Intertwining operators between one-dimensional space-homogeneous quantum walks

Source abstract

The subject of this paper is a kind of dynamical systems called quantum walks. We study one-dimensional homogeneous analytic quantum walks $U$. We explain how to identify the space of all the uniform intertwining operators between these walks. We can also determine whether $U$ can be realized by a (not necessarily homogeneous) continuous-time uniform quantum walk on $\mathbb{Z}$. Several examples of quantum walks, which can not be realized by continuous-time uniform quantum walks, are presented. The 4-state Grover walk is one of them. Before stating the main theorems, we clarify the definition of one-dimensional quantum walks. For the first half of this paper, we study basic properties of one-dimensional quantum walks, which are not necessarily homogeneous. An equivalence relation between quantum walks called similarity is also introduced. This allows us to manipulate quantum walks in a flexible manner.