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arXiv:2304.02342

Eigenvalues and threshold rezonances of a two-dimensional split-step quantum walk with strong shift

Toru Fuda, Daiju Funakawa, Sasayama, Satoshi, Akito Suzuki

Published Apr 5, 2023• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2304.02342

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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 fff 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.
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Mathematical analysis of the spectral properties of the time-evolution operator in quantum walks is essential for understanding key dynamical behaviors such as localization and long-term evolution. The inhomogeneous three-state case, in particular, poses substantial analytical challenges due to its higher internal degrees of freedom and the absence of translational invariance. We develop a general framework for the spectral analysis of three-state quantum walks on the one-dimensional lattice with arbitrary time-evolution operators. Our approach is based on a transfer-matrix formulation that reduces the infinite-dimensional eigenvalue problem to a tractable system of two-dimensional recursions, enabling exact characterization of eigenstates. This framework applies broadly to space-inhomogeneous models, including those with finite defects and two-phase structures. We rigorously derive necessary and sufficient conditions for the existence of point spectrum, along with a complete description of the corresponding eigenvalues and eigenstates, which are known to underlie quantum localization phenomena. Furthermore, we give a complete spectral decomposition—discrete spectrum, flat-band eigenvalues (of infinite multiplicity), and absolutely continuous spectrum—with explicit characterization of each component. Using this method, we perform exact numerical analyses of the Fourier walk with spatial inhomogeneity, revealing the emergence of localization despite its delocalized nature in the homogeneous case. Our results provide mathematical tools and physical insights into the structure of quantum walks, offering a systematic path for identifying and characterizing localized quantum states in complex quantum systems.
Abstract Mathematical analysis on the existence of eigenvalues is essential because it is deeply related to localization, which is an exceptionally crucial property of quantum walks (QWs). We construct the method for the eigenvalue problem via the transfer matrix for space-inhomogeneous three-state QWs in one dimension with a self-loop, which is an extension of the technique in a previous study (Kiumi and Saito 2021 Quantum Inf. Process. 20 171). This method reveals the necessary and sufficient condition for the eigenvalue problem of a two-phase three-state QW with one defect whose time evolution varies in the negative part, positive part, and at the origin.
Mathematical analysis on the existence of eigenvalues is essential because it is equivalent to the occurrence of localization, which is an exceptionally crucial property of quantum walks. We construct the method for eigenvalue problem via the transfer matrix for space-inhomogeneous nnn-state quantum walks in one dimension with n−2n-2n−2 self-loops, which is an extension of the technique in a previous study (Quantum Inf. Process 20(5), 2021). By this method, we reveal the necessary and sufficient condition for the eigenvalue problem of a two-phase three-state quantum walk with one defect whose time evolution varies in the negative part, positive part, and at the origin.
We investigate one-dimensional three-state quantum walks. We find a formula for the moments of the weak limit distribution via a vacuum expectation of powers of a self-adjoint operator. We use this formula to fully characterize the localization of three-state quantum walks in one dimension. The localization is also characterized by investing the eigenvectors of the evolution operator for the quantum walk. As a byproduct we clarify the concepts of localization differently used in the literature. We also study the continuous part of the limit distribution. For typical examples we show that the continuous part is the same kind as that of two-state quantum walks. We provide with explicit expressions for the density of the weak limits of some three-state quantum walks.