Periodicity for the Hadamard Walk on Cycles
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The present paper treats the period <i>T</i><sub><i>N</i></sub> of the Hadamard walk on a cycle <i>C</i><sub><i>N</i></sub> with <i>N</i> vertices. Dukes (2014) considered the periodicity of more general quantum walks on <i>C</i><sub><i>N</i></sub> and showed <i>T</i><sub>2</sub>=2, <i>T</i><sub>4</sub>=8, <i>T</i><sub>8</sub>=24 for the Hadamard walk case. We prove that the Hadamard walk does not have any period except for his case, i.e., <i>N</i> = 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.
- Interdisciplinary Information Sciences
Interdisciplinary Information Sciences 23(1), 1-8, 2017
The Editorial Committee of the Interdisciplinary Information Sciences