Interdisciplinary Information Sciences
Online ISSN : 1347-6157
Print ISSN : 1340-9050
ISSN-L : 1340-9050
Quantum Simulation and Quantum Walks
The Discrete-time Quaternionic Quantum Walk and the Second Weighted Zeta Function on a Graph
Norio KONNOHideo MITSUHASHIIwao SATO
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2017 Volume 23 Issue 1 Pages 9-17

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Abstract

We define the quaternionic quantum walk on a finite graph and investigate its properties. This walk can be considered as a natural quaternionic extension of the Grover walk on a graph. We explain the way to obtain all the right eigenvalues of a quaternionic matrix and a notable property derived from the unitarity condition for the quaternionic quantum walk. Our main results determine all the right eigenvalues of the quaternionic quantum walk by using complex eigenvalues of the quaternionic weighted matrix which is easily derivable from the walk. Since our derivation is owing to a quaternionic generalization of the determinant expression of the second weighted zeta function, we explain the second weighted zeta function and the relationship between the walk and the second weighted zeta function.

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© 2017 by the Graduate School of Information Sciences (GSIS), Tohoku University
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