On an infinite family of integral Cayley graphs of Pauli groups

The classical Pauli group can be obtained as the central product of the dihedral group of $8$ elements with the cyclic group of order $4$. Inspired by this characterization, we introduce the notion of central product of Cayley graphs, which allows to regard the Cayley graph of a central product of groups as a quotient of the Cartesian product of the Cayley graphs of the factor groups. We focus our attention on the Cayley graph $Cay(P_n,S_{P_n})$ of the generalized Pauli group $P_n$ on $n$-qubits; in fact, $P_n$ may be decomposed as the central product of finite $2$-groups, and a suitable choice of the generating set $S_{P_n}$ allows us to recognize the structure of central product of graphs in $Cay(P_n,S_{P_n})$. Using this approach, we are able to recursively construct the adjacency matrix of $Cay(P_n,S_{P_n})$ for each $n\geq 1$, and to explicitly describe its spectrum and the associated eigenvectors. It turns out that $Cay(P_n,S_{P_n})$ is a $(3n+2)$-regular bipartite graph on $4^{n+1}$ vertices, and it has integral spectrum. This is a highly nontrivial property if one considers that, by choosing as a generating set for $P_1$ the three classical Pauli matrices, one gets the so-called M\"obius-Kantor graph, belonging to the class of generalized Petersen graphs, whose spectrum is not integral.