Dissertations, Theses, and Capstone Projects

Date of Degree

6-2024

Document Type

Dissertation

Degree Name

Ph.D.

Program

Physics

Advisor

Vadim Oganesyan

Committee Members

Sarang Gopalakrishnan

Vladimir Rosenhaus

Tzu-Chieh Wei

Sriram Ganeshan

Miles E. Stoudenmire

Subject Categories

Condensed Matter Physics | Fluid Dynamics | Quantum Physics | Statistical, Nonlinear, and Soft Matter Physics

Keywords

Tensor Networks, Statistical Mechanics, XY model, Quantics Tensor Trains, Information theory, Partial Differential Equation

Abstract

This four-chapter dissertation studies the efficient discretization of continuous variable functions with tensor train representation. The first chapter describes all the methodology used to discretize functions and store them efficiently. In this section, the algorithm tensor renormalization group is explained for self-containment purposes. The second chapter centers around the XY model. Quantics tensor trains are used to describe the transfer matrix of the model and compute one and two-dimensional quantities. The one dimensional magnitudes are compared to analytical results with an agreement close to machine precision. As for two dimensions, the analytical results cannot be computed. However, the critical temperature of the BKT transition is calculated and compared to state-of-the-art results, with good agreement between them. The third chapter describes how to solve differential and integral equations in one dimensions with tensor trains. Specifically, the heat equation and the Schrödinger’s equation with Ekham potential are solved with high accuracy. In the final chapter, the focus shifts to the study of the ability to guess the initial state of a qubit with a string of T measurements when it is evolving under an external field. The projective measurement limit behaves as expected, but the weak measurement displays a universal behavior independent of the field applied. There are 3 different appendices containing technical discussions about the dependence of the bond dimension with temperature, and the compressibility of certain functions.

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