Spectral mapping theorem of an abstract non-unitary quantum walk

TL;DR

This paper extends the spectral mapping theorem to certain non-unitary quantum walks with chiral symmetry, providing insights into their spectra and applications to graph theory and random walks.

Contribution

It generalizes previous spectral mapping results to non-unitary operators with specific eigenvalue constraints, broadening the understanding of quantum walks in open systems.

Findings

Spectra of non-unitary operators lie in the circle and real axis in the complex plane.

Includes examples of non-unitary quantum walks and applications to Ihara zeta functions.

Provides a framework for analyzing open quantum systems and related graph structures.

Abstract

This paper continues the previous work (Quantum Inf. Process (2019)) by two authors of the present paper about a spectral mapping property of chiral symmetric unitary operators. In physics, they treat non-unitary time-evolution operators to consider quantum walks in open systems. In this paper, we generalize the above result to include a chiral symmetric non-unitary operator whose coin operator only has two eigenvalues. As a result, the spectra of such non-unitary operators are included in the (possibly non-unit) circle and the real axis in the complex plane. We also give some examples of our abstract results, such as non-unitary quantum walks defined by Mochizuki et al. Moreover, we present an application to the Ihara zeta functions and correlated random walks on regular graphs, which are not quantum walks.