Instructor: Zhiyan Ding (zyding at umich.edu)
Course description:
Quantum computers promise to transform computational science. At the core of this paradigm are quantum algorithms, which often differ substantially from their classical counterparts. This advanced graduate course introduces key quantum algorithms for scientific computation, organized into two main parts:
Fundamental quantum algorithms, with a specific focus on quantum linear algebra. This includes the basics of quantum computation, eigenvalue estimation, Hamiltonian simulation, and matrix representation and manipulation on quantum computers.
Applications of these algorithms to scientific problems such as ground-state energy estimation and preparation, Shor’s algorithm, and the solution of linear systems and differential equations.
The first part emphasizes foundational, textbook-based material, while the second explores traditional algorithms alongside recent advances and state-of-the-art results, including selected open problems. Overall, the course focuses on the design, structure, and analysis of quantum algorithms rather than hardware-specific implementations. By the end of the course, students will have a solid understanding of core quantum algorithmic techniques and be prepared to analyze or develop new quantum algorithms in research.
Textbook:
There is no official textbook in this topic class because the material is not restrictive to textbook-based material. However, there are several recommend lecture notes and textbooks:
1. (Highly recommended) Lin Lin's Lecture notes (https://arxiv.org/abs/2201.08309)
The first category of the material will mainly be introduced following the chapters with Lin's lecture notes.
During the semester, I will also create and update the lecture note for this class and posted it online.
2. Michael Nielsen, Issac Chuang, Quantum computation and quantum information, 10th anniversary edition, ISBN-13: 978-1107002173
This is a basic yet comprehensive textbook for beginners in quantum computing. If you have not studied quantum mechanics before, it is recommended to quickly go through the first part of this book—especially Chapter 2—to gain a general understanding of the subject. The first part is introductory and accessible to students with diverse academic backgrounds.
3. John Preskill's Lecture notes
4. Andrew Childs, Lecture Notes on Quantum Algorithms
Prerequisite:
Due to the interdisciplinary nature of this course, the material draws on a broad range of topics. At a minimum, students should have a solid foundation in linear algebra, along with basic knowledge of probability theory and numerical analysis at the undergraduate level. While prior exposure to quantum mechanics is recommended, it is not required. Below is a list of relevant courses that students may have previously taken or encountered:
Linear algebra: One of MATH 417, MATH 419, MATH 420, MATH 571
Probability: One of MATH 425, MATH 525, MATH 526
Numerical analysis: One of MATH 317, MATH 471, MATH 571
Note: These courses are not mandatory prerequisites. If you are unsure whether you meet the background requirements, please read the first chapter of Lin Lin’s lecture notes. If you feel comfortable with the material there, this course should be at an appropriate level of difficulty for you.
Evaluation:
Since this is a topic-based course, there will be no exams. A few written homework assignments will be given during the semester, primarily covering the first category of material. Undergraduate students are required to complete and submit these assignments for grading, while graduate students are highly encouraged—but not required—to do so.
At the end of the semester, a final project will be assigned. This project typically takes about half a day to complete, and detailed instructions will be provided later in the semester.
Tentative outline of the class:
1. Basic quantum computing language and quantum operation (First category)
Lecture note Chapter 1-2, Chapter 5, and Lieb-Robinson bound
2. Traditional quantum phase estimation algorithm and application (First category)
Lecture note Chapters 3, 4
3. Hadamard test revisit/ Recent quantum phase estimation algorithm and classical signal processing (Second category)
Recent works on quantum phase estimation (A summary can be found in https://quantum-journal.org/papers/q-2024-10-02-1487/ Section 2.1)
4. Basic quantum linear matrix operation (First category)
Lecture note Chapter 6
Basic matrix operation: summation, subtractaction, multiplication, tensorization, etc
5. Quantum ordinary or partial differential equation solver (Second category)
Dilation idea and HHL
Linear combination of unitary and LCHS
Nonlinear equation solver
7. Matrix function (First category)
Lecture note Chapter 7 and 8
8. Application of matrix function (Second category)
Previous task revisit