Quantum symmetries of Cayley graphs of abelian groups

Abstract

We study Cayley graphs of abelian groups from the perspective of quantum symmetries. We develop a general strategy for determining the quantum automorphism groups of such graphs. Applying this procedure, we find the quantum symmetries of the halved cube graph, the folded cube graph, and the Hamming graphs.

Quantum isomorphic strongly regular graphs from the E 8 root system

“…since we have M * M = 1 for all M associated to the permutations in L. Since P wi and P wj are the rank-1 projections associated to w i and w j , respectively, we have ⟨w i , w j ⟩ = 0 if and only if P wi P wj = 0. We see that the latter is equivalent to u (i) ks u (j) lt = 0 by multiplying M * from the left and M from the right in (7). For (ii), recall from Lemma 3.5 that u (i) ks and u (j) lt are the rank-1 projections of the vectors associated to the vertices in V i and V j , respectively.…”

“…They could find the graphs of [1] in this way, but they did not construct new examples. Recently, Chan and Martin [5] and Gromada [7] have shown that Hadamard graphs of the same order are quantum isomorphic.…”