Eigenvalues of Quantum Walks of Grover and Fourier Types

Takashi Komatsu, Tatsuya Tate

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

Abstract

A necessary and sufficient conditions for a certain class of periodic unitary transition operators to have eigenvalues are given. Applying this, it is shown that Grover walks in any dimension has both of ±1 as eigenvalues and it has no other eigenvalues. It is also shown that the lazy Grover walks in any dimension has 1 as an eigenvalue, and it has no other eigenvalues. As a result, a localization phenomenon occurs for these quantum walks. A general conditions for the existence of eigenvalues can be applied also to certain quantum walks of Fourier type. It is shown that the two-dimensional Fourier walk does not have eigenvalues and hence it is not localized at any point. Some other topics, such as Grover walks on the triangular lattice, products and deformations of Grover walks, are also discussed.

Original languageEnglish
Pages (from-to)1293-1318
Number of pages26
JournalJournal of Fourier Analysis and Applications
Volume25
Issue number4
DOIs
Publication statusPublished - 2019 Aug 15

Keywords

  • Eigenvalue
  • Grover and Fourier walks
  • Localization
  • Quantum walk

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)
  • Applied Mathematics

Fingerprint Dive into the research topics of 'Eigenvalues of Quantum Walks of Grover and Fourier Types'. Together they form a unique fingerprint.

Cite this