$ε$-Uniform Mixing in Discrete Quantum Walks

TL;DR

This work analyzes local $\epsilon$-uniform mixing in discrete quantum walks started from a vertex, linking the mixing behavior to the spectral data of regular non-bipartite graphs through the transition matrix $U$ and its eigenprojections. It develops a comprehensive spectral-structural framework (including strong cospectrality, PGST, and Kronecker approximation) and leverages equitable partitions and association schemes to classify when uniform or $\epsilon$-uniform mixing occurs in two central families: complete graphs and strongly regular graphs. The main results show that complete graphs admit very limited uniform mixing (only $K_2$ and $K_4$), while strongly regular graphs admit $\epsilon$-uniform mixing precisely when $X$ or its complement has parameter sets $(4m^2,2m^2\pm m,m^2\pm m,m^2\pm m)$ for $m\ge 2$, with several infinite families and special cases identified via Hadamard structures. These findings connect quantum walk mixing phenomena to Hadamard matrices in Bose–Mesner algebras and provide a rigorous, spectrum-driven classification with implications for quantum information transfer on regular graphs.

Abstract

We study whether the probability distribution of a discrete quantum walk can get arbitrarily close to uniform, given that the walk starts with a uniform superposition of the outgoing arcs of some vertex. We establish a characterization of this phenomenon on regular non-bipartite graphs in terms of their adjacency eigenvalues and eigenprojections. Using theory from association schemes, we show this phenomenon happens on a strongly regular graph $X$ if and only if $X$ or $\overline{X}$ has parameters $(4m^2, 2m^2\pm m, m^2\pm m, m^2\pm m)$ where $m\ge 2$.

Theorems & Definitions (39)

• Theorem 2.1
• Lemma 2.2
• Theorem 2.3
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 4.1
• Corollary 4.2
• Theorem 4.3
• Lemma 5.1