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Recent developments in the index theory of discrete-time quantum walks allowus to assign a certain well-defined supersymmetric index to a pair of a unitarytime-evolution $U$ and a $\mathbb{Z}_2$-grading operator $\varGamma$ satisfyingthe chiral symmetry condition $U^* = \varGamma U \varGamma.$ In this paper,this index theory will be extended to encompass non-unitary $U$. The existingliterature for unitary $U$ makes use of the indispensable assumption that $U$is essentially gapped; that is, we require that the essential spectrum of $U$contains neither $-1$ nor $+1$ to define the associated index. It turns outthat this assumption is no longer necessary, if the given time-evolution $U$ isnon-unitary. As a concrete example, we shall consider a well-known non-unitaryquantum walk model on the one-dimensional integer lattice, introduced byMochizuki-Kim-Obuse.

The Witten Index for One-dimensional Non-unitary Quantum Walks with Gapless Time-evolution