Limit theorems for the discrete-time quantum walk on a graph with joined half lines

Kota Chisaki, Norio Konno, Etsuo Segawa

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)

Abstract

We consider a discrete-time quantum walk W t,k at time t on a graph with joined half lines J k, which is composed of k half lines with the same origin. Our analysis is based on a reduction of the walk on a half line. The idea plays an important role to analyze the walks on some class of graphs with symmetric initial states. In this paper, we introduce a quantum walk with an enlarged basis and show that W t,k can be reduced to the walk on a half line even if the initial state is asymmetric. For W t,k, we obtain two types of limit theorems. The first one is an asymptotic behavior of W t,k which corresponds to localization. For some conditions, we find that the asymptotic behavior oscillates. The second one is the weak convergence theorem for W t,k. On each half line, W t,k converges to a density function like the case of the one-dimensional lattice with a scaling order of t. The results contain the cases of quantum walks starting from the general initial state on a half line with the general coin and homogeneous trees with the Grover coin.

Original languageEnglish
Pages (from-to)314-333
Number of pages20
JournalQuantum Information and Computation
Volume12
Issue number3-4
Publication statusPublished - 2012 Mar 1

Keywords

  • Homogeneous tree
  • Localization
  • Quantum walk
  • Weak convergence

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Statistical and Nonlinear Physics
  • Nuclear and High Energy Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Computational Theory and Mathematics

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