Welcome to

MATH 595: Quantum Learning Theory

Spring 2026 — University of Illinois, Urbana-Champaign

Course Information

Welcome to MATH 595: Quantum Learning Theory! This course will survey the burgeoning field of quantum learning theory (QLT), which lies at the intersection of mathematics, theoretical computer science, and quantum information theory. QLT can be viewed as a quantum generalization of classical statistical learning theory. At its core, the goal of QLT is to quantify the resources (samples, measurements, memory, time, etc.) needed to test or learn a desired quantity. In this course, we will place a particular emphasis on understanding the effects that randomness, adaptivity, and measurement restrictions have on the sample complexity of various quantum state learning and testing tasks.

Instructor: Jacob Beckey
Meeting times: Tues/Thurs, 3:30–4:50 PM
Location: 132 Davenport Hall
Prerequisites: Math 416 Abstract Linear Algebra (or equivalent) and, ideally, Intro to Quantum Mechanics/Information (e.g., Phys 486/487, Phys 513, or ECE 404)

Contact & Communication

Instructor's email: jbeckey@illinois.edu
My office: 400B Harker Hall

Please reach out if you have questions about course content, assignments, or logistics!

Assignments & Grading

There will be no mandatory homework or exams. We will work through proofs together in most classes, and optional additional problems will provided at the end of each section of the notes.

Final Project

Grading will be based upon a final project. All students will do a final project that distills a QLT paper into a guided tutorial-style problem that their peers could complete by that point in the course, highlighting open questions left by the paper. An example tutorial can be found here.

Classroom Climate & Inclusivity

People learn mathematics best when they feel respected and comfortable asking questions. In this course, we are committed to creating a supportive environment where everyone can engage fully in discourse and grow as mathematical thinkers.

All students are expected to treat one another with respect, patience, and kindness.
Questions, confusion, and mistakes are a normal and valuable part of learning mathematics.
Disruptive, dismissive, or disrespectful behavior will not be tolerated.
If something in the course environment makes it harder for you to participate, please let me know so we can address it.

Course Notes

Primary Notes: These notes will be uploaded continuously throughout the semester.

Relevant Lecture Notes and Review Articles

Quantum Learning Theory

Course Notes: John Wright's Quantum Learning Theory course notes (UC Berkeley, Spring 2024)
Course Notes: Sitan Chen and Jordan Cotler's Quantum Learning Theory course notes (Harvard, Fall 2025)
Course Notes: Robert Huangs's Quantum Learning Theory course notes (Caltech, Fall 2025)
Survey: Montanaro and de Wolf, "A Survey of Quantum Property Testing" (2016)
Survey: Anshu and Arunachalam, "A Survey on the Complexity of Learning Quantum States" (2024)

Representation Theoretical Methods in QIT

Course Notes: Felix Leditzky's "Representation-theoretic methods in quantum information theory" (UIUC, Fall 2025)
Course Notes: Michael Walter's "Symmetry in Quantum Information" (University of Amsterdam, Spring 2018)
Tutorial: Antonio Anna Mele, "Introduction to Haar Measure Tools in Quantum Information" (2023)

Course Outline

The course will focus on the following canonical testing/learning problems and associated mathematical techniques:

Quantum State Discrimination: Pure state and mixed state discrimination
Quantum State Tomography: Representation theory review, mixed state tomography with multi-copy and restricted measurements
Shadow Tomography: Classical shadow tomography, exponential separations, applications, and open questions
Quantum Property Testing: State certification, purity testing (and exponential separations), entanglement testing
Advanced Topics: Open problems and class-voted topics

Weekly Outline

Date	Lecture	Topics Covered	Additional Resources
1/20	1	Introduction; pure state discrimination
1/22	2	Mixed state discrimination; total variation distance; trace distance
1/27	3	Problem Session
1/29	4	Cancelled (QIP)
2/3	5	Distinguishing distriubtions with multiple samples
2/5	6	Distinguishing distriubtions with multiple samples (cont.)
2/10	7	Distinguishing states with multiple samples; fidelity; Start Pauli tomography
2/12	8	Textbook Pauli tomography algorithm
2/17	9	Measurement classes and simulation; Intro to Haar Measure	Simulating measurments [Wri25]; Intro to Haar [Mel24], Pennylane Haar Tutorial, Generating Haar Random Matrices [Ozo09]
2/19	10	The Church of the Symmetric Subspace	Symmetric Subspace [Har13], [Mel24]
2/24	11	Optimal Single-copy, global QST	Algorithm we covered (Sec. 5.1 of [Wri16] or Lecture 13 in [Wri25]); Original papers [KRT14], [Gut+18]
2/26	12	Multi-copy, global pure state tomography	Pure state algorithm (Lecture 12 in [Wri25] or Sec. 4.3 in [Wal18]); Continuous POVMs (Sec. 4.1 in [Wal18]); Original paper [Hay98]
3/3	13	Multi-copy, global mixed state tomography	Original papers [HHJ+16], [OW15]; Recent optimal algorithms via pure state tomography reduction [PSTW25], lower bound [SSW25]; Random purification paper [TWZ25], simplified treatment [GML25]
3/5		Class cancelled due to travel
3/10	14	Universal Tomography Lower Bound	Original papers [HHJ+16], [OW15]; Angus Lowe's masters thesis [Low21]; John Wright's Thesis [Wri16]
3/12	15	Universal Tomography Lower Bound	Original papers [HHJ+16], [OW15]; Angus Lowe's masters thesis [Low21]; John Wright's Thesis [Wri16]
3/24	16	Classical Shadows I	Original paper [HKP20]; Chen and Cotler's lecture notes [CC25]; Robert's lecture notes [Hua25]
3/26	17	Classical Shadows II	Original paper [HKP20]; Chen and Cotler's lecture notes [CC25]; Robert's lecture notes [Hua25]
3/31	18	Classical Shadows III	Original paper [HKP20]; Chen and Cotler's lecture notes [CC25]; Robert's lecture notes [Hua25]
4/2	19	Single-copy Lower Bound for Pauli Shadows	Original paper [CCHL22]; Chen and Cotler's lecture notes [CC25]
4/7	20	Multi-copy Upper Bound for Pauli Shadow Tomography	Learning Absolute Values [HKP21] and Learning Signs [CGY24]; Chen and Cotler's lecture notes [CC25]
4/9	21	Shadow Tomography Wrap-up
4/14	22	Property Testing and the SWAP Test
4/16	23	Purity Testing Exponential Separation
4/21	24	Product Testing
4/23	25	Product Testing Wrap-up + Course recap/open questions
4/28	26	Visitor Talk: Jack Spilecki (UC Berkeley)
4/30	27	In-person Final Project Peer Review
5/5	27	Class Cancelled -- Final Project Work