Geometrical aspects of quantum walks on random two-dimensional structures

TL;DR

This paper investigates how the geometry of two-dimensional structures affects quantum and classical transport properties, revealing that quantum walks require more bonds for transport in elongated structures and exhibit localization effects.

Contribution

It provides a comparative analysis of quantum and classical transport on random 2D structures with varying aspect ratios, highlighting the impact of geometry on quantum localization and transport efficiency.

Findings

Quantum walks need more bonds than classical walks in elongated structures.

Quantum eigenstates are strongly localized in configurations with facilitated classical transport.

Transport efficiency varies non-monotonically with aspect ratio in classical cases.

Abstract

We study the transport properties of continuous-time quantum walks (CTQW) over finite two-dimensional structures with a given number of randomly placed bonds and with different aspect ratios (AR). Here, we focus on the transport from, say, the left side to the right side of the structure where absorbing sites are placed. We do so by analyzing the long-time average of the survival probability of CTQW. We compare the results to the classical continuous-time random walk case (CTRW). For small AR (landscape configurations) we observe only small differences between the quantum and the classical transport properties, i.e., roughly the same number of bonds is needed to facilitate the transport. However, with increasing AR (portrait configurations) a much larger number of bonds is needed in the CTQW case than in the CTRW case. While for CTRW the number of bonds needed decreases when going from small AR to large AR, for CTRW this number is large for small AR, has a minimum for the square configuration, and increases again for increasing AR. We corroborate our findings for large AR by showing that the corresponding quantum eigenstates are strongly localized in situations in which the transport is facilitated in the CTRW case.