  The index theory for the space of finite propagation unitary operators was
developed by Gross, Nesme, Vogts and Werner from the viewpoint of quantum walks
in mathematical physics. In particular, they proved that $\pi_0$ of the space
is determined by the index. However, nothing is known about the higher homotopy
groups. In this article, we describe the homotopy type of the space of finite
propagation unitary operators on the Hilbert space of square summable
$\mathbb{C}$-valued $\mathbb{Z}$-sequences, so we can determine its homotopy
groups. We also study the space of (end-)periodic finite propagation unitary
operators.
