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

TL;DR

This paper develops a framework connecting quantum walks on regular graphs with anyon systems by utilizing association schemes, Fock spaces, and modular tensor categories, and demonstrates it with Grover walks.

Contribution

It introduces a novel approach linking quantum walks on regular graphs to anyon systems through association schemes and modular tensor categories.

Findings

Framework successfully models quantum walks with anyons.

Demonstration with Grover quantum walk on distance-regular graphs.

Interacting Fock spaces are well-defined asymptotically for growing graphs.

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 graphs 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 any