Generator of an abstract quantum walk

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 $U=S(2d_A^{\ast }{d}_A-1)$, a generalization of the Szegedy evolution operator. Here $d_A$ is a boundary operator and $S$ a shift operator.