Quantum walks on blow-up graphs

• URL: http://arxiv.org/abs/2308.13887v2
• Date: Sun, 14 Jan 2024 19:45:42 GMT
• Title: Quantum walks on blow-up graphs
• Authors: Bikash Bhattacharjya, Hermie Monterde, Hiranmoy Pal
• Abstract summary: A blow-up of $n$ copies of a graph $G$ is the graph $oversetnuplusG$. We investigate the existence of quantum state transfer on a blow-up graph $oversetnuplusG$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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