  We show that any two Hadamard graphs on the same number of vertices are
quantum isomorphic. This follows from a more general recipe for showing quantum
isomorphism of graphs arising from certain association schemes. The main result
is built from three tools. A remarkable recent result of Man\v{c}inska and
Roberson shows that graphs $G$ and $H$ are quantum isomorphic if and only if,
for any planar graph $F$, the number of graph homomorphisms from $F$ to $G$ is
equal to the number of graph homomorphisms from $F$ to $H$. A generalization of
partition functions called "scaffolds" affords some basic reduction rules such
as series-parallel reduction and can be applied to counting homomorphisms. The
final tool is the classical theorem of Epifanov showing that any plane graph
can be reduced to a single vertex and no edges by extended series-parallel
reductions and Delta-Wye transformations. This last sort of transformation is
available to us in the case of exactly triply regular association schemes. The
paper includes open problems and directions for future research.
