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arXiv · 2205.11046

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

Chusei Kiumi·Kei Saito·Yohei Tanaka
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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.

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Chusei Kiumi, Kei Saito, Yohei Tanaka. 2022-05-23. Spectral Analysis of Non-unitary Two-phase Quantum Walks in One Dimension. https://arxiv.org/abs/2205.11046

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