Generator of an abstract quantum walk
Abstract
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.
 Publication:

arXiv eprints
 Pub Date:
 August 2015
 arXiv:
 arXiv:1508.07473
 Bibcode:
 2015arXiv150807473S
 Keywords:

 Mathematical Physics;
 Mathematics  Spectral Theory;
 81S25 (Primary) 81P68;
 82B41 (Secondary)
 EPrint:
 Quantum Studies: Mathematics and Foundations, 2016, Vol. 3, Issue 1, pp. 1130