  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.
