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

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 $\mathrm{Cay}(\mathrm{\Gamma },S)$ exhibits perfect state transfer if and only if $S=\{a,-a,a+h,-a+h\}$, where $h$ has order $2$, and $4\nmid \mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }\mathrm{/}2$. Combining this characterization with the valency-one and valency-two cases, we explicitly determine all pairs $(\mathrm{\Gamma },S)$ for which $\mathrm{Cay}(\mathrm{\Gamma },S)$ 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.