  We clarify the relations between the mathematical structures that enable
fashioning quantum walks on regular graphs and their realizations in anyonic
systems. Our protagonist is association schemes that may be synthesized from
type-II matrices which have a canonical construction of subfactors. This way we
set up quantum walks on growing distance-regular graphs induced by association
schemes via interacting Fock spaces and relate them to anyon systems described
by subfactors. We discuss in detail a large family of graphs that may be
treated within this approach. Classification of association schemes and
realizable anyon systems are complex combinatorial problems and we tackle a
part of it with a quantum walk application based approach.
