Dynamical Localization for d-Dimensional Random Quantum Walks

TL;DR

This paper proves dynamical localization in d-dimensional random quantum walks with site-dependent randomness, showing that under certain conditions, the walk exhibits Anderson localization with bounded moments and spectral localization.

Contribution

It demonstrates that when deterministic transition amplitudes are near those preventing propagation, dynamical localization occurs for almost all random phases in high-dimensional quantum walks.

Findings

Dynamical localization is proven under specific amplitude conditions.

Quantum moments remain uniformly bounded over time.

Spectral localization holds almost surely.

Abstract

We consider a d-dimensional random quantum walk with site-dependent random coin operators. The corresponding transition coefficients are characterized by deterministic amplitudes times independent identically distributed site-dependent random phases. When the deterministic transition amplitudes are close enough to those of a quantum walk which forbids propagation, we prove that dynamical localization holds for almost all random phases. This instance of Anderson localization implies that all quantum mechanical moments of the position operator are uniformly bounded in time and that spectral localization holds, almost surely.