Connecting the discrete and continuous-time quantum walks

Abstract

Recently, quantized versions of random walks have been explored as effective elements for quantum algorithms. In the simplest case of one dimension, the theory has remained divided into the discrete-time quantum walk and the continuous-time quantum walk. Though the properties of these two walks have shown similarities, it has remained an open problem to find the exact relation between the two. The precise connection of these two processes, both quantally and classically, is presented. Extension to higher dimensions is also discussed.

5 pages, 1 figure

References in corpus (15)

• Quantum Computation and Decision Trees
• From quantum cellular automata to quantum lattice gases
• An example of the difference between quantum and classical random walks
• Weak limits for quantum random walks
• The quantum to classical transition for random walks
• Dirac and Weyl Equations on a Lattice as Quantum Cellular Automata
• Quantum walk on the line as an interference phenomenon
• Limit Theorem for Continuous-Time Quantum Walk on the Line
• Aperiodic Quantum Random Walks
• Simulating quantum mechanics on a quantum computer
• Relativistic quantum walks
• Quantum random walk on the line as a markovian process
• Quantum Random Walks do not need a Coin Toss
• Diffusive-ballistic crossover in 1D quantum walks
• Convergence of continuous-time quantum walks on the line

Cited by in corpus (94)

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• Random Walks: A Review of Algorithms and Applications
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• Quantum stochastic walks: A generalization of classical random walks and quantum walks
• Limit distributions of two-dimensional quantum walks
• Quantum Walks in artificial electric and gravitational Fields
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• Recurrence properties of unbiased coined quantum walks on infinite $d$-dimensional lattices
• Discrete Time Quantum Walk Approach to State Transfer
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• Photonic Simulation of Entanglement Growth After a Spin Chain Quench
• The Dirac equation as a quantum walk: higher dimensions, observational convergence
• Quantum Walks as simulators of neutrino oscillations in vacuum and matter
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• Universally Optimal Noisy Quantum Walks on Complex Networks
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• Review on Quantum Walk Computing: Theory, Implementation, and Application
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• Quantum walks on graphs representing the firing patterns of a quantum neural network
• Perfect state transfer by means of discrete-time quantum walk on complete bipartite graphs
• Mixing Times in Quantum Walks on the Hypercube
• Quantum walks as massless Dirac Fermions in curved Space-Time
• Dynamics of continuous-time quantum walks in restricted geometries
• Asymptotic entanglement in a two-dimensional quantum walk
• Non-Markovian continuous-time quantum walks on lattices with dynamical noise
• Non-interacting multi-particle quantum random walks applied to the graph isomorphism problem for strongly regular graphs
• Quantum simulation of bosonic-fermionic non-interacting particles in disordered systems via quantum walk
• Tailoring discrete quantum walk dynamics via extended initial conditions: Towards homogeneous probability distributions
• Directional correlations in quantum walks with two particles
• Spatial entanglement using a quantum walk on a many-body system
• Quantum Search with Multiple Walk Steps per Oracle Query
• Aperiodic space-inhomogeneous quantum walks: localization properties, energy spectra and enhancement of entanglement
• Full-revivals in 2-D Quantum Walks
• Equivalence between discrete quantum walk models in arbitrary topologies
• Crossovers induced by discrete-time quantum walks
• Pólya number of continuous-time quantum walks
• Eigenvalue Measurement of Topologically Protected Edge states in Split-Step Quantum Walks
• Limit theorem for a time-dependent coined quantum walk on the line
• Coherent exciton transport and trapping on long-range interacting cycles
• Quantum walks in weak electric fields and Bloch oscillations
• Quantum cellular automata and quantum field theory in two spatial dimensions
• Localization in Quantum Walks on a Honeycomb Network
• Discrete spacetime, quantum walks and relativistic wave equations
• Connection Between Continuous and Discrete Time Quantum Walks on d-Dimensional Lattices; Extensions to General Graphs
• Quantum walk transport properties on graphene structures
• Continuous Limit of Discrete Quantum Walks
• Fractional recurrence in discrete-time quantum walk
• Detection of discrete spacetime by matter interferometry
• Discrete-time quantum walks on one-dimensional lattices
• Continuous-Time Quantum Random Walks Require Discrete Space
• Interacting quantum walkers: Two-body bosonic and fermionic bound states
• Discrete-time quantum walks as fermions of lattice gauge theory
• Quantum field theory from a quantum cellular automaton in one spatial dimension and a no-go theorem in higher dimensions
• Framework for discrete-time quantum walks and a symmetric walk on a binary tree
• Generic quantum walk using a coin-embedded shift operator
• Unveiling and exemplifying the unitary equivalence of discrete time quantum walk models
• Discrete dynamics and non-Markovianity
• Green function approach for scattering quantum walks
• Twisted quantum walks, generalised Dirac equation and Fermion doubling
• Encoding qubits into harmonic-oscillator modes via quantum walks in phase space
• Tunneling effects in a one-dimensional quantum walk
• The Heun differential equation and the Gauss differential equation related to quantum walks
• A Quantum Walk Enhanced Grover Search Algorithm for Global Optimization
• Steepest Entropy Ascent Solution for a Continuous-Time Quantum Walker
• How to Suppress Dark States in Quantum Networks and Bio-Engineered Structures
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• Quantum centipedes: collective dynamics of interacting quantum walkers
• Phase space patterns of quantum transport on ordered and disordered networks
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• Transport and Localization in Quantum Walks on a Random Hierarchy of Barriers
• Discretization of continuous-time quantum walks via the staggered model with Hamiltonians
• Time averaged distribution of a discrete-time quantum walk on the path
• An analytic solution for one-dimensional quantum walks
• Experiments with Schrödinger Cellular Automata
• An overview of Quantum Cellular Automata
• Exact simulation of coined quantum walks with the continuous-time model
• Mass and Proper Time as Conjugated Observables
• The Hamiltonians generating one-dimensional discrete-time quantum walks
• Directionality and quantum backfire in continuous-time quantum walks from delocalized states: Exact results
• On the von Neumann entropy of certain quantum walks subject to decoherence
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• A Novel Bulk-Optics Scheme for Quantum Walk with High Phase Stability
• Eigenbasis of the Evolution Operator of 2-Tessellable Quantum Walks
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