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Quantum walks have attracted attention as a promising platform realizing topological phenomena and many physicists have introduced various types of indices to characterize topologically protected bound states that are robust against perturbations. In this paper, we introduce an index from a supersymmetric point of view. This allows us to define indices for all chiral symmetric quantum walks such as multi-dimensional split-step quantum walks and quantum walks on graphs, for which there has been no index theory. Moreover, the index gives a lower bound on the number of bound states robust against compact perturbations. We also calculate the index for several concrete examples including the unitary transformation that appears in Grovers search algorithm.
Given its importance to many other areas of physics, from condensed matter physics to thermodynamics, time-reversal symmetry has had relatively little influence on quantum information science. Here we develop a network-based picture of time-reversal
We classify periodically driven quantum systems on a one-dimensional lattice, where the driving process is local and subject to a chiral symmetry condition. The analysis is in terms of the unitary operator at a half-period and also covers systems in
Chirally symmetric discrete-time quantum walks possess supersymmetry, and their Witten indices can be naturally defined. The Witten index gives a lower bound for the number of topologically protected bound states. The purpose of this paper is to give
We give a new determinant expression for the characteristic polynomial of the bond scattering matrix of a quantum graph G. Also, we give a decomposition formula for the characteristic polynomial of the bond scattering matrix of a regular covering of
In this paper we consider Schr{o}dinger operators on $M times mathbb{Z}^{d_2}$, with $M={M_{1}, ldots, M_{2}}^{d_1}$ (`quantum wave guides) with a `$Gamma$-trimmed random potential, namely a potential which vanishes outside a subset $Gamma$ which is