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 problems at the end of most classes, and optional additional problems will provided at the end of each section of the notes.

Final Group Project

Grading will be based upon a final group 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. More details forthcoming!

Literature

We will utilize the following notes and surveys (many more to be added soon):

• 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)
• Course Notes: Felix Leditzky's "Representation-theoretic methods in quantum information theory"
• Tutorial: Antonio Anna Mele, "Introduction to Haar Measure Tools in Quantum Information" (2023)

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 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