The Quantum Walk Characteristic Polynomial Distinguishes All Strongly Regular Graphs of Prime Orde

TL;DR

This paper proves that the quantum walk characteristic polynomial uniquely identifies strongly regular graphs of prime order, enabling polynomial-time graph isomorphism testing within this class.

Contribution

It introduces a method to distinguish strongly regular graphs of prime order using quantum walk spectra, improving upon previous algorithms.

Findings

Quantum walk characteristic polynomial determines graph up to isomorphism.

Explicit formula links polynomial roots to Fourier coefficients of the adjacency matrix.

Graph isomorphism testing becomes polynomial-time for this class using quantum spectra.

Abstract

Let $G$ be a strongly regular graph of prime order $p$ with connection degree $k \geq 6$. We prove that the \emph{quantum walk characteristic polynomial} $\chi_q(G,\lambda) \coloneqq \det(\lambda I - U_G)$, where $U_G$ is the coined quantum walk operator on $G$, completely determines $G$ up to isomorphism within the class of strongly regular graphs of the same order. The proof proceeds in three steps. First, we show that $U_G$ block-diagonalizes under the discrete Fourier transform over $\Z_p$, yielding $p$ blocks $U_G^{(j)}$ of size $k \times k$. Second, we prove an explicit formula \[ \chi_q\!\bigl(U_G^{(j)}, \lambda\bigr) = (\lambda-1)^{(k-2)/2}(\lambda+1)^{(k-2)/2} \!\left(\lambda^2 - \tfrac{2\widehat{A}_G(j)}{k}\,\lambda + 1\right), \] from which the Fourier coefficient $\widehat{A}_G(j)$ is recovered as the unique real part of an eigenvalue of $U_G^{(j)}$ distinct from $\pm 1$. Third, the inverse discrete Fourier transform recovers the connection set $S$ of $G$, and Turner's theorem (1967) identifies $G$ up to isomorphism. As a consequence, graph isomorphism is decidable in polynomial time within this class using the quantum walk spectrum, without resorting to the general quasi-polynomial algorithm of Babai (2016).