Periodicity for the Hadamard walk on cycles

2015/04/24 by Norio Konno, Yuki Shimizu, Konno, Norio +3
Computer Science · Mathematics · Physics and Astronomy · #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum-Dot Cellular Automata #math-ph #math.MP #math.PR #quant-ph

paper · pdf · doi:10.48550/arxiv.1504.06396

12 pages

arxiv created 2015/04/24 · arxiv updated 2015/04/27

Abstract

The present paper treats the period T_N of the Hadamard walk on a cycle C_N with N vertices. Dukes (2014) considered the periodicity of more general quantum walks on C_N and showed T₂ =2, T₄=8, T₈=24 for the Hadamard walk case. We prove that the Hadamard walk does not have any period except for his case, i.e., N=2, 4, 8. Our method is based on a path counting and cyclotomic polynomials which is different from his approach based on the property of eigenvalues for unitary matrix that determines the evolution of the walk.

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