The Staggered Quantum Walk Model

TL;DR

This paper establishes the formal equivalence between Szegedy's and staggered quantum walk models, demonstrating how search algorithms and spectral properties can be transferred between them, thus unifying different quantum walk frameworks.

Contribution

It provides a formal definition of the staggered quantum walk model and proves its equivalence to Szegedy's model, including extensions for marked vertices and spectral analysis methods.

Findings

Any Szegedy's model instance is equivalent to a staggered model instance.

Some staggered model instances cannot be represented by Szegedy's model.

Szegedy's search algorithms can be adapted to the staggered model.

Abstract

There are at least three models of discrete-time quantum walks (QWs) on graphs currently under active development. In this work we focus on the equivalence of two of them, known as Szegedy's and staggered QWs. We give a formal definition of the staggered model and discuss generalized versions for searching marked vertices. Using this formal definition, we prove that any instance of Szegedy's model is equivalent to an instance of the staggered model. On the other hand, we show that there are instances of the staggered model that cannot be cast into Szegedy's framework. Our analysis also works when there are marked vertices. We show that Szegedy's spatial search algorithms can be converted into search algorithms in staggered QWs. We take advantage of the similarity of those models to define the quantum hitting time in the staggered model and to describe a method to calculate the eigenvalues and eigenvectors of the evolution operator of staggered QWs.