Coined Quantum Walks Lift the Cospectrality of Graphs and Trees

In this paper we consider the problem of distinguishing graphs that are cospectral with respect to the standard adjacency and Laplacian matrix representations. Borrowing ideas from the field of quantum computing, we define a new matrix based on paths of the coined quantum walk. Quantum walks exhibit interference effects and their behaviour is markedly different to that of classical random walks. We show that the spectrum of this new matrix is able to distinguish many graphs which cannot be distinguished by standard spectral methods. We pay particular attention to strongly regular graphs; if a pair of strongly regular graphs share the same parameter set then there is no efficient algorithm that is proven to be able distinguish them. We have tested the method on large families of co-parametric strongly regular graphs and found it to be successful in every case. We have also tested the spectra’s performance when used to give a distance measure for inexact graph matching tasks.