Quantum walks on blow-up graphs

TL;DR

This paper investigates quantum state transfer properties on blow-up graphs, establishing conditions for perfect and pretty good state transfer, and finds that such transfer occurs only when the blow-up size is two.

Contribution

It provides necessary and sufficient conditions for quantum state transfer on blow-up graphs and characterizes when PST and PGST occur, especially highlighting the case when n=2.

Findings

PST and PGST on blow-up graphs only occur when n=2.

Vertices in blow-up graphs with invertible adjacency matrices exhibit strong cospectrality.

Infinite families of graphs with PST and PGST are identified through blow-up constructions.

Abstract

A blow-up of $n$ copies of a graph $G$ is the graph $\overset{n}\uplus~G$ obtained by replacing every vertex of $G$ by an independent set of size $n$, where the copies of vertices in $G$ are adjacent in the blow-up if and only if the vertices adjacent in $G$. Our goal is to investigate the existence of quantum state transfer on a blow-up graph $\overset{n}\uplus~G$, where the adjacency matrix is taken to be the time-independent Hamiltonian of the quantum system represented by $\overset{n}\uplus~G$. In particular, we establish necessary and sufficient conditions for vertices in a blow-up graph to exhibit strong cospectrality and various types of high probability quantum transport, such as periodicity, perfect state transfer (PST) and pretty good state transfer (PGST). It turns out, if $\overset{n}\uplus~G$ admits PST or PGST, then one must have $n=2.$ Moreover, if $G$ has an invertible adjacency matrix, then we show that every vertex in $\overset{2}\uplus~G$ pairs up with a unique vertex to exhibit strong cospectrality. We then apply our results to determine infinite families of graphs whose blow-ups admit PST and PGST.