© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2311.06468 · Physical Review Research

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

Chusei Kiumi, Jirô Akahori, Takuya Watanabe, Norio Konno

Published Apr 2, 2026• 0 citations

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.

DOI: 10.1103/w8q6-zh9q

Related papers

Localization of space-inhomogeneous three-state quantum walks

Chusei Kiumi

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.

arXiv:2111.14300•Journal of Physics A Mathematical and Theoretical•Apr 22, 2022•4 citations

Eigenvalues of two-phase three-state quantum walks with one defect

Chusei Kiumi

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 $n$-state quantum walks in one dimension with $n-2$ 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.

•Nov 29, 2021•1 citation

Localization in quantum walks with periodically arranged coin matrices

Chusei Kiumi

There is a property called localization, which is essential for applications of quantum walks. From a mathematical point of view, the occurrence of localization is known to be equivalent to the existence of eigenvalues of the time evolution operators, which are defined by coin matrices. A previous study proposed an approach to the eigenvalue problem for space-inhomogeneous models using transfer matrices. However, the approach was restricted to models whose coin matrices are the same in positions sufficiently far to the left and right, respectively. This study shows that the method can be applied to extended models with periodically arranged coin matrices. Moreover, we investigate localization by performing the eigenvalue analysis and deriving their time-averaged limit distribution.

arXiv:2111.15131•International Journal of Quantum Information•Apr 28, 2022•1 citation

Localization and fractality in inhomogeneous quantum walks with self-duality

Yutaka Shikano, Hosho Katsura

We introduce and study a class of discrete-time quantum walks on a one-dimensional lattice. In contrast to the standard homogeneous quantum walks, coin operators are inhomogeneous and depend on their positions in this class of models. The models are shown to be self-dual with respect to the Fourier transform, which is analogous to the Aubry-André model describing the one-dimensional tight-binding model with a quasiperiodic potential. When the period of coin operators is incommensurate to the lattice spacing, we rigorously show that the limit distribution of the quantum walk is localized at the origin. We also numerically study the eigenvalues of the one-step time evolution operator and find the Hofstadter butterfly spectrum which indicates the fractal nature of this class of quantum walks.

arXiv:1004.5394•Physical Review E•Sep 16, 2010•120 citations

Aperiodic space-inhomogeneous quantum walks: Localization properties, energy spectra, and enhancement of entanglement

A. R. C. Buarque, Wandearley S. Dias

We study the localization properties, energy spectra, and coin-position entanglement of the aperiodic discrete-time quantum walks. The aperiodicity is described by spatially dependent quantum coins distributed on the lattice, whose distribution is neither periodic (Bloch-like) nor random (Anderson-like). Within transport properties we identified delocalized and localized quantum walks mediated by a proper adjusting of aperiodic parameter. Both scenarios are studied by exploring typical quantities (inverse participation ratio, survival probability, and wave packet width), as well as the energy spectra of an associated effective Hamiltonian. By using the energy spectra analysis, we show that the early stage the inhomogeneity leads to a vanishing gap between two main bands, which justifies the predominantly delocalized character observed for ν<0.5. With increase of ν arise gaps and flat bands on the energy spectra, which corroborates the suppression of transport detected for ν>0.5. For ν high enough, we observe an energy spectra, which resembles that described by the one-dimensional Anderson model. Within coin-position entanglement, we show many settings in which an enhancement in the ability to entangle is observed. This behavior brings new information about the role played by aperiodicity on the coin-position entanglement for static inhomogeneous systems, reported before as almost always reducing the entanglement when comparing with the homogeneous case. We extend the analysis in order to show that systems with static inhomogeneity are able to exhibit asymptotic limit of entanglement.

