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arXiv:2006.12678

Parameterization of quantum walks on cycles

Shuji Kuriki, Md Sams Afif Nirjhor, Hiromichi Ohno

Published Jun 23, 2020• 1 citation
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Abstract

This study investigate the unitary equivalence classes of quantum walks on cycles. We show that unitary equivalence classes of quantum walks on a cycle with $N$ vertices are parameterized by $2N$ real parameters. Moreover, the ranges of two of the parameters are restricted, and the ranges depend on the parity of $N$.

DOI
10.48550/arxiv.2006.12678

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This study investigate the unitary equivalence classes of quantum walks on cycles. We show that unitary equivalence classes of quantum walks on a cycle with NNN vertices are parameterized by 2N2N2N real parameters. Moreover, the ranges of two of the parameters are restricted, and the ranges depend on the parity of NNN.
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We introduce eigen-independency of certain unitary matrices of degree 2. We show that this property characterizes stationary measures of quantum walks on cycles with odd number of vertices. We also characterize the stationary measures of the two-state Hadamard walk (both moving-shift and flip-flop shit) on cycles with ppp vertices, where ppp is an odd prime number.