Design for implementation of discrete-time quantum walk with circulant matrix on graph by optical polarizing elements

TL;DR

This paper proposes a method to implement discrete-time quantum walks with circulant matrices on graphs using optical polarizing elements, enabling practical realization of quantum walk dynamics.

Contribution

It introduces a novel optical circuit design for implementing stationary states of circulant quantum walks on graphs, linking quantum walk theory with optical implementation techniques.

Findings

Optical circuit can realize stationary states of circulant quantum walks.

A condition is provided for implementing circulant quantum walks with the optical circuit.

The method connects quantum walk theory with optical engineering for potential experimental applications.

Abstract

In this paper, we introduce a quantum walk whose local scattering at each vertex is denoted by a unitary circulant matrix; namely the circulant quantum walk. We also introduce another quantum walk induced by the circulant quantum walk; namely the optical quantum walk, whose underlying graph is a $2$-regular directed graph and obtained by blowing up the original graph in some way. We propose a design of an optical circuit which implements the stationary state of the optical quantum walk. We show that if the induced optical quantum walk does not have $+1$ eigenvalue, then the stationary state of the optical quantum walk gives that of the original circulant quantum walk. From this result, we give a useful condition for the setting of the circulant quantum walks which can be implemented by this optical circuit.