PaperlayerView exact revision on arXiv

Source-linked AI summary

Quantum walks: a comprehensive review

Salvador E. Venegas-Andraca

arXiv:1201.4780v2quant-phmath-ph

TL;DR

Quantum walks raise foundational and algorithmic questions about quantum counterparts of classical random walks, including how discrete and continuous models are related and which properties are genuinely quantum. This review synthesizes their theory, algorithms, experiments, and entanglement, and reports computational universality for both continuous- and discrete-time quantum walks.

  • Problem

    The field needs a unified account of quantum-walk foundations, algorithms, model connections, experimental work, and the quantum properties underlying possible computational advantages.

  • Method

    The paper reviews theoretical foundations, randomness, connections between coined discrete and continuous models, quantumness, entanglement, experiments, and algorithms based on both walk types.

  • Results

    The review reports that both continuous- and discrete-time quantum walks are computationally universal.

  • Takeaways & Limitations

    Quantum walks and quantum circuits have essentially the same computational power, according to the reviewed universality results.

  • Takeaways & Limitations

    The review notes that enhanced variance can be reproduced by classical-physics implementations, leaving which properties are exclusively quantum as an open question.

Abstract

from arXiv · show

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.

1 Introduction

The introduction positions quantum walks as quantum-mechanical counterparts of classical random walks and as tools for quantum algorithms, while emphasizing the importance of computational complexity and physical models of computation. The review surveys foundations, algorithms, experimental work, and the computational universality of discrete- and continuous-time quantum walks.

  • Computational complexity helps estimate implementation costs and compare problems through the resources required to solve them.
  • Quantum walks are quantum-mechanical counterparts of classical random walks and have been shown to constitute a universal model of quantum computation.
  • The review develops a computer-science perspective on quantum-mechanical and computational concepts relevant to quantum walks.
  • The foundations survey covers discrete- and continuous-time quantum walks, randomness, their mathematical connections, quantumness, entanglement, and experiments.
  • The algorithms survey examines search, element distinctness, triangle problems, quantized stochastic matrices, continuous-time algorithms, and non-unitary extensions.
  • The final section reviews the computational universality of both continuous- and discrete-time quantum walks.

2 Fundamentals of Quantum Walks

This section introduces quantum walks as quantum counterparts of classical random walks and distinguishes discrete from continuous models by their evolution timing. It reviews their mathematical foundations, algorithmic relevance, physical interpretation, and experimental realizations.

  • Quantum walks are quantum counterparts of classical random walks and form a relatively new research topic linked to quantum computation and natural-phenomena modeling.
  • Discrete quantum walks use a walker, a coin, and a unitary evolution operator applied at discrete time steps.
  • Continuous quantum walks use a walker and a Hamiltonian evolution operator that can be applied without timing restrictions.
  • Quantum-walk models are generally studied on discrete graphs because graphs are widely used in computer science and quantum-algorithm development.
  • The section reviews classical foundations, line and graph walks, barriers, decoherence, limit theorems, localization, and other discrete-time results.
  • It also points readers to related reviews covering the mathematical, physical, and algorithmic properties of quantum walks.

2.1 Classical random walk on an unrestricted line

The section introduces classical discrete random walks on graphs and analyzes an unrestricted walk on a line through its binomial position distribution and hitting times. Hitting time depends strongly on the target’s location: central positions are reached much sooner than positions near the distribution’s tails.

  • A classical walk on a line moves one unit left or right at each step according to probabilities p and q = 1 − p.
  • After n steps, the walker’s position is described by a binomial distribution, with mean np and variance npq for the step count.
  • The position variance satisfies V[Z_n] = 4npq, so it grows as O(n).
  • Cayley graphs are k-regular graphs whose group structure can support algorithm development, and regular connected graphs have a uniform stationary distribution.
  • Hitting time is the expected number of steps before a target node is visited, while mixing time measures convergence toward the limiting distribution.
  • For an n-step unrestricted walk, reaching k << n takes √n steps on average, whereas reaching k ≈ n takes exponentially many steps.

