Quantum Walks on Generalized Quadrangles

TL;DR

This paper investigates the spectral properties of quantum walks on strongly regular graphs, providing counterexamples to a conjecture by analyzing line intersection graphs of generalized quadrangles.

Contribution

It offers the first probabilistic computation of the spectrum of a quantum walk matrix on specific strongly regular graphs, challenging a previous conjecture.

Findings

Counterexamples to the conjecture were found.

Spectral analysis distinguishes non-isomorphic graphs.

Probabilistic methods effectively compute spectra.

Abstract

We study the transition matrix of a quantum walk on strongly regular graphs. It is proposed by Emms, Hancock, Severini and Wilson in 2006, that the spectrum of $S^+(U^3)$, a matrix based on the amplitudes of walks in the quantum walk, distinguishes strongly regular graphs. We probabilistically compute the spectrum of the line intersection graphs of two non-isomorphic generalized quadrangles of order $(5^2,5)$ under this matrix and thus provide strongly regular counter-examples to the conjecture.