Parameterization of quantum walks on cycles

TL;DR

This paper characterizes the parameter space of quantum walks on cycle graphs, revealing how their unitary equivalence classes are parameterized by 2N real parameters with restrictions depending on the cycle size's parity.

Contribution

It provides a complete parameterization of quantum walks on cycles, highlighting the dependence of parameter ranges on the parity of the number of vertices.

Findings

Unitary equivalence classes are parameterized by 2N real parameters.

Parameter ranges depend on whether N is even or odd.

The study advances understanding of quantum walk classifications on cyclic structures.

Abstract

This study investigate the unitary equivalence classes of quantum walks on cycles. We show that unitary equivalence classes of quantum walks on a cycle with $N$ vertices are parameterized by $2N$ real parameters. Moreover, the ranges of two of the parameters are restricted, and the ranges depend on the parity of $N$.