Quantum walks: a comprehensive review
arXiv:1201.4780 · doi:10.1007/s11128-012-0432-5
Abstract
Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.
Paper accepted for publication in Quantum Information Processing Journal
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- Gossip Algorithms in Quantum Networks
- Fermionic walkers driven out of equilibrium
- On the equivalence between quantum and random walks on finite graphs
- Quantum walk processes in quantum devices
- Accelerated first detection in discrete-time quantum walks using sharp restarts
- Dynamical Phase Transitions in Open Quantum Walks
- New theory of diffusive and coherent nature of optical wave via a quantum walk
- Borromean states in discrete-time quantum walks
- Spectral Properties of Non-Unitary Band Matrices
- Phase transition of an open quantum walk
- Superdiffusivity of quantum walks: A Feynman sum-over-paths description
- QReach: A Reachability Analysis Tool for Quantum Markov Chains
- A limit law of the return probability for a quantum walk on a hexagonal lattice
- Photonic random walks with traps
- Finite-element assembly approach of optical quantum walk networks
- High-fidelity state transfer via quantum walks from delocalized states
- Experimental realization of universal quantum gates and six-qubit entangled state using photonic quantum walk
- Weak limits for quantum walks on the half-line
- Quantum Walk on Orbit Spaces
- Anderson localization without eigenstates in photonic quantum walks
- Scattering Expansion for Localization in One Dimension: from Disordered Wires to Quantum Walks
- History states of one-dimensional quantum walks
- Anomalous transport in periodic photonic chains with designed loss
- Optical realization of one-dimensional generalized split-step quantum walks
- Experimental quantum state transfer of an arbitrary single-qubit state on a cycle with four vertices using a coined quantum random walk
- Universal dynamical scaling laws in three-state quantum walks
- Exponential speedups for quantum walks in random hierarchical graphs
- Revivals in One-dimensional Quantum Walks with a Time and Spin-dependent Phase Shift
- Nonclassical oscillations in pre- and post-selected quantum walks
- Feynman checkers: the probability to find an electron vanishes nowhere inside the light cone
- Quantum Stochastic Walk Models for Quantum State Discrimination
- On the potential of quantum walks for modeling financial return distributions
- Quantum walks and entanglement in cavity networks
- Non-Markovianity in Discrete-Time Open Quantum Random Walk on Arbitrary Graphs
- Hitting time for quantum walks of identical particles
- Two-particle coined-quantum walk with long-range interaction
- Nonstabilizerness generation in a multiparticle quantum walk
- Analytical expression for variance of homogeneous-position quantum walk with decoherent position
- Hitting statistics from quantum jumps
- Spin-1 Quantum Walks
- Quantum centipedes: collective dynamics of interacting quantum walkers
- Dequantization via quantum channels
- Escaping Local Minima with Quantum Coherent Cooling
- Analytical solutions for quantum walks on 1D chain with different shift operators
- Influence of coin symmetry on infinite hitting times in quantum walks
- Dynamics and computation in linear open quantum walks
- Strong analog classical simulation of coherent quantum dynamics
- Directivity of quantum walk via its random walk replica
- Disorder-free localisation in continuous-time quantum walks : Role of symmetries
- Transport and Localization in Quantum Walks on a Random Hierarchy of Barriers
- Quantum walk for SU(1,1)
- The syntactic side of autonomous categories enriched over generalised metric spaces
- Wigner formalism for a particle on an infinite lattice: dynamics and spin
- On Hitting Times for General Quantum Markov Processes
- Scattering Expansion for Localization in One Dimension
- Quantum walks as thermalizations, with application to fullerene graphs
- Lower Bounds on the Localisation Length of Balanced Random Quantum Walks
- Effective simulation of state distribution in qubit chains
- Information scrambling in quantum walks: Discrete-time formulation of Krylov complexity
- Attractors and asymptotic dynamics of open discrete-time quantum walks on cycles
- Quantum vs Classical Birth and Death Processes; Exactly Solvable Examples
- One-dimensional Continuous-Time Quantum Markov Chains: qubit probabilities and measures
- Quantum Ultra-Walks: Walks on a Line with Hierarchical Spatial Heterogeneity
- Optimal quantum transport on a ring via locally monitored chiral quantum walks
- Quantum transfer through small networks coupled to phonons: effects of topology vs phonons
- Deterministic generation of hybrid entangled states using quantum walks
- Programmable non-Hermitian photonic quantum walks via dichroic metasurfaces
- Entanglement transfer, accumulation and retrieval via quantum-walk-based qubit-qudit dynamics
- A quantum walk on the half line with a particular initial state
- Optimization for the propagation of a multiparticle quantum walk in a one-dimensional lattice