arXiv:1812.03003•Physical review. E•Sep 4, 2019•28 citations

Role of the coin in the spectrum of quantum walks

Lauri Lehman

The most elementary quantum walk is characterized by a 2-dimensional unitary coin flip matrix, which can be parameterized by 4 real variables. The influence of the choice of the coin flip matrix on the time evolution operator is analysed in a systematic way. By changing the coin parameters, the dispersion and asymmetry of eigenvalues of the time evolution operator can be tuned in a controlled way. The reduced eigenvectors in coin space are distributed along trajectories on the surface or inside the Bloch sphere, depending on the degeneracy of the spectrum. At certain values of the coin parameters the spectrum of the time evolution operator becomes 2-fold degenerate, but there might exist unique eigenvalues at the top and bottom of each quasi-energy band. The eigenstates corresponding to such eigenvalues are robust against arbitrary temporal variations in the bias parameter of the coin, as long as rest of the parameters remain unchanged.

arXiv:1501.07294•Jan 28, 2015•0 citations

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

n

-state quantum walks in one dimension with

n-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.

There is a property called localization, which is essential for applications of quantum walks. From a mathematical point of view, the occurrence of localization is known to be equivalent to the existence of eigenvalues of the time evolution operators, which are defined by coin matrices. A previous study proposed an approach to the eigenvalue problem for space-inhomogeneous models using transfer matrices. However, the approach was restricted to models whose coin matrices are the same in positions sufficiently far to the left and right, respectively. This study shows that the method can be applied to extended models with periodically arranged coin matrices. Moreover, we investigate localization by performing the eigenvalue analysis and deriving their time-averaged limit distribution.

We introduce and study a class of discrete-time quantum walks on a one-dimensional lattice. In contrast to the standard homogeneous quantum walks, coin operators are inhomogeneous and depend on their positions in this class of models. The models are shown to be self-dual with respect to the Fourier transform, which is analogous to the Aubry-André model describing the one-dimensional tight-binding model with a quasiperiodic potential. When the period of coin operators is incommensurate to the lattice spacing, we rigorously show that the limit distribution of the quantum walk is localized at the origin. We also numerically study the eigenvalues of the one-step time evolution operator and find the Hofstadter butterfly spectrum which indicates the fractal nature of this class of quantum walks.

We study the localization properties, energy spectra, and coin-position entanglement of the aperiodic discrete-time quantum walks. The aperiodicity is described by spatially dependent quantum coins distributed on the lattice, whose distribution is neither periodic (Bloch-like) nor random (Anderson-like). Within transport properties we identified delocalized and localized quantum walks mediated by a proper adjusting of aperiodic parameter. Both scenarios are studied by exploring typical quantities (inverse participation ratio, survival probability, and wave packet width), as well as the energy spectra of an associated effective Hamiltonian. By using the energy spectra analysis, we show that the early stage the inhomogeneity leads to a vanishing gap between two main bands, which justifies the predominantly delocalized character observed for ν<0.5. With increase of ν arise gaps and flat bands on the energy spectra, which corroborates the suppression of transport detected for ν>0.5. For ν high enough, we observe an energy spectra, which resembles that described by the one-dimensional Anderson model. Within coin-position entanglement, we show many settings in which an enhancement in the ability to entangle is observed. This behavior brings new information about the role played by aperiodicity on the coin-position entanglement for static inhomogeneous systems, reported before as almost always reducing the entanglement when comparing with the homogeneous case. We extend the analysis in order to show that systems with static inhomogeneity are able to exhibit asymptotic limit of entanglement.

The most elementary quantum walk is characterized by a 2-dimensional unitary coin flip matrix, which can be parameterized by 4 real variables. The influence of the choice of the coin flip matrix on the time evolution operator is analysed in a systematic way. By changing the coin parameters, the dispersion and asymmetry of eigenvalues of the time evolution operator can be tuned in a controlled way. The reduced eigenvectors in coin space are distributed along trajectories on the surface or inside the Bloch sphere, depending on the degeneracy of the spectrum. At certain values of the coin parameters the spectrum of the time evolution operator becomes 2-fold degenerate, but there might exist unique eigenvalues at the top and bottom of each quasi-energy band. The eigenstates corresponding to such eigenvalues are robust against arbitrary temporal variations in the bias parameter of the coin, as long as rest of the parameters remain unchanged.