State transfer in strongly regular graphs with an edge perturbation

TL;DR

This paper investigates how specific edge perturbations in strongly regular graphs can induce perfect or pretty good quantum state transfer, expanding understanding of quantum walks in algebraic graph structures.

Contribution

It characterizes conditions for state transfer in perturbed strongly regular graphs and provides infinite families where perfect transfer occurs, also showing the independence of certain graph invariants.

Findings

Certain perturbations induce perfect state transfer in strongly regular graphs.

Every strongly regular graph admits some perturbation leading to pretty good state transfer.

The graph invariant (X e) is independent of the chosen edge e.

Abstract

Quantum walks, an important tool in quantum computing, have been very successfully investigated using techniques in algebraic graph theory. We are motivated by the study of state transfer in continuous-time quantum walks, which is understood to be a rare and interesting phenomenon. We consider a perturbation on an edge $uv$ of a graph where we add a weight $\beta$ to the edge and a loop of weight $\gamma$ to each of $u$ and $v$. We characterize when for this perturbation results in strongly cospectral vertices $u$ and $v$. Applying this to strongly regular graphs, we give infinite families of strongly regular graphs where some perturbation results in perfect state transfer. Further, we show that, for every strongly regular graph, there is some perturbation which results in pretty good state transfer. We also show for any strongly regular graph $X$ and edge $e \in E(X)$, that $\phi(X\setminus e)$ does not depend on the choice of $e$.