2.2 Discrete quantum walk on a line

Discrete quantum walks on a line use a coin and conditional shift to produce nontrivial walker evolution. The Hadamard walk exhibits skewed, rapidly spreading distributions and has quantum mixing behavior distinct from classical walks.

  • Motivation: DQWLs provide a simple model for constructing walks on circles or general graphs and for testing the quantumness of experimental quantum computers.Their simplicity also supports exploration of properties relevant to quantum-algorithm development.
  • Structure of a basic coined DQWL: A DQWL uses a coin, a walker, unitary coin and shift operators, and measurements of the resulting quantum state.The conditional shift moves the walker left or right according to the coin state, while the total evolution is U = S · (C ⊗ I_p).
  • Related formulations: The coin-based formulation follows earlier work showing that an additional coin system enables nontrivial discrete evolution.Alternative coinless formulations rearrange the Hamiltonian but sacrifice translation invariance.
  • Schrödinger approach: Two 100-step Hadamard DQWLs have skewed probability distributions whose skewness symmetry depends on the initial coin state.The compared initial states are |0⟩c ⊗ |0⟩p and |1⟩c ⊗ |0⟩p.
  • Hadamard walk properties: The Hadamard walk’s wave function is nearly uniform within its main propagation region and decreases rapidly outside it.The position probability is concentrated in an interval whose width grows linearly with the number of steps.
  • Hadamard walk properties: The unrestricted Hadamard walk has standard deviation O(t), whereas the corresponding classical walk has standard deviation O(√t).Its mixing time is O(t), compared with the classical line-walk mixing time O(t^2).

2.2.3 Discrete Path Integral Analysis of the Hadamard Walk

The discrete path-integral approach analyzes the Hadamard walk by counting paths between positions and translating those counts into quantum amplitudes. Its resulting probabilities can be represented with Jacobi polynomials, and it is equivalent to the Schrödinger approach.

  • Discrete path-integral method: The combinatorial approach counts paths carrying a walker from one position to another, functioning as a discrete path-integral method.The method quantifies quantum-state amplitudes through path counts.
  • Amplitude calculation: For the Hadamard walk, a lemma gives the amplitudes of position n after t steps using the path-counting formulation.The stated conditions relate t, n, and the path-count parameter l.
  • Polynomial representation: The probabilities derived from the Hadamard-walk amplitudes can be expressed using Jacobi polynomials.The polynomials provide a mathematical representation of the resulting position probabilities.
  • Relation to other methods: The Schrödinger and combinatorial approaches to the Hadamard walk are equivalent.Other combinatorial methods based on unitary-matrix decompositions and group theory were also proposed for quantum-walk analysis.

2.2.4 Unrestricted DQWL with a general coin

The general-coin analysis extends the Hadamard-walk treatment to unrestricted quantum walks on a line. By varying the initial coin state’s phase while fixing the coin operator, the study captures the relevant phase-factor behavior of general walks.

  • Generalization: Important properties of the Hadamard walk, including standard deviation and mixing time, are shared by quantum walks on the line more generally.This motivates using the Hadamard walk as a representative model.
  • General coin formulation: A general unrestricted walk is analyzed through the Fourier-transformed form of a two-dimensional coin operator and its t-step evolution.The coin is parameterized by θ, φ, and ρ.
  • Phase dependence: Fixing the coin operator while varying the initial coin-state phase allows different values of the combined phase factor r = α + θ.Multiple pairs of α and θ can yield the same phase factor.
  • Generalization: The analysis argues that studying a Hadamard walk suffices to analyze the properties of all unrestricted quantum walks on a line.The correspondence is framed through the phase factor determined by the initial state and coin operator.
  • Related results: For a general symmetric SU(2) coin operator, a closed form exists for the probability that a quantum walk reaches a given vertex after n steps.This provides an additional result for coined walks on a line.

2.2.5 Discrete Quantum walk with boundaries

The review examines discrete quantum walks with one or two absorbing boundaries, emphasizing absorption probabilities and their contrast with classical walks.

