Quantum Algorithms and Applications: A Scaffolding Approach

• ISBN 978-1-961880-11-5 (vol 1+2 combined, ebook, color)
• ISBN 978-1-961880-12-2 (vol 1, paperback, b/w)
• ISBN 978-1-961880-13-9 (vol 2, paperback, b/w)
• ISBN 978-1-961880-14-6 (vol 1, hardcover, b/w)
• ISBN 978-1-961880-15-3 (vol 2, hardcover, b/w)

Quantum Algorithms and Applications: A Scaffolding Approach develops quantum computing through the lens of algorithms and the workflows they enable. It guides readers in a carefully staged progression from core primitives—such as Fourier-based methods, amplitude techniques, Hamiltonian simulation, and modern polynomial-approximation frameworks including block encoding and quantum singular value transformation—to practical application motifs in physics and chemistry simulation, optimization, quantum machine learning, and quantum methods for linear systems and differential equations.

Throughout, the emphasis is on mastering durable concepts and principles: what is being computed, what is assumed about data access and outputs, how resources and error shape performance, and how to evaluate claims of quantum advantage with clarity. The book is written for senior undergraduates, beginning graduate students, and practitioners, and is reinforced with in-text exercises, end-of-chapter problem sets, and recurring design patterns that help readers build a coherent, long-lasting mental model of quantum computation.

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

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.

Dr. 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.

Dr. Huiwen Ji holds a Ph.D. in Chemistry from Princeton University. She is a materials chemist whose research spans solid-state functional materials, quantum materials, and energy-related materials. After appointments at the University of California, Berkeley and Lawrence Berkeley National Laboratory, she joined the University of Utah as a faculty member in Materials Science and Engineering. Her honors include an NSF CAREER Award.

This book is organized into four parts.

Part I: Foundations of Quantum Algorithms.

We establish the computational viewpoint used throughout the book, including algorithmic efficiency, the circuit and query models, and the complexity language needed to reason about scaling, precision, and success probability.

Part II: Core Quantum Algorithms.

We develop the core algorithmic primitives that reappear across quantum speedups: Fourier-based methods (quantum Fourier transform and quantum phase estimation), period finding and Shor’s algorithm, amplitude amplification and estimation, and modern polynomial-approximation frameworks centered on block encodings, including linear combination of unitaries, quantum signal processing, and quantum singular value transformation.

Part III: Quantum Simulation in Physics and Chemistry.

We show how the toolkit is used for simulation tasks motivated by physics and chemistry. The emphasis is on how physical structure becomes computational structure through Hamiltonians, encodings, and measurement strategies, and on how accuracy, cost, and feasibility trade off in practice.

Part IV: Other Applications.

We survey additional areas where the same primitives recur, including optimization, quantum machine learning, and quantum methods for linear systems and differential equations, with attention to input/output models, verification, and resource accounting.

The appendices provide supporting reference material, including brief refreshers on quantum computing fundamentals and curated overviews of problem landscapes with quantum potential.

For print production, this single book is issued as two physical volumes due to length limits. Volume 1 contains Parts I–II, and Volume 2 contains Parts III–IV and the appendices.

VOLUME 1

Part I Foundations of Quantum Algorithms

1 Algorithmic Thinking

2 Quantum Algorithms: A New Paradigm

3 Quantum Circuit Model and Query Model

4 Classification of Computational Complexity

5 FTQC vs. NISQ Algorithms

Part II Core Quantum Algorithms

6 Quantum Fourier Transform (QFT)

7 Quantum Phase Estimation (QPE)

8 Shor’s Algorithm and Period Finding

9 Amplitude Amplification and Estimation

10 Foundations of Hamiltonian Simulation

11 Polynomial Approximation Algorithms

12 Quantum Linear System Algorithms (QLSA)

13 Adiabatic and Variational Quantum Algorithms

14 Measurement Primitives

VOLUME 2

Part III Quantum Physics & Chemistry Simulations

15 Simulation Formulations and Encodings

16 Foundations of Quantum Chemistry Simulations

17 Static Properties and Spectra

18 Dynamics and Correlation Functions

19 Many-Body and Materials Models

Part IV Other Applications

20 Quantum Optimization and Heuristics

21 Quantum Machine Learning

22 Quantum Solvers for Differential Equations

23 The Quest for Quantum Advantage

Part V Supporting Materials

Appendices

A Quantum Computing Fundamentals

B Error Correction and Fault-Tolerance Formalism

C Quantum Software Platforms and Frameworks

D Problem Landscapes with Quantum Potential

For print production, this single book is issued as two physical volumes due to length limits: Volume 1 contains Parts I-II, and Volume 2 contains Parts III-IV and the Appendices.

Zlatko Minev

Google Quantum AI; CIFAR; Formerly: IBM Quantum, Yale, and UC Berkeley

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.

Omar Alnaseri (Jan)

Adjunct Professor at DHBW, Germany; Researcher in Quantum Communication Systems and Quantum ML/AI; SMIEEE

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.

Steven Frankel

Rosenblatt Professor of Mechanical Engineering, Technion - Israel Institute of Technology

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.

Naoki Yamamoto

Professor, Department of Applied Physics and Physico-Informatics, Keio University, Japan

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.

Jaewan Kim

National Distinguished Research Fellow, Korea Research Institute of Standards and Science (KRISS); Professor Emeritus, Yonsei University and Korea Institute for Advanced Study (KIAS)

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.

Fujio Yamamoto, 

Professor Emeritus, Information and Computer Sciences Department, Kanagawa Institute of Technology, Japan

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.