Implementation of a discrete-time quantum walk with a circulant matrix on a graph by optical polarizing elements

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.