  • One absorbing boundary: The semi-infinite walk’s eventual absorption probability differs sharply from the classical value of unity.The cited theorem gives the quantum probability, although the extracted passage truncates its numerical expression.
  • One absorbing boundary: Quantum absorption decays faster than classical absorption, yielding a finite conditional expectation unlike the classical case.This comparison is explicitly reported for the semi-infinite walk.
  • Numerical illustrations: The section also reports 100-step Hadamard-walk graphs using specified coin-position initial states and an equation-defined shift operator.The extracted caption fragments identify distinct initial states for the graphs and the computational setup.
  • Two absorbing boundaries: For two absorbing boundaries, the review defines p_n as the probability of exiting left and q_n as the probability of exiting right.The displayed theorem introduces these probabilities for each n > 1.
  • Related analyses: Subsequent work revisited the one- and two-barrier results using Fourier-transform and path-counting proofs and extended studies of absorption probabilities.The review cites Bach, Borisov, and Konno among these follow-up analyses.

2.2.6 Unrestricted quantum walks on a line with several coins

This section surveys unrestricted quantum walks with multiple, position-dependent, time-dependent, and higher-dimensional coins, alongside decoherence and disorder as routes toward classical or localized behavior.

  • Multiple coins: Four-state and maximally entangled coins are associated with localization through eigenvalue degeneracies or concentration near the origin.The review presents these as related localization phenomena in one-dimensional walks.
  • Inhomogeneous coins: Inhomogeneous quantum walks allow coin operators to depend on position and coin registers, enabling studies of self-duality, localization, fractality, and return probabilities.The cited works analyze several structural properties of this generalized model.
  • Multiple and time-dependent coins: Multiple two-dimensional coins have been analyzed for limit distributions and for conditions under which their behavior becomes classical.The review also mentions time-dependent coins and corresponding analytical limit distributions.
  • Classical–quantum connections: Research connects classical and quantum walks through classical-walk simulation, quantum-to-classical transitions, and comparisons of the laws governing both processes.The section frames these connections as computational and foundational questions.
  • Decoherence: Decoherence, arising from interaction with the environment, is studied as both a transition mechanism toward classical walks and a potentially useful resource for quantum algorithms.The review identifies measurement, classical-environment interaction, coin decoherence, and higher-dimensional coins as relevant mechanisms.
  • Disorder: Disorder can produce diffusive spread or Anderson localization, depending on whether it is dynamic or static in photonic quantum-walk implementations.The cited experimental analysis distinguishes dynamic disorder from static disorder.

2.2.8 Limit theorems for quantum walks

The review surveys weak-limit and localization theorems for discrete and continuous quantum walks, emphasizing analytical distributions, moment behavior, and dependence on the initial state.

  • Discrete-time limit theorems: Konno’s weak-limit theorem provides a foundational limit-distribution result for one-dimensional discrete quantum walks.The theorem is presented for initial qubit states and general unitary evolution operators.
  • Konno’s density: Konno’s density function describes the limiting variable Z_ϕ obtained from a scaled one-dimensional quantum walk.The review introduces it after a path-integral derivation and gives its density over a bounded interval.
  • Discrete-time limit theorems: Theorem 9 states that three measures, Φ_s, Φ_0, and Φ_⊥, coincide when abcd ≠ 0.The review identifies this as a generalization of the Hadamard-walk result.
  • Moment behavior: Even moments of the walk position are independent of the initial qubit state, whereas odd moments depend on that state.The review reports this contrast as a central result of the cited limit-theorem work.
  • Hadamard walks: For Hadamard walks, the limiting distribution is compared with the classical symmetric random walk through an indicator-function formulation.The comparison is stated after the Hadamard specialization of the general theorem.
  • Continuous-time limit theorems: Continuous-time quantum walks on the integer line also admit weak-convergence theorems for their position distributions.Theorem 11 is stated for a walk with probability distribution P(k,t) at location k and time t.
  • Extensions: The literature extends limit and localization results to higher dimensions, time-dependent coins, random walks, many coins, qudits, quantum Markov chains, Cayley trees, and inhomogeneous walks.The review presents these as subsequent analytical generalizations using Fourier-transform, path-counting, and related methods.

