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

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