Quantum walks on regular graphs with realizations in a system of anyons

Abstract

We build interacting Fock spaces from association schemes and set up quantum walks on the resulting regular graphs (distance-regular and distance-transitive). The construction is valid for growing graphs and the interacting Fock space is well defined asymptotically for the growing graph. To realize the quantum walks defined on the spaces in terms of anyons we switch to the dual view of the association schemes and identify the corresponding modular tensor categories from the Bose–Mesner algebra. Informally, the fusion ring induced by the association scheme and a topological twist can be the basis for developing a modular tensor category and thus a system of anyons. Finally, we demonstrate the framework in the case of Grover quantum walk on distance-regular graph in terms of anyon systems for the graphs considered. In the dual perspective interacting Fock spaces gather a new meaning in terms of anyon collisions for the case of distance-regular graphs.