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 C a y (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 C a y (P n , S P n ). Using this approach, we are able to recursively construct the adjacency matrix of C a y (P n , S P n ) for each n ≥ 1 , and to explicitly describe its spectrum and the associated eigenvectors. It turns out that C a y (P n , S P n ) is a (3 n + 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 Möbius-Kantor graph, belonging to the class of generalized Petersen graphs, whose spectrum is not integral. [ABSTRACT FROM AUTHOR]