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  • Open Access

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

Chusei Kiumi1,*, Jirô Akahori2,†, Takuya Watanabe2,‡, and Norio Konno2,§

  • 1Center for Quantum Information and Quantum Biology, Osaka University, 1-2 Machikaneyama, Toyonaka 560-0043, Japan
  • 2College of Science and Engineering, Ritsumeikan University, 1-1-1 Noji-higashi, Kusatsu 525-8577, Japan

  • *Contact author: c.kiumi.qiqb@osaka-u.ac.jp
  • †Contact author: akahori@se.ritsumei.ac.jp
  • ‡Contact author: t-watana@se.ritsumei.ac.jp
  • §Contact author: n-konno@fc.ritsumei.ac.jp
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Phys. Rev. Research 8, 023201 – Published 22 May, 2026

DOI: https://doi.org/10.1103/w8q6-zh9q

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.

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Phys. Rev. Research 8, 023201– Published 22 May, 2026
  • Received 4 June 2025
  • Accepted 2 April 2026
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