Date of Defense
9-4-2026 4:00 PM
Location
Room F3-136
Document Type
Thesis Defense
Degree Name
Master of Science in Mathematics
College
College of Science
Department
Mathematical Sciences
First Advisor
Houssam Abdul-Rahman
Keywords
Discrete-time quantum walks, quantum transport, velocity bounds, electric fields
Abstract
This thesis presents an analysis of transport in one-dimensional discrete-time quantum walks (DTQWs) on the Hilbert space . Quantum walks serve as fundamental models of coherent quantum transport and exhibit ballistic spreading driven by superposition and interference. The primary focus of this work is the review and derivation of sharp maximal velocity bounds for several classes of quantum walk step operators, including the shift-coin walk, the split-step walk, and models with constant as well as position-dependent coin operators. We establish general a priori bounds that remain valid beyond the translation-invariant regime. For homogeneous models, Fourier and spectral analysis yield explicit dispersion relations, allowing us to confirm the sharpness of the velocity bounds through precise group velocity formulas. In the presence of periodic electric fields, we demonstrate that translation invariance is restored after finitely many steps, leading to exact expressions for effective propagation velocities and revealing regimes of velocity suppression. The thesis further reviews the structural correspondence between split-step quantum walks and generalized extended CMV matrices and introduces lifted operator representations on enlarged Hilbert spaces to provide a framework for future extensions. These results contribute to a unified mathematical understanding of quantum walk transport in both field-free and field-driven settings.
Included in
Transport of Quantum Walks in Electric Fields
Room F3-136
This thesis presents an analysis of transport in one-dimensional discrete-time quantum walks (DTQWs) on the Hilbert space . Quantum walks serve as fundamental models of coherent quantum transport and exhibit ballistic spreading driven by superposition and interference. The primary focus of this work is the review and derivation of sharp maximal velocity bounds for several classes of quantum walk step operators, including the shift-coin walk, the split-step walk, and models with constant as well as position-dependent coin operators. We establish general a priori bounds that remain valid beyond the translation-invariant regime. For homogeneous models, Fourier and spectral analysis yield explicit dispersion relations, allowing us to confirm the sharpness of the velocity bounds through precise group velocity formulas. In the presence of periodic electric fields, we demonstrate that translation invariance is restored after finitely many steps, leading to exact expressions for effective propagation velocities and revealing regimes of velocity suppression. The thesis further reviews the structural correspondence between split-step quantum walks and generalized extended CMV matrices and introduces lifted operator representations on enlarged Hilbert spaces to provide a framework for future extensions. These results contribute to a unified mathematical understanding of quantum walk transport in both field-free and field-driven settings.