Hypergroup structures of open quantum random walks on distance sets

TL;DR

This paper explores how open quantum random walks on distance sets can generate a wide variety of hypergroup structures, including non-hermitian ones, expanding the understanding of quantum walk distributions through hypergroup theory.

Contribution

It demonstrates that any discrete hypergroup, not necessarily hermitian, can be realized by an open quantum random walk on a distance set, applying Wildberger's hypergroup construction method.

Findings

Any discrete hypergroup can be realized by an OQRW on a distance set.

The method applies to both hermitian and non-hermitian hypergroups.

Distributions of OQRWs are analyzed through the lens of hypergroups.

Abstract

Wildberger has introduced the method to construct a hermitian discrete hypergroup from a random walk on a graph. We will apply his method to an open quantum random walk (OQRW) on a distance set, and show that any discrete hypergroup which is not necessary hermitian is realized by an OQRW on a distance set. We will investigate distributions of OQRWs on distance sets in the view point of hypergroups.