A complete classification of perfect state transfer in Grover walks on abelian Cayley graphs

Abstract:We completely classify connected Cayley graphs over finite abelian groups that exhibit perfect state transfer in Grover walks. First, we show that a Cayley graph over any finite group exhibiting perfect state transfer has valency $1$, $2$, or $4$. For valency $4$, we prove that an abelian Cayley graph $\operatorname{Cay}(\Gamma,S)$ exhibits perfect state transfer if and only if $S=\{a,-a,a+h,-a+h\}$, where $h$ has order $2$, and $4\nmid |\Gamma|/2$. Combining this characterization with the valency-one and valency-two cases, we explicitly determine all pairs $(\Gamma,S)$ for which $\operatorname{Cay}(\Gamma,S)$ exhibits perfect state transfer. Our results extend the classification of circulant graphs of valency at most four obtained by Kubota and Yoshino [Circulant graphs with valency up to 4 that admit perfect state transfer in Grover walks, \emph{J. Comb. Theory Ser. A.}, 216 (2025) 106064] to Cayley graphs over arbitrary finite abelian groups and to all valencies.

Combinatorics (math.CO); Quantum Physics (quant-ph)

arXiv:2610.21946 [math.CO]

(or arXiv:2610.21946v1 [math.CO] for this version)

https://doi.org/10.48550/arXiv.2610.21946 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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