Programmable Quantum Simulation of Topological Matter and Learned Many-Body Diagonalization

Abstract

Quantum many-body systems pose fundamental challenges in our pursuit to understand and control the complex phenomena that emerge from the collective behavior of interacting particles. This thesis investigates how programmable quantum platforms, together with hybrid quantum-classical algorithms, can advance the study of strongly correlated quantum materials. We develop approaches for sensing magnetic order, preparing topologically ordered states, and improving quantum subspace methods for many-body simulation on quantum computers. First, we study spin defects in hexagonal boron nitride as local probes of magnetic order at material interfaces. We employ first-principles calculations to identify the electronic conditions required to preserve the negatively charged boron-vacancy defect near a magnetic substrate and show that proximity-induced exchange can produce exceptionally large effective magnetic fields. These results establish a potential route to probing microscopic magnetic structure beyond the capabilities of conventional stray-field sensing. Second, we realize a fermionic $\nu=1/3$ Laughlin state on a trapped-ion quantum processor. By constructing a symmetry-preserving Hamiltonian variational ansatz, we prepare the state on 16 qubits and characterize it using density observables, correlations, and entanglement measurements. The measured topological entanglement entropy provides compelling evidence for the first realization of a fermionic Laughlin state on a quantum processor. Third, we introduce generative Krylov quantum diagonalization, a learned reference-state proposal framework for stabilized quantum subspace diagonalization methods. We formulate the selection of Krylov reference states as a learning problem and develop a graph-conditioned multimodal autoregressive transformer that generates reference state circuits for interacting spinless Fermi--Hubbard models. The model learns to propose reference states that simultaneously yield low projected energies and well-conditioned subspace representations. Across the Hamiltonian instances considered, the learned proposals provide systematic improvements in energy and subspace stability relative to matched benchmark strategies.