In partial fulfillment of the requirements for the degree of 

Doctor of Philosophy in Physics 

School of Physics Thesis Dissertation Defense

Zhixin (Jack) Song

Dr. Spencer Bryngelson, CSE, Georgia Institute of Technology (Advisor)

Solving Differential Equations on Quantum Computers

Date: Monday April 13, 2026

Time: 9:00 a.m.

Location: Howey Physics Building, Room N201/202

Zoom Link: https://gatech.zoom.us/j/91272230097?pwd=eU4AkZyuaKgRhQcUZ7kfRe38BCPmpq.1

Committee members:

Dr. Bryan Gard, CIPHER, Georgia Tech Research Institute

Dr. Fenton Flavio, School of Physics, Georgia Institute of Technology

Dr. Brian Kennedy, School of Physics, Georgia Institute of Technology

Dr. Xiangyu Li, Physical and Computational Sciences, Pacific Northwest National Laboratory

Abstract:

Quantum computing has emerged as a transformative computational paradigm, promising superpolynomial speedups for problems that remain intractable on classical hardware. One compelling application is the solution of differential equations, which underpin the modeling of many physical, chemical, and engineering systems but often incur prohibitive computational costs as problem dimensionality grows. This thesis investigates near-term and fault-tolerant quantum algorithms for solving partial differential equations, addressing important bottlenecks such as efficient state preparation through matrix-product-state approximation and readout through real-valued tomography. We develop and analyze end-to-end quantum workflows built upon quantum linear system algorithms (QLSAs) and the Linear Combination of Hamiltonian Simulation (LCHS) framework, which maps non-unitary dynamics onto a unitary dilation amenable to quantum simulation. We benchmark and validate the near-term quantum algorithms on real IBM quantum hardware. Although fault-tolerant quantum computers remain far from realization, we present resource estimates for fault-tolerant implementations targeting surface-code architectures, offering concrete guidance on the problem sizes and precision regimes in which quantum speedups over state-of-the-art classical solvers may become achievable.