Relation between coined quantum walks and quantum cellular automata

Very recently Patel et al. [1] constructed a quantum walk (QW) on a line without using a coin toss instruction, and analyzed the asymptotic behavior of the walk on the line and its escape probability with an absorbing wall. In fact the QW investigated by them can be considered as a class of quantum cellular automata on the line (see [2, 3, 4], for examples), so we call their non-coined QW a quantum cellular automaton (QCA) in this paper. On the other hand, a usual QW with a coin toss instruction was introduced and intensively studied by Ambainis et al. [5], (see Refs. [6, 7, 8] for reviews of the QW). Here we call the QW a coined QW. At a first glance, the QCA looks different from the coined QW. However, we show that there exists a one-toone correspondence between them in a more general setting. The purpose of the present paper is to clarify this connection. Once the connection is well understood, the result by Patel et al. [1] is straightforwardly obtained. The rest of this paper is organized as follows. Section 2 is devoted to the definition of the QCA. In Section 3, the definitions of A-type and B-type QWs are presented. In Section 4 (resp. 5), we describe a connection between QCAs and A-type (resp. B-type) QWs. Section 6 gives a relation between QCAs and two-step coined QWs. Finally, we discuss the case given by Patel et al. [1].