  We classify instances of quantum pseudo-telepathy in the graph isomorphism
game, exploiting the recently discovered connection between quantum information
and the theory of quantum automorphism groups. Specifically, we show that
graphs quantum isomorphic to a given graph are in bijective correspondence with
Morita equivalence classes of certain Frobenius algebras in the category of
finite-dimensional representations of the quantum automorphism algebra of that
graph. We show that such a Frobenius algebra may be constructed from a central
type subgroup of the classical automorphism group, whose action on the graph
has coisotropic vertex stabilisers. In particular, if the original graph has no
quantum symmetries, quantum isomorphic graphs are classified by such subgroups.
We show that all quantum isomorphic graph pairs corresponding to a well-known
family of binary constraint systems arise from this group-theoretical
construction. We use our classification to show that, of the small order
vertex-transitive graphs with no quantum symmetry, none is quantum isomorphic
to a non-isomorphic graph. We show that this is in fact asymptotically almost
surely true of all graphs.
