Up and down Grover walks on simplicial complexes

Xin Luo, Tatsuya Tate

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

Notions 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 almost 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 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.

Original languageEnglish
Pages (from-to)174-206
Number of pages33
JournalLinear Algebra and Its Applications
Volume545
DOIs
Publication statusPublished - 2018 May 15

Keywords

  • Combinatorial Laplacian
  • Grover walks
  • Simplicial complex

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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