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

• URL: http://arxiv.org/abs/2311.06468v4
• Date: Wed, 04 Jun 2025 10:39:21 GMT
• Title: Eigenvalue analysis of three-state quantum walks with general coin matrices
• Authors: Chusei Kiumi, Jirô Akahori, Takuya Watanabe, Norio Konno,
• Abstract summary: We develop a general framework for the spectral analysis of three-state quantum walks on the one-dimensional lattice with arbitrary time evolution operators.<n>We rigorously derive necessary and sufficient conditions for the existence of discrete eigenvalues, along with a complete description of the corresponding eigenvalues and eigenstates.<n>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.
• Score: 0.7343478057671141
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 novel 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 localized 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 discrete eigenvalues, along with a complete description of the corresponding eigenvalues and eigenstates, which are known to underlie quantum localization phenomena. 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.

Related papers

• Path integral analysis of Schrödinger-type eigenvalue problems in the complex plane: Establishing the relation between instantons and resonant states [0.0]
  We study eigenvalue problems for which the boundary conditions are provided in specific angular sectors of the complex plane.<n>We find that the arising functional integrals are naturally defined on a complexified integration contour.<n>We identify the previously elusive one-to-one correspondence between decay rates derived from real-time quantum tunneling dynamics.
  arXiv  Detail & Related papers  (2025-07-30T22:05:02Z)
• Grassmann Variational Monte Carlo with neural wave functions [45.935798913942904]
  We formalize the framework introduced by Pfau et al.citepfau2024accurate in terms of Grassmann geometry of the Hilbert space.<n>We validate our approach on the Heisenberg quantum spin model on the square lattice, achieving highly accurate energies and physical observables for a large number of excited states.
  arXiv  Detail & Related papers  (2025-07-14T13:53:13Z)
• Simulating matrix models with tensor networks [0.0]
  Matrix models, as quantum mechanical systems without explicit spatial dependence, provide valuable insights into gauge and gravitational theories. Simulating these models allows for exploration of their kinematic and dynamic properties. We construct ground states as matrix product states and analyse features such as their entanglement structure.
  arXiv  Detail & Related papers  (2024-12-05T12:57:58Z)
• Simulating NMR Spectra with a Quantum Computer [49.1574468325115]
  This paper provides a formalization of the complete procedure of the simulation of a spin system's NMR spectrum. We also explain how to diagonalize the Hamiltonian matrix with a quantum computer, thus enhancing the overall process's performance.
  arXiv  Detail & Related papers  (2024-10-28T08:43:40Z)
• Resolvent-based quantum phase estimation: Towards estimation of parametrized eigenvalues [0.0]
  We propose a novel approach for estimating the eigenvalues of non-normal matrices based on the matrix resolvent formalism. We construct the first efficient algorithm for estimating the phases of the unit-norm eigenvalues of a given non-unitary matrix. We then construct an efficient algorithm for estimating the real eigenvalues of a given non-Hermitian matrix.
  arXiv  Detail & Related papers  (2024-10-07T08:51:05Z)
• Exploring entanglement in finite-size quantum systems with degenerate ground state [0.0]
  We develop an approach for characterizing non-local quantum correlations in spin systems with exactly or nearly degenerate ground states.<n> Estimating the von Neumann entropy of the random wave functions helps to reveal previously unknown features of the quantum correlations in the phases with degeneracy of the ground state.<n>Digital quantum simulations performed with quantum computers can provide accurate description of the entanglement of the degenerate systems even in the presence of noise.
  arXiv  Detail & Related papers  (2024-10-01T08:56:34Z)
• Quantum tomography of helicity states for general scattering processes [55.2480439325792]
  Quantum tomography has become an indispensable tool in order to compute the density matrix $rho$ of quantum systems in Physics. We present the theoretical framework for reconstructing the helicity quantum initial state of a general scattering process.
  arXiv  Detail & Related papers  (2023-10-16T21:23:42Z)
• Distinguishing dynamical quantum criticality through local fidelity distances [0.0]
  We study the dynamical quantum phase transition in integrable and non-integrable Ising chains. The non-analyticities in the quantum distance between two subsystem density matrices identify the critical time. We propose a distance measure from the upper bound of the local quantum fidelity for certain quench protocols.
  arXiv  Detail & Related papers  (2023-08-01T10:27:35Z)
• Third quantization of open quantum systems: new dissipative symmetries and connections to phase-space and Keldysh field theory formulations [77.34726150561087]
  We reformulate the technique of third quantization in a way that explicitly connects all three methods. We first show that our formulation reveals a fundamental dissipative symmetry present in all quadratic bosonic or fermionic Lindbladians. For bosons, we then show that the Wigner function and the characteristic function can be thought of as ''wavefunctions'' of the density matrix.
  arXiv  Detail & Related papers  (2023-02-27T18:56:40Z)
• Universality of critical dynamics with finite entanglement [68.8204255655161]
  We study how low-energy dynamics of quantum systems near criticality are modified by finite entanglement. Our result establishes the precise role played by entanglement in time-dependent critical phenomena.
  arXiv  Detail & Related papers  (2023-01-23T19:23:54Z)
• Irrational quantum walks [0.0]
  We develop theory to study any quantum walk generated by an integral Hamiltonian. We use our methods to study geometric properties of beautiful curves arising from entries of the quantum walk matrix.
