Quantum walks on simplicial complexes

TL;DR

This paper introduces a novel quantum walk model on simplicial complexes, extending Szegedy walks on graphs, and explores its spreading and localization properties influenced by topological and geometric structures.

Contribution

It develops a new quantum walk framework on simplicial complexes, revealing unique behaviors and potential topological and geometric influences not seen in traditional quantum walks.

Findings

Quantum walks exhibit linear spreading and localization.

Localization reflects topological and geometric structures.

The model shows nontrivial behaviors absent in standard quantum walks.

Abstract

We construct a new type of quantum walks on simplicial complexes as a natural extension of the well-known Szegedy walk on graphs. One can numerically observe that our proposing quantum walks possess linear spreading and localization as in the case of the Grover walk on lattices. Moreover, our numerical simulation suggests that localization of our quantum walks reflect not only topological but also geometric structures. On the other hand, our proposing quantum walk contains an intrinsic problem concerning exhibition of nontrivial behavior, which is not seen in typical quantum walks such as Grover walks on graphs.