Local subgraph structure can cause localization in continuous-time quantum walk

TL;DR

This paper investigates how local subgraph structures influence the localization phenomenon in continuous-time quantum walks on graphs, providing a decomposition method and conditions for localization as graph size grows.

Contribution

It introduces a novel graph decomposition technique and establishes conditions under which quantum walk localization occurs in large graphs.

Findings

Decomposition method for Laplacian matrices based on graph structure

Conditions for localization probability tending to 1 in large graphs

Insights into the influence of local subgraph structures on quantum walk behavior

Abstract

In this paper, we consider continuous-time quantum walks (CTQWs) on finite graphs determined by the Laplacian matrices. By introducing fully interconnected graph decomposition of given graphs, we show a decomposition method for the Laplacian matrices. Using the decomposition method, we show several conditions for graph structure which return probability of CTQW tends to 1 while the number of vertices tends to infinity.