Staggered Quantum Walk on Hexagonal Lattices

TL;DR

This paper investigates the properties and dynamics of the staggered quantum walk on hexagonal lattices, including its potential for spatial search algorithms, expanding understanding of quantum walks beyond previous models.

Contribution

It introduces the staggered quantum walk on hexagonal lattices, analyzing its dynamics and providing analytic results for spatial search complexity.

Findings

Analyzed position standard deviation and localization in the walk.

Derived analytic time complexity for spatial search with one marked node.

Extended the framework of quantum walks to hexagonal lattice structures.

Abstract

A discrete-time staggered quantum walk was recently introduced as a generalization that allows to unify other versions, such as the coined and Szegedy's walk. However, it also produces new forms of quantum walks not covered by previous versions. To explore their properties, we study here the staggered walk on a hexagonal lattice. Such a walk is defined using a set of overlapping tessellations that cover the graph edges, and each tessellation is a partition of the node set into cliques. The hexagonal lattice requires at least three tessellations. Each tessellation is associated with a local unitary operator and the product of the local operators defines the evolution operator of the staggered walk on the graph. After defining the evolution operator on the hexagonal lattice, we analyze the quantum walk dynamics with the focus on the position standard deviation and localization. We also obtain analytic results for the time complexity of spatial search algorithms with one marked node using cyclic boundary conditions.