Generator of an abstract quantum walk

TL;DR

This paper introduces a framework for abstract quantum walks using a unitary operator that defines a directed graph and analyzes the walk's asymptotic behavior through its generator, generalizing Szegedy's model.

Contribution

It provides a method to derive the generator of an abstract quantum walk, extending the Szegedy evolution operator to a broader class of quantum walks.

Findings

Defines a directed graph from the unitary operator of a quantum walk.

Derives the generator of the quantum walk's evolution, linking it to asymptotic behavior.

Generalizes Szegedy's evolution operator to a new class of quantum walks.

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.