Math275 Spring 2026: Topics in Applied Mathematics
Quantum algorithms for scientific computation

Instructor: Lin Lin
Lecture hours: TTh 9:30A-10:59A, Evans 748 (7th floor)
Office Hours: T 11AM-12PM, Evans 817
Course catalog: https://classes.berkeley.edu/content/2026-spring-math-275-002-lec-002

Lecture notes:

https://math.berkeley.edu/~linlin/qasc/

1/21: Live notes updated Ch 5 and 9
1/15: New lecture notes ("live notes") are uploaded. Please read Ch 1-2 for background materials.

Quantum computers have the potential to revolutionize how we think about computing. Central to quantum computation are quantum algorithms, which often differ considerably from classical algorithms. This is an advanced graduate course course that introduces quantum algorithms essential for scientific computation. Topics include phase estimation, Hamiltonian simulation, block encoding, quantum singular value transformation, and their applications in tasks like solving linear systems, eigenvalue problems, and differential equations. The focus is on algorithmic components, design, and analysis. The quantum algorithms discussed are largely independent of the specific physical hardware on which they're implemented. Upon completing the course, students will have a solid understanding of the primary quantum algorithmic techniques for scientific computation and will be prepared to engage with technical discussions and design novel quantum algorithms in their research.

Prerequisites

Due to the interdisciplinary nature of the topic, the course material requires a broad knowledge base. At a minimum, students should have a solid understanding of linear algebra, as well as basic knowledge of probability theory and quantum mechanics (all at the undergraduate level). Below is a reference list of relevant courses you may have taken or been exposed to:

• Linear Algebra (MATH 54 / PHYSICS 89 / EECS 16A, or MATH 110)
• Probability (MATH 55 / STAT 20 / CS 70)
• Quantum Mechanics (PHYSICS 7C, PHYSICS 137A, or CHEM 120A), or Quantum Information Theory (CHEM/CS/PHYS 191, or CS 294-66)

Before the first class: please read Chapter 1 (Preliminaries of quantum computation) 1.1-1.6 of the Notes. Ensure that you either have prior knowledge of the material or can comprehend it upon reading.

Enrollment Instructor consent is not needed. However, undergraduate students need to submit Sp26 Graduate Enrollment Request Form to be granted a permission. Additionally, please also fill this Google form

Resources:

• A significant portion of the course materials are related to the IPAM Tutorial (Tuesday and Wednesday) in Fall 2023 (see presentations and slides)
• Andrew Childs, Lecture Notes on Quantum Algorithms
• John Preskill's Lecture notes
• Eleanor Rieffel and Wolfgang Polak, Quantum Computing: A Gentle Introduction, 2014 ISBN-13 : 978-0262526678
• Michael Nielsen, Issac Chuang, Quantum computation and quantum information, 10th anniversary edition, ISBN-13: 978-1107002173
• Ruizhe Zhang's slides Algorithms for Data Science, Fall 2025 (Quantum algorithm part).
• Quantum Algorithm Zoo. This should be viewed as a dictionary.

The formal grade evaluation will be entirely based on the final project. Details and guidelines for the project will be released during the semester.

This is a tentative weekly schedule, subject to (likely) changes.
Ch refers to chapters in the "live notes"

#	Date	Content	Comments
1	Tue. 1/20	Course information. Review of background materials.	Ch 1, Ch 2.2-2.6
2	Thu. 1/22	Block encoding
3	Tue. 1/27	Linear combination of unitaries
4	Thu. 1/29	Foundation: Quantum processing of classical information; Block encoding of sparse matrices
5	Tue. 2/3	Qubitization and cosine-sine decomposition
6	Thu. 2/5	Grover type algorithms
7	Tue. 2/10	Classical Markov chains and quantum walk
8	Thu. 2/12	Continuous time quantum walk
9	Tue. 2/17	Quantum signal processing and nonlinear Fourier transform
10	Thu. 2/19	Quantum singular value transformation
11	Tue. 2/24	Foundation: perturbation theory
12	Thu. 2/26	Block encoding based Hamiltonian simulation
13	Tue. 3/3	Block encoding based Hamiltonian simulation
14	Thu. 3/5	Operator splitting based Hamiltonian simulation
15	Tue. 3/10	Operator splitting based Hamiltonian simulation
16	Thu. 3/12	Foundation: probability, density operator and quantum channel
17	Tue. 3/17	Foundation: distance measures
18	Thu. 3/19	qDRIFT algorithm
19	Tue. 3/24	Spring break. No class.
20	Thu. 3/26	Spring break. No class.
21	Tue. 3/31	Quantum Fourier transform and quantum phase estimation
22	Thu. 4/2	Foundation: query lower bound
23	Tue. 4/7	Eigenvalue problems
24	Thu. 4/9	Linear systems of equations
25	Tue. 4/14	Differential equations
26	Thu. 4/16	Open quantum systems
27	Tue. 4/21	Project presentation.
28	Thu. 4/23	TBD
29	Tue. 4/28	Project presentation.
30	Thu. 4/30	Project presentation.