Up and down grover walks on simplicial complexes

TL;DR

This paper introduces up and down Grover walks on simplicial complexes, exploring their spectral properties, relationships with combinatorial Laplacians, and implications for topology and geometry.

Contribution

It proposes a novel type of quantum walk on simplicial complexes that incorporates orientation information and analyzes their spectral and topological properties.

Findings

Spectral structures are determined by discriminant operators.

Relations between spectra and orientability are established.

Examples of finite and infinite complexes are provided.

Abstract

A notion of up and down Grover walks on simplicial complexes are proposed and their properties are investigated. These are abstract Szegedy walks, which is a special kind of unitary operators on a Hilbert space. The operators introduced in the present paper are usual Grover walks on graphs defined by using combinatorial structures of simplicial complexes. But the shift operators are modified so that it can contain information of orientations of each simplex in the simplicial complex. It is well-known that the spectral structures of this kind of unitary operators are completely determined by its discriminant operators. It has strong relationship with combinatorial Laplacian on simplicial complexes and geometry, even topology, of simplicial complexes. In particular, theorems on a relation between spectrum of up and down discriminants and orientability, on a relation between symmetry of spectrum of discriminants and combinatorial structure of simplicial complex are given. Some examples, both of finite and infinite simplicial complexes, are also given. Finally, some aspects of finding probability and stationary measures are discussed.