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arXiv:1703.04969 · Quantum Information and Computation

Quaternionic quantum walks of Szegedy type and zeta functions of graphs

Norio Konno, Kaname Matsue, Hideo Mitsuhashi, Iwao Sato

Published Dec 1, 2017• 0 citations
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Abstract

We define a quaternionic extension of the Szegedy walk on a graph and study its right spectral properties. The condition for the transition matrix of the quaternionic Szegedy walk on a graph to be quaternionic unitary is given. In order to derive the spectral mapping theorem for the quaternionic Szegedy walk, we derive a quaternionic extension of the determinant expression of the second weighted zeta function of a graph. Our main results determine explicitly all the right eigenvalues of the quaternionic Szegedy walk by using complex right eigenvalues of the corresponding doubly weighted matrix. We also show the way to obtain eigenvectors corresponding to right eigenvalues derived from those of doubly weighted matrix.

DOI
10.26421/qic17.15-16-6

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We define a quaternionic extension of the Szegedy walk on a graph and study its right spectral properties. The condition for the transition matrix of the quaternionic Szegedy walk on a graph to be quaternionic unitary is given. In order to derive the spectral mapping theorem for the quaternionic Szegedy walk, we derive a quaternionic extension of the determinant expression of the second weighted zeta function of a graph. Our main results determine explicitly all the right eigenvalues of the quaternionic Szegedy walk by using complex right eigenvalues of the corresponding doubly weighted matrix. We also show the way to obtain eigenvectors corresponding to right eigenvalues derived from those of doubly weighted matrix.
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.
Given two Hilbert spaces, H\mathcal{H}H and K\mathcal{K}K, we introduce an abstract unitary operator UUU on H\mathcal{H}H and its discriminant TTT on K\mathcal{K}K induced by a coisometry from H\mathcal{H}H to K\mathcal{K}K and a unitary involution on H\mathcal{H}H. In a particular case, these operators UUU and TTT become the evolution operator of the Szegedy walk on a graph, possibly infinite, and the transition probability operator thereon. We show the spectral mapping theorem between UUU and TTT via the Joukowsky transform. Using this result, we have completely detemined the spectrum of the Grover walk on the Sierpiński lattice, which is pure point and has a Cantor-like structure.