## Physics & Astronomy ETDs

## Publication Date

Spring 5-1-2024

## Abstract

Fermions are fundamental particles which obey seemingly bizarre quantum-mechanical principles, yet constitute all the ordinary matter that we inhabit. As such, their study is heavily motivated from both fundamental and practical incentives. In this dissertation, we will explore how the tools of quantum information and computation can assist us on both of these fronts. We primarily do so through the task of partial state learning: tomographic protocols for acquiring a reduced, but sufficient, classical description of a quantum system. Developing fast methods for partial tomography addresses a critical bottleneck in quantum simulation algorithms, which is a particularly pressing issue for currently available, imperfect quantum machines. At the same time, in the search for such protocols, we also refine our notion of what it means to learn quantum states. One important example is the ability to articulate, from a computational perspective, how the learning of fermions contrasts with other types of particles.

The main results of this dissertation are as follows. First, we introduce a protocol called fermionic classical shadows that efficiently learns all local properties of a many-fermion system. We furthermore show that this protocol is optimal, in the sense that information-theoretic arguments imply matching lower bounds for the fundamental complexity of this learning task. Next, we extend the theory to a noise-robust formulation called symmetry-adjusted classical shadows, which uses symmetry in the quantum system to mitigate errors that occur during the quantum computation. We establish a rigorous theory for this protocol, as well as demonstrate realistic, near-term performance with comprehensive numerical experiments. We then study unitary partitioning, an alternative strategy for easing the measurement complexity of Hamiltonians for interacting electrons. Finally, we draw surprising connections between a hard classical optimization problem, known as the noncommutative Grothendieck problem, and the simulation of fermions. We establish a natural embedding of the classical problem into a fermionic one and provide both theoretical and numerical evidence that quantum computers can provide high-quality solutions over classical approximations through this embedding.

## Degree Name

Physics

## Level of Degree

Doctoral

## Department Name

Physics & Astronomy

## First Committee Member (Chair)

Akimasa Miyake

## Second Committee Member

Ivan H. Deutsch

## Third Committee Member

Milad Marvian

## Fourth Committee Member

Andrew J. Landahl

## Language

English

## Keywords

Quantum computation, Quantum algorithms, Fermionic simulation, Quantum tomography, Error mitigation

## Document Type

Dissertation

## Recommended Citation

Zhao, Andrew. "Learning, Optimizing, and Simulating Fermions with Quantum Computers." (2024). https://digitalrepository.unm.edu/phyc_etds/319