- Limit distribution of a continuous-time quantum walk with a spatially 2-periodic Hamiltonian
- Perfect state transfer using Markovian quantum walk
- Experimental measurement of quantum first-passage-time distributions
- Discovery of energy landscapes towards optimized quantum transport: Environmental effects and long-range tunneling
- Deutsch-Jozsa and Bernstein-Vazirani algorithm using single-particle discrete-time quantum walk
- Bidirectional quantum teleportation using quantum walks
- Bosonic Entanglement and Quantum Sensing from Energy Transfer in two-tone Floquet Systems
- Two-dimensional quantum central limit theorem by quantum walks
- Quantum walk with coherent multiple translations induces fast quantum gate operations
- Compact and programmable large-scale optical processor in free space
- Dissipative realization of a quantum distance-based classifier using open quantum walks
- Quantum Hitting Time according to a given distribution
- Hitting time expressions for quantum channels: beyond the irreducible case and applications to unitary walks
- Repeated measurements and random scattering in quantum walks
- Model Reduction for Quantum Systems: Discrete-time Quantum Walks and Open Markov Dynamics
- State preparation and evolution in quantum computing: a perspective from Hamiltonian moments
- Directional quantum random walk induced by coherence
- Invariance in Quantum Walks
- Spectral Property of Magnetic Quantum Walk on Hypercube
- Dynamical Localization and Transport properties of Quantum Walks on the hexagonal lattice
- The role of coherence on two-particle quantum walks
- Hypercube Quantum Search: Exact Computation of the Probability of Success in Polynomial Time
- Quantum Magic in Discrete-Time Quantum Walk
- Simulating the discrete-time quantum walk dynamics with simultaneous coin and shift operators
- Decoherence and classicalization of continuous-time quantum walks on graphs
- Mimicking the Hadamard discrete-time quantum walk with a time-independent Hamiltonian
- Realizing non-trivial doublon formation using a quantum computer
- Complementarity in quantum walks
- Exact solutions and symmetry analysis for the limiting probability distribution of quantum walks
- Quantum walks assisted by particle number fluctuations
- Influence of generic quantum coins on the spreading and entanglement in binary aperiodic quantum walks
- A limit theorem for a splitting distribution of a quantum walk
- The localization of quantum random walks on sierpinski gaskets
- Conserved Quantities in Linear and Nonlinear Quantum Search
- Quantum walk mixing is faster than classical on periodic lattices
- Quantum spatial best-arm identification via quantum walks
- Continuous-time quantum walks on a defective lattice: Boosting the spreading of delocalized states through Parrondo's strategy
- Absorption-based qubit estimation in discrete-time quantum walks
- Crystalline Spectral Form Factors
- Quantum state transfer and periodicity in discrete-time quantum walks under non--Markovian dephasing noise
- Temporal nonclassicality in continuous-time quantum walks
- Correlated dynamics of three-particle bound states induced by emergent impurities in Bose-Hubbard model
- A simple model of quantum walk with a gap in distribution
- Transition in Splitting Probabilities of Quantum Walks
- Quantum states supported by matroids
- Many-Body Effects in a Molecular Quantum NAND Tree
- Hybrid atom-molecule quantum walks in one-dimensional optical lattice
- Efficient Scheme of Experimental Quantifying non-Markovianity in High-Dimension Systems
- Robustness and classical proxy of entanglement in variants of quantum walk
- Feynman checkers: through the looking-glass
- Phase transition of a continuous-time quantum walk on the half line
- Realizing Topological Quantum Walks on NISQ Digital Quantum Computer
- Return probability of quantum and correlated random walks
- Quantum walks advantage on the dihedral group for uniform sampling problem
- Comparison among Classical, Probabilistic and Quantum Algorithms for Hamiltonian Cycle problem
- Limit distribution of a time-dependent quantum walk on the half line
- Does Quantum Mechanics Breed Larger, More Intricate Quantum Theories? The Case for Experience-Centric Quantum Theory and the Interactome of Quantum Theories
- Search Via quantum walks with intermediate measurements
- Quantum Coin Flipping, Qubit Measurement and Generalized Fibonacci Numbers
- Faster Search of Clustered Marked States with Lackadaisical Quantum Walks
- Harmonic oscillator kicked by spin measurements: a Floquet-like system without classical analogous
- Topology identification of autonomous quantum dynamical networks
- Response to glassy disorder in coin on spread of quantum walker
- Complexity for one-dimensional discrete time quantum walk circuits
- Key graph properties affecting transport efficiency of flip-flop Grover percolated quantum walks
- Quantum walks under superposition of causal order
- Feynman checkers: lattice quantum field theory with real time
- Skeleton structure inherent in discrete-time quantum walks
- Optimal detection of quantum states via projective measurements