Sedentary quantum walks

TL;DR

This paper investigates quantum walks on specific graph structures, showing conditions under which the walk remains localized or achieves uniform mixing, revealing how graph properties influence quantum walk behavior.

Contribution

It provides new results on localization and mixing in quantum walks on cones over regular graphs and strongly regular graphs, depending on graph parameters.

Findings

Quantum walks on complete graphs tend to stay at the starting vertex with high probability.

Quantum walks on cones over graphs with large degree-to-vertex ratio tend to remain localized at the apex.

For small valency graphs, uniform mixing can occur at specific times.

Abstract

Let $X$ be a graph with adjacency matrix $A$. The \textsl{continuous quantum walk} on $X$ is determined by the unitary matrices $U(t)=\exp(itA)$. If $X$ is the complete graph $K_n$ and $a\in V(X)$, then \[1-|U(t)_{a,a}|\le2/n. \] In a sense, this means that a quantum walk on a complete graph stay home with high probability. In this paper we consider quantum walks on cones over an $\ell$-regular graph on $n$ vertices. We prove that if $\ell^2/n\to\infty$ as $n$ increases, than a quantum walk that starts on the apex of the cone will remain on it with probability tending to $1$ as $n$ increases. On the other hand, if $\ell\le2$ we prove that there is a time $t$ such that local uniform mixing occurs, i.e., all vertices are equally likely. We investigate when a quantum walk on strongly regular graph has a high probability of "staying at home", producing large families of examples with the stay-at-home property where the valency is small compared to the number of vertices.