Partition-based discrete-time quantum walks

TL;DR

This paper introduces a unified framework for discrete-time quantum walks based on two-partition models, demonstrating their equivalence and encompassing various existing models like coined, Szegedy's, and staggered walks.

Contribution

The paper formalizes a family of quantum walks based on two partitions, unifying several existing models and proving their unitary equivalence, thus providing a comprehensive framework.

Findings

The two-step coined model is unitarily equivalent to Szegedy's and staggered models.

Most known discrete-time quantum walks can be described within the two-partition framework.

Selecting a specific model among these is a matter of preference, not limitation.

Abstract

We introduce a family of discrete-time quantum walks, called two-partition model, based on two equivalence-class partitions of the computational basis, which establish the notion of local dynamics. This family encompasses most versions of unitary discrete-time quantum walks driven by two local operators studied in literature, such as the coined model, Szegedy's model, and the 2-tessellable staggered model. We also analyze the connection of those models with the two-step coined model, which is driven by the square of the evolution operator of the standard discrete-time coined walk. We prove formally that the two-step coined model, an extension of Szegedy model for multigraphs, and the two-tessellable staggered model are unitarily equivalent. Then, selecting one specific model among those families is a matter of taste not generality.