellectually ambitious frontiers in modern science and engineering. Its long-term impact will depend on the co-evolution of hardware scalability, fault-tolerant architectures, and algorithm design. Contributions that strengthen the algorithmic foundation of the field are therefore essential.

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.

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.

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.

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.

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.

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.

Quantum Computing is definitely going to impact our future lives. This book adheres to a pedagogical methodology that balances theoretical rigor with accessibility. The scaffolding approach that the authors use guides the reader through the learning journey. This makes the book not only academically rigorous but also effective as a teaching tool.

This impressive book covers the burgeoning field of quantum information, bridging the fundamentals of quantum mechanics and its present and future applications in secure communication and quantum computing. The author’s approach is rigorous—including all the necessary linear algebra—while the book is highly readable and accessible. It will benefit a wide range of audiences with different backgrounds, from undergraduate students learning quantum mechanics to experts who want a deep understanding of quantum information protocols.

This textbook is elegantly crafted, utilizing a unique scaffolding approach to render complex topics in quantum computing easily comprehensible for newcomers to the field. It is invaluable for both educators and students of quantum computing. The authors employ a lucid and engaging style, ensuring that difficult material remains accessible. Their original illustrations and tables, designed to complement the text, enhance comprehension. In addition, the book provides both concise and detailed examples, helping entry-level students grasp fundamental concepts. A thoughtful balance between straightforward exercises and broader integrative problems is maintained throughout.

This text presents Quantum Computing and Information in a measured format. The reader is exposed to concepts, notation, and calculations in qubit-sized pieces, that are further described in later parts of the text in evolving detail. This scaffolding approach does not demand full comprehension of one topic before going on to the next. Each iteration reviews and adds to the previous. Reading this from a teaching perspective, I found insights about how I can improve my course for mid-career professionals wanting to pivot to a career in quantum information science. I also find this text a useful quick teaching reference to target specific topics with helpful graphics and tables. I highly recommend this text for those teaching first-time quantum students.

Clarity in content, clarity in style. That perfectly sums up this new, soon-to-be classic, textbook on quantum computing. From the basic postulates, to single- and multi-qubits, gates, circuits, and foundational and modern algorithms, this book is your one-stop-shop for all things quantum. The writing is clear, the mathematics elegant, and the presentation is beautiful. As you read, it is obvious the author’s had you and your comprehension in mind, providing a scaffold, or support, for building your quantum computing knowledge. Highly recommended!

While many books have been written about quantum computing, this text by Lee et al. offers a refreshing perspective. Most other comparable books require the reader to be mathematically mature, for example a physics graduate student. In contrast, this text is designed to be approachable. The scaffolding pedagogical theories that the authors draw from put the student at the center of the learning process. Each layer of material is presented only after previous layers have been mastered. To help convey the material, numerous examples that clarify tricky parts have been provided.

It may sound like this book is only for beginners, yet I was impressed by the latter chapters on quantum error correction and quantum information. These segments rely on somewhat abstract ideas, and the authors managed to convey them with a soft touch. Quantum computing and information has been transitioning from an advanced graduate topic to undergraduate courses, and many students will find this book an invaluable resource.

This book offers a structured and intuitive introduction to quantum mechanics, computation, and information theory. Its distinctive feature is the scaffolding approach: a progressive path that guides readers step by step from fundamental principles toward more advanced topics. This approach breaks down complex concepts, making them accessible to students, self-learners, and professionals transitioning into the field. The 2nd edition expands on quantum error correction, post-quantum cryptography, and recent algorithmic advances. It also refines explanations, introduces new exercises, and incorporates updated examples, making it more informative, structured, and engaging.

This comprehensive and eminently readable, timely and sumptuously illustrated book is a tour-de-force through the world of quantum computing and information (QCI). It takes readers on a vivid and accessible journey from the foundations of quantum computing using photons, all the way to quantum algorithms and error correction by way of quantum gates, circuits, superposition, entanglement, and all points in between. It is rare to find a book that speaks so directly to the experience of the reader as they tackle new concepts and techniques, and this is another of its excellent qualities.

This book is structured to introduce students from many different technical interests to quantum information science. The careful computational and conceptual development at the beginning of the book is oriented toward students at the freshman or sophomore levels who have the appropriate mathematical background in algebra and linear algebra. Unfortunately, many students who could benefit from this do not yet have the needed training in linear algebra.

The next part of the book, on quantum entanglement and Bell inequalities, offers a superb historical orientation to how quantum information science became influential by about 2000. It provides a strong culmination for a sophomore-level course based on this text.

The final section explores quantum information science as it existed in 2022–2024 with many openings for research and exploratory student projects. This makes the book attractive for upper-division undergraduate courses and also for first-year graduate students. Overall, it is a very important text: well-written, timely, and highly recommended. The field of quantum information science is likely to increase in importance in the next ten years.