  We consider an abstract quantum walk defined by a unitary evolution operator
$U$, which acts on a Hilbert space decomposed into a direct sum of Hilbert
spaces $\{\mathcal{H}_v \}_{v \in V}$. We show that such $U$ naturally defines
a directed graph $G_U$ and the probability of finding a quantum walker on
$G_U$. The asymptotic property of an abstract quantum walker is governed by the
generator $H$ of $U$ such that $U^n = e^{inH}$. We derive the generator of an
evolution of the form $U = S(2d_A^* d_A -1)$, a generalization of the Szegedy
evolution operator. Here $d_A$ is a boundary operator and $S$ a shift operator.
