The role of tessellation intersection in staggered quantum walks

TL;DR

This paper investigates how the size of polygon intersections in the staggered quantum walk model affects quantum walk dynamics, introducing processes to modify intersections and relate different graph structures.

Contribution

It introduces intersection reduction and expansion processes that relate SQWs on different graphs and simplify their analysis, including establishing equivalences with Szegedy's model.

Findings

Intersection size influences SQW dynamics.

Processes relate SQWs on different graphs.

Simplifies analysis and establishes equivalences.

Abstract

The staggered quantum walk (SQW) model is defined by partitioning the graph into cliques, which are called polygons. We analyze the role that the size of the polygon intersection plays on the dynamics of SQWs on graphs. We introduce two processes (intersection reduction and intersection expansion), that change the number of vertices in some intersection of polygons, and we compare the behavior of the SQW on the reduced or expanded graph in relation to the SQW on the original graph. We describe how the eigenvectors and eigenvalues of the evolution operators relate to each other. This processes can help to establish the equivalence between SQWs on different graphs and to simplify the analysis of SQWs. We also show an example of a SQW on a graph that is not included in Szegedy's model, but which is equivalent to an instance of Szegedy's model after applying the intersection reduction.