Subfactors from regular graphs induced by association schemes

TL;DR

This paper explores the connection between association schemes, subfactors, and quantum walks on regular graphs, providing a framework that links combinatorial structures with anyonic systems in quantum physics.

Contribution

It introduces a novel approach to construct quantum walks on distance-regular graphs using association schemes and subfactors, bridging combinatorics and quantum physics.

Findings

Established a method to realize quantum walks via association schemes and subfactors.

Linked quantum walks on graphs to anyon systems through subfactor theory.

Analyzed a large family of graphs within this framework.

Abstract

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.