  arXiv  Detail & Related papers  (2022-08-18T17:21:47Z)
• Transfer-matrix summation of path integrals for transport through nanostructures [62.997667081978825]
  We develop a transfer-matrix method to describe the nonequilibrium properties of interacting quantum-dot systems. The method is referred to as "transfer-matrix summation of path integrals" (TraSPI)
  arXiv  Detail & Related papers  (2022-08-16T09:13:19Z)
• Quantum circuits for the preparation of spin eigenfunctions on quantum computers [63.52264764099532]
  Hamiltonian symmetries are an important instrument to classify relevant many-particle wavefunctions. This work presents quantum circuits for the exact and approximate preparation of total spin eigenfunctions on quantum computers.
  arXiv  Detail & Related papers  (2022-02-19T00:21:46Z)
• Quantum algorithms for matrix operations and linear systems of equations [65.62256987706128]
  We propose quantum algorithms for matrix operations using the "Sender-Receiver" model. These quantum protocols can be used as subroutines in other quantum schemes.
  arXiv  Detail & Related papers  (2022-02-10T08:12:20Z)
• Three-fold way of entanglement dynamics in monitored quantum circuits [68.8204255655161]
  We investigate the measurement-induced entanglement transition in quantum circuits built upon Dyson's three circular ensembles. We obtain insights into the interplay between the local entanglement generation by the gates and the entanglement reduction by the measurements.
  arXiv  Detail & Related papers  (2022-01-28T17:21:15Z)
• Localization in quantum walks with periodically arranged coin matrices [0.0]
  The occurrence of localization is known to be equivalent to the existence of eigenvalues of the time evolution operators. This study shows that the method can be applied to extended models with periodically arranged coin matrices.
  arXiv  Detail & Related papers  (2021-11-30T05:22:28Z)
• Localization of space-inhomogeneous three-state quantum walks [0.0]
  We construct the method for the eigenvalue problem via the transfer matrix for space-inhomogeneous $n$-state quantum walks in one dimension with $n-2$ self-loops. This method reveals 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.
  arXiv  Detail & Related papers  (2021-11-29T02:13:30Z)
• Vertex Lattice Models Simulated with Quantum Circuits [0.0]
  We build quantum circuits that implement the actions of transfer matrices on arbitrary many-qubit states. The number of qubits and the depth of the circuits grow linearly with the size of the system. We present tests using quantum simulators and demonstrate that important physical quantities can be extracted.
  arXiv  Detail & Related papers  (2021-10-31T14:24:20Z)
• Entanglement dynamics of spins using a few complex trajectories [77.34726150561087]
  We consider two spins initially prepared in a product of coherent states and study their entanglement dynamics. We adopt an approach that allowed the derivation of a semiclassical formula for the linear entropy of the reduced density operator.
  arXiv  Detail & Related papers  (2021-08-13T01:44:24Z)
• Strongly trapped space-inhomogeneous quantum walks in one dimension [0.30458514384586394]
  localization is a characteristic phenomenon of space-inhomogeneous quantum walks in one dimension. In this paper, we introduce the analytical method to calculate eigenvectors using the transfer matrix. We also extend our results to characterize eigenvalues not only for two-phase quantum walks with one defect but also for a more general space-inhomogeneous model.
  arXiv  Detail & Related papers  (2021-05-23T15:36:54Z)
• General expressions for the quantum Fisher information matrix with applications to discrete quantum imaging [0.28675177318965034]
  We derive general expressions for the quantum Fisher information matrix which bypass matrix diagonalization and do not require the expansion of operators on an orthonormal set of states. We demonstrate the power of our approach by deriving novel results in the timely field of discrete quantum imaging.
  arXiv  Detail & Related papers  (2020-12-02T22:18:22Z)
• Quantum particle across Grushin singularity [77.34726150561087]
  We study the phenomenon of transmission across the singularity that separates the two half-cylinders. All the local realisations of the free (Laplace-Beltrami) quantum Hamiltonian are examined as non-equivalent protocols of transmission/reflection. This allows to comprehend the distinguished status of the so-called bridging' transmission protocol previously identified in the literature.
  arXiv  Detail & Related papers  (2020-11-27T12:53:23Z)
• Hilbert-space geometry of random-matrix eigenstates [55.41644538483948]
  We discuss the Hilbert-space geometry of eigenstates of parameter-dependent random-matrix ensembles. Our results give the exact joint distribution function of the Fubini-Study metric and the Berry curvature. We compare our results to numerical simulations of random-matrix ensembles as well as electrons in a random magnetic field.
  arXiv  Detail & Related papers  (2020-11-06T19:00:07Z)
• Unraveling the topology of dissipative quantum systems [58.720142291102135]
  We discuss topology in dissipative quantum systems from the perspective of quantum trajectories. We show for a broad family of translation-invariant collapse models that the set of dark state-inducing Hamiltonians imposes a nontrivial topological structure on the space of Hamiltonians.
  arXiv  Detail & Related papers  (2020-07-12T11:26:02Z)
• State preparation and measurement in a quantum simulation of the O(3) sigma model [65.01359242860215]
  We show that fixed points of the non-linear O(3) sigma model can be reproduced near a quantum phase transition of a spin model with just two qubits per lattice site. We apply Trotter methods to obtain results for the complexity of adiabatic ground state preparation in both the weak-coupling and quantum-critical regimes. We present and analyze a quantum algorithm based on non-unitary randomized simulation methods.
  arXiv  Detail & Related papers  (2020-06-28T23:44:12Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.