The spreading behavior of quantum walks induced by drifted random walks on some magnifier graph

TL;DR

This paper investigates how attaching finite paths to a one-dimensional lattice induces localization in quantum walks, linking it to the properties of an underlying random walk and revealing altered velocity and distribution behaviors.

Contribution

It demonstrates that localization in quantum walks on modified graphs is caused by eigenvalues related to the underlying random walk, and shows how transience affects the walk's dynamics.

Findings

Localization is caused by eigenvalues from finite round trip paths.

Transience of the underlying random walk slows the quantum walk's pseudo velocity.

The limit distribution differs from the standard Konno distribution.

Abstract

In this paper, we consider the quantum walk on $\mathbb{Z}$ with attachment of one-length path periodically. This small modification to $\mathbb{Z}$ provides localization of the quantum walk. The eigenspace causing this localization is generated by finite length round trip paths. We find that the localization is due to the eigenvalues of an underlying random walk. Moreover we find that the transience of the underlying random walk provides a slow down of the pseudo velocity of the induced quantum walk and a different limit distribution from the Konno distribution.