Periodicity of quantum walks defined by mixed paths and mixed cycles

Abstract

In this paper, we determine periodicity of quantum walks defined by mixed paths and mixed cycles. By the spectral mapping theorem of quantum walks, consideration of periodicity is reduced to eigenvalue analysis of $\eta$-Hermitian adjacency matrices. First, we investigate coefficients of the characteristic polynomials of $\eta$-Hermitian adjacency matrices. We show that the characteristic polynomials of mixed trees and their underlying graphs are same. We also define $n+1$ types of mixed cycles and show that every mixed cycle is switching equivalent to one of them. We use these results to discuss periodicity. We show that the mixed paths are periodic for any $\eta$. In addition, we provide a necessary and sufficient condition for a mixed cycle to be periodic and determine their periods.

arXiv comment22 pages, 12 figures

References in corpus (5)

• Spectra of Perfect State Transfer Hamiltonians on Fractal-Like Graphs
• Quantum walks defined by digraphs and generalized Hermitian adjacency matrices
• Periodicity of Grover walks on distance-regular graphs
• On the spectrum of complex unit gain graph
• On the multiplicities of digraph eigenvalues

Cited by in corpus (5)

• Complex networks with complex weights
• Combinatorial necessary conditions for regular graphs to induce periodic quantum walks
• Periodicity and perfect state transfer of Grover walks on quadratic unitary Cayley graphs
• State transfer in Grover walks on unitary and quadratic unitary Cayley graphs over finite commutative rings
• The trace formula with respect to the twisted Grover matrix of a mixed digraph