Perfect state transfer, graph products and equitable partitions

TL;DR

This paper introduces new graph constructions, including variants of double cones and Cartesian products, that exhibit perfect state transfer in quantum walks, expanding the understanding of quantum information transfer in complex networks.

Contribution

The paper presents novel methods for constructing graphs with perfect state transfer using graph products, double cones, and equitable partitions, generalizing previous results.

Findings

Double cone graphs with weighted edges can have perfect state transfer.

Certain graph products preserve perfect state transfer under specific spectral conditions.

A generalized path collapsing technique simplifies analysis of perfect state transfer.

Abstract

We describe new constructions of graphs which exhibit perfect state transfer on continuous-time quantum walks. Our constructions are based on variants of the double cones [BCMS09,ANOPRT10,ANOPRT09] and the Cartesian graph products (which includes the n-cube) [CDDEKL05]. Some of our results include: (1) If $G$ is a graph with perfect state transfer at time $t_{G}$, where $t_{G}\Spec(G) \subseteq \ZZ\pi$, and $H$ is a circulant with odd eigenvalues, their weak product $G \times H$ has perfect state transfer. Also, if $H$ is a regular graph with perfect state transfer at time $t_{H}$ and $G$ is a graph where $t_{H}|V_{H}|\Spec(G) \subseteq 2\ZZ\pi$, their lexicographic product $G[H]$ has perfect state transfer. (2) The double cone $\overline{K}_{2} + G$ on any connected graph $G$, has perfect state transfer if the weights of the cone edges are proportional to the Perron eigenvector of $G$. This generalizes results for double cone on regular graphs studied in [BCMS09,ANOPRT10,ANOPRT09]. (3) For an infinite family $\GG$ of regular graphs, there is a circulant connection so the graph $K_{1}+\GG\circ\GG+K_{1}$ has perfect state transfer. In contrast, no perfect state transfer exists if a complete bipartite connection is used (even in the presence of weights) [ANOPRT09]. We also describe a generalization of the path collapsing argument [CCDFGS03,CDDEKL05], which reduces questions about perfect state transfer to simpler (weighted) multigraphs, for graphs with equitable distance partitions.