Quantum state transfer on Q-graphs

TL;DR

This paper investigates quantum state transfer properties in Q-graphs derived from regular graphs, establishing conditions under which perfect or pretty good state transfer occurs, and identifying new families of such graphs.

Contribution

It provides new theoretical results linking eigenvalues of regular graphs to quantum state transfer capabilities in their Q-graphs, including conditions for no perfect transfer and the existence of pretty good transfer.

Findings

Regular graphs with all integer eigenvalues have Q-graphs with no perfect state transfer.

Q-graphs of regular graphs can exhibit pretty good state transfer under mild conditions.

Many new families of Q-graphs are identified that lack perfect transfer but have pretty good transfer.

Abstract

We study the existence of quantum state transfer in $\mathcal{Q}$-graphs in this paper. The $\mathcal{Q}$-graph of a graph $G$, denoted by $\mathcal{Q}(G)$, is the graph derived from $G$ by plugging a new vertex to each edge of $G$ and joining two new vertices which lie on adjacent edges of $G$ by an edge. We show that, if all eigenvalues of a regular graph $G$ are integers, then its $\mathcal{Q}$-graph $\mathcal{Q}(G)$ has no perfect state transfer. In contrast, we also prove that the $\mathcal{Q}$-graph of a regular graph has pretty good state transfer under some mild conditions. Finally, applying the obtained results, we also exhibit many new families of $\mathcal{Q}$-graphs having no perfect state transfer, but admitting pretty good state transfer.