  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 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.
