Author ORCID Identifier

https://orcid.org/0009-0002-8856-4427

Date Available

11-21-2025

Year of Publication

2025

Document Type

Doctoral Dissertation

Degree Name

Doctor of Philosophy (PhD)

College

Arts and Sciences

Department/School/Program

Physics and Astronomy

Faculty

Anatoly Dymarsky

Abstract

This dissertation investigates computational techniques addressing critical problems arising within quantum chaos and quantum error correction (QEC). First, we investigate quantum dynamics via the Lanczos algorithm, establishing how its computational features — branching patterns, convergence, and complexity saturation — relate to physical properties like spectral statistics and operator matrix elements. We derive a formula connecting Lanczos coefficients to spectral densities for two-branched Lanczos sequences, revealing how branching patterns encode level repulsion—a hallmark of quantum chaos. Furthermore, we develop an analytic framework predicting Krylov complexity saturation from static system properties alone. Specifically, we decompose the saturation value into a spectral density-dependent baseline and a correction term governed by operator matrix element statistics, which in chaotic systems follows the Eigenstate Thermalization Hypothesis (ETH). Second, we address two key computational problems in QEC: computing the minimum code distance and performing syndrome decoding. We introduce an efficient reformulation of both problems as Quadratic Unconstrained Binary Optimization (QUBO) instances. This QUBO approach allows for the adoption of quantum algorithms to find numerical solutions. We demonstrate our method’s viability through numerical experiments conducted using classical and quantum devices.

Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2025.510

Funding Information

This research is supported by the NSF grant PHY 2310426 in 2023.

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