Quantum walks defined by digraphs and generalized Hermitian adjacency matrices

TL;DR

This paper introduces a new quantum walk model based on digraphs and generalized Hermitian adjacency matrices, providing explicit formulas, support analysis, and computational tables for digraph identification via eigenvalues.

Contribution

It defines a novel quantum walk framework using digraphs and generalized Hermitian matrices, expanding the theoretical understanding and computational tools for quantum walks.

Findings

Derived explicit formulas for supports of transfer matrices.

Provided computational tables for digraph identification.

Connected quantum walk properties with eigenvalues of digraphs.

Abstract

We propose a quantum walk defined by digraphs (mixed graphs). This is like Grover walk that is perturbed by a certain complex-valued function defined by digraphs. The discriminant of this quantum walk is a matrix that is a certain normalization of generalized Hermitian adjacency matrices. Furthermore, we give definitions of the positive and negative supports of the transfer matrix, and clarify explicit formulas of their supports of the square. In addition, we give tables by computer on the identification of digraphs by their eigenvalues.