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Spectral analysis of three-state quantum walks with general coin matrices

Chusei Kiumi, Jirô Akahori, Takuya Watanabe, Norio Konno

TL;DR

This work develops a general transfer-matrix framework for space-inhomogeneous three-state quantum walks on the one-dimensional lattice with arbitrary coin matrices, reducing the eigenproblem for the evolution operator $U$ to a two-dimensional recurrence. It establishes necessary and sufficient conditions for eigenvalues, and provides a complete spectral decomposition $\sigma(U)=\sigma_{\mathrm{disc}}(U)\cup\sigma_{\mathrm{ess}}(U)$ with discrete, flat-band (infinite multiplicity), and absolutely continuous components, including a rigorous treatment of the essential spectrum under finite defects and two-phase tails. The authors apply the method to Fourier walks to numerically reveal localization in one-defect and two-phase settings, despite delocalization in the homogeneous case. These results yield mathematical tools for identifying localized states and clarifying the spectral structure that underpins localization, topological-like edge modes, and potential implications for quantum algorithms.

Abstract

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.

Spectral analysis of three-state quantum walks with general coin matrices

TL;DR

This work develops a general transfer-matrix framework for space-inhomogeneous three-state quantum walks on the one-dimensional lattice with arbitrary coin matrices, reducing the eigenproblem for the evolution operator to a two-dimensional recurrence. It establishes necessary and sufficient conditions for eigenvalues, and provides a complete spectral decomposition with discrete, flat-band (infinite multiplicity), and absolutely continuous components, including a rigorous treatment of the essential spectrum under finite defects and two-phase tails. The authors apply the method to Fourier walks to numerically reveal localization in one-defect and two-phase settings, despite delocalization in the homogeneous case. These results yield mathematical tools for identifying localized states and clarifying the spectral structure that underpins localization, topological-like edge modes, and potential implications for quantum algorithms.

Abstract

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.
Paper Structure (24 sections, 14 theorems, 103 equations, 16 figures)

This paper contains 24 sections, 14 theorems, 103 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Three state quantum walk and transfer matrix
  3. Definition of three-state quantum walk
  4. Eigenvalue analysis
  5. Reducing the dimensionality
  6. Transfer matrix on reduced space
  7. Spectral analysis for two-phase QW with defects
  8. Eigenvalue and Eigenvector
  9. Essential spectrum
  10. Example: Homogeneous, one-defect, two-phase model
  11. Homogeneous model
  12. One-defect model
  13. Two-phase model
  14. Numerical analysis on Fourier walk
  15. One-defect model
  16. ...and 9 more sections

Key Result

Lemma 1

For $\lambda \in [0,2\pi )$, $e^{i\lambda}$ is an eigenvalue of $U$ if and only if there exists $\tilde{\Psi} \in \ell ^{2} (\mathbb{Z} ;\mathbb{C}^{2} )$ such that and the corresponding eigenvector is given by $\iota ^{-1}\tilde{\Psi}$.

Figures (16)

  • Figure 1: $\theta=\frac{\pi}{12}$
  • Figure 2: $\theta=\frac{3\pi}{12}$
  • Figure 3: $\theta=\frac{7\pi}{12}$
  • Figure 4: $\theta=\frac{11\pi}{12}$
  • Figure 6: $\theta=\frac{\pi}{12}$
  • ...and 11 more figures

Theorems & Definitions (22)

  • Lemma 1
  • Proposition 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • Proposition 2
  • proof
  • Corollary 1
  • Theorem 2
  • proof
  • ...and 12 more