Advancing Applications of Quantum Computers in Quantum Simulation, Optimization, Learning, and Topological Data Analysis

Author: King, William Robert

Year: 2025

Degree: Dissertation (Ph.D.)

Advisor: Schulman, Leonard J.

Committee Members: Vidick, Thomas Georges; Schulman, Leonard J.; Preskill, John P.; Huang, Hsin-Yuan (Robert); Umans, Christopher M.

Option: Computer Science

DOI: 10.7907/w905-b372

Abstract

This thesis investigates novel directions for harnessing the potential of quantum computers in future applications. It is structured into three sections.

Quantum Simulation.
We address two key questions: what systems exhibit quantum advantage in predicting ground state properties, and how can we reduce the cost of quantum simulations? For the former, we find that strongly interacting fermionic systems have promising characteristics for quantum advantage. For the latter, we develop an improved method for compiling block encodings using sum-of-squares optimization.

Learning with Entangled Measurements.
We explore the benefits of leveraging entangled measurements on quantum states stored in quantum memory. These learning algorithms can be applied to the readout stage of quantum simulations, or to learn from quantum data from nature.

Topological Data Analysis.

Using complexity-theoretic insights, we demonstrate that certain problems in topological data analysis possess a quantum mechanical structure, suggesting opportunities for quantum algorithms in this area.

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