Trapping and spreading properties of quantum walk in homological structure

TL;DR

This paper investigates how quantum walks on specific graph structures exhibit localization due to homological features, revealing spectral properties and differences in spreading behavior through eigenvalue analysis and density functions.

Contribution

It identifies the spectral and localization properties of Grover walks on graphs with homological structures, highlighting differences in their spectral measures and spreading behaviors.

Findings

Eigenvalues are induced by cycles in the graphs.

Localization occurs as part of the eigenspace related to homological structures.

Different spectral measures lead to distinct spreading behaviors.

Abstract

We attempt to extract a homological structure of two kinds of graphs by the Grover walk. The first one consists of a cycle and two semi-infinite lines and the second one is assembled by a periodic embedding of the cycles in $\mathbb{Z}$. We show that both of them have essentially the same eigenvalues induced by the existence of cycles in the infinite graphs. The eigenspace of the homological structure appears as so called {\it localization} in the Grover walks, in that the walk is partially trapped by the homological structure. On the other hand, the difference of the absolutely continuous part of spectrum between them provides different behaviors. We characterize the behaviors by the density functions in the weak convergence theorem: the first one is the delta measure at the bottom while the second one is expressed by two kinds of continuous functions which have different finite supports $(-1/\sqrt{10},1/\sqrt{10})$ and $(-2/7,2/7)$, respectively.