On the behavior of quantum walks confined to a cycle coupled with a half line

TL;DR

This paper investigates the long-term behavior of quantum walks on a combined cycle and half-line environment, revealing distinct non-classical distribution patterns that differ significantly from classical random walks.

Contribution

It provides a theoretical analysis and numerical evidence of the unique distributional tendencies of quantum walks in a topologically confined setting.

Findings

Quantum walks maintain a non-zero probability on the cycle.

Quantum walks exhibit ballistic spreading on the half-line.

Classical walks tend to vanish on the cycle and spread diffusively on the half-line.

Abstract

When confined to a topological environment consisting of a cycle coupled with a half-line, quantum walks exhibit long-term statistical tendencies which differ dramatically from the tendencies of classical random walks in the same environment. In particular, as suggested by numerical simulations, the probability distribution of the walker's position resolves, in part, into a non-vanishing distribution on the cycle and, in part, into a ballistic distribution on the half-line. By contrast, for a classical random walk, the probability distribution of the walker's position tends always to vanish on the cycle and to migrate completely to the half-line as a purely diffusive process.