Part 2 Quantum Hamiltonian complexity | Sandy Irani (University of California, Irvine)

Lecture 2 notes
Problem session 2

One of the goals of quantum information theory is to understand quantum systems from the standpoint of computational complexity. How difficult is it to compute fundamental properties of a quantum system or simulate a particular system over time? Physicists have been using computers for decades to understand various aspects of quantum systems, but these methods are typically heuristic and achieve success on only limited classes of systems. In this course, we will approach these problems from a formal, complexity-theoretic point of view. In particular, one of the most basic properties of a system is its lowest energy state or ground state. We will focus on the complexity of ground states of finite and infinite systems and the computational resources required to compute them.

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The 2023 Program: Quantum Computation
Organizers: David Gosset, University of Waterloo; Aram Harrow, MIT; Stacey Jeffery, CWI and QuSoft; Ryan O'Donnell, Carnegie Mellon University; and Thomas Vidick, Caltech.

Very recently we have seen experiments at the boundary of the "quantum computing advantage", where quantum computers can massively outperform classical ones at certain tasks. These advances highlight the need for further mathematical understanding of the computational power of near-term quantum devices. The goal of the 2023 GSS is to dive deeply into the mathematics relevant for building near-term quantum computers, analyzing their power, and putting them to use. Minicourses will include: overviews of quantum learning, information theory, and linear-algebraic algorithms; recent advances in quantum error-correcting codes; and, the complexity theory of random circuits and Hamiltonians.

Structure:
The Graduate Summer School at PCMI consists of a series of several interwoven minicourses on different aspects of the main research theme of that summer. These courses are taught by leading experts in the field, chosen not only for their stature in the field but their pedagogical abilities. Each minicourse comprises three to five lectures. Each course is accompanied by a daily problem session, structured to help students develop facility with the material.

2023 Schedule
Week 1
Andras: Quantum Fourier transform beyond Shor's algorithm
Omar: Quantum information theory
Srinivasan: Overview of quantum learning theory
Week 2
Ewin: Quantum and quantum-inspired linear algebra
Nicolas: Quantum LDPC codes
Yassine: Quantum query complexity
Week 3
Bill: Computational complexity of near-term quantum experiments
Jeongwan: Topological aspects of quantum codes
Sandy: Quantum Hamiltonian complexity
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The GSS takes place within the broader structure of PCMI, so there are many researchers at all levels in the field in attendance, as well as participants in the other PCMI programs.