Spectral Analysis of Non-unitary Two-phase Quantum Walks in One Dimension

TL;DR

This paper develops a spectral analysis method for non-unitary two-phase quantum walks in one dimension, providing explicit formulas for topologically protected bound states using transfer matrices.

Contribution

It introduces a transfer matrix approach to explicitly determine topologically bound states in non-unitary quantum walks, extending spectral analysis techniques.

Findings

Explicit formula for bound states derived

Transfer matrix method applied successfully

Spectral properties characterized in complex plane

Abstract

It is recently shown by Asahara-Funakawa-Seki-Tanaka that existing index theory for chirally symmetric (discrete-time) quantum walks can be extended to the setting of non-unitary quantum walks. More precisely, they consider a certain non-unitary variant of the two-phase split-step quantum walk as a concrete one-dimensional example, and give a complete classification of the associated index in their study. Note, however, that it remains uncertain whether or not their index gives a lower bound for the number of so-called topologically protected bound states unlike the setting of unitary quantum walks. In fact, the spectrum of a non-unitary operator can be any subset of the complex plane, and so the definition of such bound states is ambiguous in the non-unitary case. The purpose of the present article is to show that the simple use of transfer matrices naturally allows us to obtain an explicit formula for a topologically bound state associated with the non-unitary split-step quantum walk model mentioned above.