Localization and Fractality in Inhomogeneous Quantum Walks with Self-Duality
arXiv:1004.5394 · doi:10.1103/PhysRevE.82.031122
Abstract
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 quasi-periodic 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.
8 pages, 4 figures
References in corpus (31)
- Quantum random walks - an introductory overview
- Environment-Assisted Quantum Walks in Photosynthetic Energy Transfer
- A Quantum Random Walk Search Algorithm
- Universal computation by quantum walk
- From quantum cellular automata to quantum lattice gases
- Universal quantum computation using the discrete time quantum walk
- Discrete single-photon quantum walks with tunable decoherence
- Controlling discrete quantum walks: coins and intitial states
- Quantum walks in higher dimensions
- Decoherence in quantum walks - a review
- Weak limits for quantum random walks
- One-Dimensional Three-State Quantum Walk
- Localization of Two-Dimensional Quantum Walks
- Recurrence and Pólya number of quantum walks
- Breakdown of an Electric-Field Driven System: a Mapping to a Quantum Walk
- Limit distributions of two-dimensional quantum walks
- Recurrence properties of unbiased coined quantum walks on infinite -dimensional lattices
- Localization of Multi-State Quantum Walk in One Dimension
- Quantum Group and Magnetic Translations. Bethe-Ansatz Solution for Azbel-Hofstadter Problem
- Quantum phase transition using quantum walks in an optical lattice
- Inhomogeneous Quantum Walks
- Explicit Solutions of the Bethe Ansatz Equations for Bloch Electrons in a Magnetic Field
- One-dimensional quantum random walks with two entangled coins
- Recurrence of biased quantum walks on a line
- Emergence of Randomness and Arrow of Time in Quantum Walks
- Quantum Group, Bethe Ansatz and Bloch Electrons in a Magnetic Field
- The Fibonacci quantum walk and its cassical trace map
- Hierarchical Structure of Azbel-Hofstader Problem: Strings and loose ends of Bethe Ansatz
- Asymptotically exact solutions of Harper equation
- Schroedinger Operators With Few Bound States
- Fractional recurrence in discrete-time quantum walk
Cited by in corpus (64)
- Quantum walks: a comprehensive review
- Mobility Edges in 1D Bichromatic Incommensurate Potentials
- Topological phases and delocalization of quantum walks in random environments
- Trapping a particle of a quantum walk on the line
- Disordered Quantum Walks in one lattice dimension
- Unveiling hidden topological phases of a one-dimensional Hadamard quantum walk
- Two-component Dirac-like Hamiltonian for generating quantum walk on one-, two- and three-dimensional lattices
- Bulk-edge correspondence of one-dimensional quantum walks
- Entangling Power of Disordered Quantum Walks
- Coined quantum walks on percolation graphs
- Observation of quasiperiodic dynamics in a one-dimensional quantum walk of single photons in space
- Spectral theory of extended Harper's model and a question by Erdős and Szekeres
- Discrete-time quantum walk with feed-forward quantum coin
- Random Time-Dependent Quantum Walks
- Directional correlations in quantum walks with two particles
- Entanglement engineering and topological protection by discrete-time quantum walks
- Localized quantum walks as secured quantum memory
- Localized State in a Two-Dimensional Quantum Walk on a Disordered Lattice
- Aperiodic space-inhomogeneous quantum walks: localization properties, energy spectra and enhancement of entanglement
- Hybrid classical-quantum linear solver using Noisy Intermediate-Scale Quantum machines
- Localization in momentum space of ultracold atoms in incommensurate lattices
- Asymptotic entanglement in 1D quantum walks with a time-dependent coined
- Electric quantum walks in two dimensions
- Entanglement dynamics and quasi-periodicity in discrete quantum walks
- Quantum and random walks as universal generators of probability distributions
- Fermion confinement via Quantum Walks in 2D+1 and 3D+1 spacetime
- Spectral Transition for Random Quantum Walks on Trees
- Localization of the Grover walks on spidernets and free Meixner laws
- Classical-like behavior in quantum walks with inhomogeneous, time-dependent coin operators
- Singular continuous Cantor spectrum for magnetic quantum walks
- Percolation induced effects in 2D coined quantum walks: analytic asymptotic solutions
- The Witten Index for 1D Supersymmetric Quantum Walks with Anisotropic Coins
- On limiting distributions of quantum Markov chains
- Accelerated quantum walk, two-particle entanglement generation and localization
- Spectral Characteristics of the Unitary Critical Almost-Mathieu Operator
- Almost Everything About the Unitary Almost Mathieu Operator
- Renormalization and Scaling in Quantum Walks
- Quantum walks on random lattices: Diffusion, localization and the absence of parametric quantum speed-up
- Massless Dirac Equation from Fibonacci Discrete-Time Quantum Walk
- Localization and Recurrence of Quantum Walk in Periodic Potential on a Line
- One-Dimensional Quantum Walks with a Position-Dependent Coin
- Correlated Markov Quantum Walks
- Ramsauer effect in one-dimensional quantum walk with multiple defects
- Symmetry Protected Topological Phases and Majorana Mode in One-dimensional Quantum Walk with Boundary
- An algorithm to factorize quantum walks into shift and coin operations
- Quantum random walk in periodic potential on a line
- Quantum walk on a chimera graph
- Limit measures of inhomogeneous discrete-time quantum walks in one dimension
- Discrete-time quantum walks generated by aperiodic fractal sequence of space coin operators
- A quantum walk simulation of extra dimensions with warped geometry
- Weak limit theorem of a two-phase quantum walk with one defect
- Asymptotic stability of small bound state of nonlinear quantum walks
- Revivals in One-dimensional Quantum Walks with a Time and Spin-dependent Phase Shift
- Transport and Localization in Quantum Walks on a Random Hierarchy of Barriers
- Limit Theorems for the Discrete-Time Quantum Walk on a Graph with Joined Half Lines
- Quantum Ultra-Walks: Walks on a Line with Hierarchical Spatial Heterogeneity
- Resolvent Methods for Quantum Walks with an Application to a Thue-Morse Quantum Walk
- Invariance in Quantum Walks
- Weak convergence of the Wojcik model
- Generator of an abstract quantum walk
- Optimization for the propagation of a multiparticle quantum walk in a one-dimensional lattice
- Unitary equivalent classes of one-dimensional quantum walks II
- Localization in quantum walks with periodically arranged coin matrices
- Landau levels for discrete-time quantum walks in artificial magnetic fields