In this paper, we derive sufficient conditions for the localization of two-dimensional split-step quantum walks with a strong shift. For this purpose, we analyze the zero points of the function $f$ introduced by Fuda et. al. (Quantum Inf Process 16(8) 203, 2017) and make these zero points explicit. These zeros provide a concrete representation of the eigenvalues and eigenvectors of the evolution operator, and in particular, clarify where localization occurs. In addition, the eigenvalues obtained here asymptotically approach threshold resonance in special cases. We also describe the display of threshold resonances and generalized eigenfunctions.In this paper, we derive sufficient conditions for the localization of two-dimensional split-step quantum walks with a strong shift. For this purpose, we analyze the zero points of the function $f$ introduced by Fuda et. al. (Quantum Inf Process 16(8) 203, 2017) and make these zero points explicit. These zeros provide a concrete representation of the eigenvalues and eigenvectors of the evolution operator, and in particular, clarify where localization occurs. In addition, the eigenvalues obtained here asymptotically approach threshold resonance in special cases. We also describe the display of threshold resonances and generalized eigenfunctions.

arXiv:2304.02342Apr 5, 20230 citations

The bulk-edge correspondence for the split-step quantum walk on the one-dimensional integer lattice

Yasumichi Matsuzawa, Motoki Seki, Yohei Tanaka

Suzuki's split-step quantum walk on the one-dimensional integer lattice can be naturally viewed as a chirally symmetric quantum walk. Given the unitary time-evolution of such a chirally symmetric quantum walk, we can separately introduce well-defined indices for the eigenvalues $\pm 1.$ The bulk-edge correspondence for Suzuki's split-step quantum walk is twofold. Firstly, we show that the multiplicities of the eigenvalues $\pm 1$ coincide with the absolute values of the associated indices. Note that this can be viewed as the symmetry protection of bound states, and that the indices we consider are robust in the sense that these depend only on the asymptotic behaviour of the parameters of the given model. Secondly, we show that that such bound states exhibit exponential decay at spatial infinity.Suzuki's split-step quantum walk on the one-dimensional integer lattice can be naturally viewed as a chirally symmetric quantum walk. Given the unitary time-evolution of such a chirally symmetric quantum walk, we can separately introduce well-defined indices for the eigenvalues $\pm 1.$ The bulk-edge correspondence for Suzuki's split-step quantum walk is twofold. Firstly, we show that the multiplicities of the eigenvalues $\pm 1$ coincide with the absolute values of the associated indices. Note that this can be viewed as the symmetry protection of bound states, and that the indices we consider are robust in the sense that these depend only on the asymptotic behaviour of the parameters of the given model. Secondly, we show that that such bound states exhibit exponential decay at spatial infinity.

arXiv:2105.06147May 13, 20212 citations

Eigenvalues of two-phase quantum walks with one defect in one dimension

Chusei Kiumi, Kei Saito

arXiv:2010.08324Quantum Information ProcessingMay 1, 202110 citations

In this paper, we derive sufficient conditions for the localization of two-dimensional split-step quantum walks with a strong shift. For this purpose, we analyze the zero points of the function

f

introduced by Fuda et. al. (Quantum Inf Process 16(8) 203, 2017) and make these zero points explicit. These zeros provide a concrete representation of the eigenvalues and eigenvectors of the evolution operator, and in particular, clarify where localization occurs. In addition, the eigenvalues obtained here asymptotically approach threshold resonance in special cases. We also describe the display of threshold resonances and generalized eigenfunctions.

\pm 1.

The bulk-edge correspondence for Suzuki's split-step quantum walk is twofold. Firstly, we show that the multiplicities of the eigenvalues

\pm 1

coincide with the absolute values of the associated indices. Note that this can be viewed as the symmetry protection of bound states, and that the indices we consider are robust in the sense that these depend only on the asymptotic behaviour of the parameters of the given model. Secondly, we show that that such bound states exhibit exponential decay at spatial infinity.