Virtually Abelian Quantum Walks

TL;DR

This paper introduces a method to analyze quantum walks on virtually Abelian groups by reducing them to walks on Abelian groups, providing exact solutions for specific cases with planar Cayley graphs.

Contribution

It presents a novel reduction technique for quantum walks on virtually Abelian groups, enabling exact solutions for certain non-Abelian cases.

Findings

Reduction technique simplifies analysis of quantum walks on virtually Abelian groups.

Exact solutions obtained for quantum walks on groups with planar Cayley graphs.

Method extends understanding of quantum walks beyond purely Abelian groups.

Abstract

We introduce quantum walks on Cayley graphs of non-Abelian groups. We focus on the easiest case of virtually Abelian groups, and introduce a technique to reduce the quantum walk to an equivalent one on an Abelian group with coin system having larger dimension. We apply the technique in the case of two quantum walks on virtually Abelian groups with planar Cayley graphs, finding the exact solution.