Quantum automorphisms of folded cube graphs

We show that the quantum automorphism group of the Clebsch graph is $S{O}_5^{-1}$. This answers a question by Banica, Bichon and Collins from 2007. More general for odd $n$, the quantum automorphism group of the folded $n$-cube graph is $S{O}_n^{-1}$. Furthermore, we show that if the automorphism group of a graph contains a pair of disjoint automorphisms this graph has quantum symmetry.