Staggered Quantum Walks on Graphs

TL;DR

This paper explores the connections between different quantum walk models using graph theory, showing how they relate and differ, and demonstrates a staggered quantum walk search method that outperforms random walks.

Contribution

It establishes a unified framework for quantum walk models via the staggered formalism and characterizes their relationships, including a novel efficient search approach.

Findings

Szegedy and coined models are special cases within the staggered model.

Some staggered quantum walks cannot be reduced to Szegedy's model.

A staggered-based search outperforms random-walk-based search in efficiency.

Abstract

The staggered quantum walk model allows to establish an unprecedented connection between discrete-time quantum walks and graph theory. We call attention to the fact that a large subclass of the coined model is included in Szegedy's model, which in its turn is entirely included in the staggered model. In order to compare those three quantum walk models, we put them in the staggered formalism and we show that the Szegedy and coined models are defined on a special subclass of graphs. This inclusion scheme is also true when the searching framework is added. We use graph theory to characterize which staggered quantum walks can be reduced to the Szegedy or coined quantum walk model. We analyze a staggered-based search that cannot be included in Szegedy's model and we show numerically that this search is more efficient than a random-walk-based search.