2.2.9 Localization in discrete quantum walks

Localization in quantum walks is the absence of diffusion and can arise from disorder, eigenvalue degeneracy, or site-dependent coin operators; the section surveys analytical, numerical, and experimental studies of it.

  • Concept and origins: Localization denotes the absence of diffusion of a quantum state and may result from random or disordered environments that break dynamical periodicity.The review places localization in the broader context of condensed-matter physics.
  • Mechanisms: In quantum walks, localization has been detected through eigenvalue degeneracy or site-dependent coin operators.The review identifies both mechanisms as sources of localization phenomena.
  • Research landscape: The broader literature combines numerical, analytical, and experimental results on localization and related quantum-walk phenomena.The review uses this range of results to characterize the field as extensive.
  • Recurrence: Recurrence studies examine monitored absorption, Pólya numbers, and fractional recurrence in discrete- and continuous-time quantum walks.One-dimensional discrete coined walks are reported to exhibit fractional recurrence characterized by the quantum Pólya number.
  • Analytical methods: The CGMV method uses matrix-valued Szegő orthogonal polynomials and CMV matrices to formulate quantum-walk theory and analyze localization.Subsequent work applies this formalism to half-line and half-plane walks through spectral measures and limit distributions.
  • Inhomogeneous walks: Inhomogeneous walks can exhibit localization, including a limit distribution localized at the origin and a fractal eigenvalue spectrum in a studied class.The fractal spectrum is reported from numerical studies of the inhomogeneous model.
  • Related directions: Additional studies address quantum-foundational measurements, relativistic connections, memory, entanglement, sojourn time, and anyonic walkers.These works extend the review beyond localization to broader structural and physical properties of discrete quantum walks.

2.3 Discrete quantum walks on graphs

This section develops discrete-time quantum walks on graphs, defining their state space, node distributions, limiting behavior, and mixing times. It highlights uniform limiting distributions on suitable Cayley graphs and quadratic-or-better speedups over classical mixing.

  • Distributions: Quantum walks generally do not converge to a stationary distribution when successive node distributions differ, motivating time-averaged distributions.The averaged distribution is introduced to obtain a limiting object.
  • Model: A coined quantum walk on a d-regular graph uses a walker space, a d-dimensional coin space, a unitary coin, and a shift operator.The shift maps a vertex and coin state to the corresponding neighboring vertex.
  • Limiting distributions: For coined walks on Abelian Cayley graphs with distinct eigenvalues, the limiting distribution is uniform over graph nodes and independent of the initial state.The cycle result applies this theorem to odd cycles using the Hadamard coin.
  • Mixing times: For an odd n-cycle with the Hadamard coin, quantum mixing takes O(n log n), compared with O(n^2) for the corresponding classical random walk.The general bounded-degree result limits the quantum advantage to at most a quadratic factor.
  • Mixing times: On bounded-degree graphs, quantum-walk mixing time is at most quadratically faster than simple classical random-walk mixing time, not exponentially faster.This motivates studying other algorithmically relevant parameters, including hitting time.
  • Hypercube: For the discrete quantum walk on a hypercube, instantaneous mixing occurs at t = kπ/4 n, giving t = O(n) with ϵ = O(n−7/6) for odd k.The walk uses Grover’s operator as its coin, and its spectral expressions support later search-algorithm design.

2.4 Continuous quantum walks

Continuous quantum walks evolve on discrete graph spaces under Hamiltonians at unrestricted continuous times. The section reviews their stochastic analogues, mathematical formulation, hitting and mixing behavior, and applications across graph topologies.

  • Classical and quantum formulations: Continuous-time classical walks are described by an infinitesimal generator matrix, while quantum walks replace this evolution with Hamiltonian dynamics.The review presents the classical generator before defining the corresponding quantum Hamiltonian.
  • Definition: A continuous quantum walk uses a graph-based Hamiltonian and evolves according to Schrödinger dynamics in continuous time.The walker occupies a Hilbert space with basis states associated with graph vertices.
  • Results and applications: Continuous quantum-walk research studies mixing times, hitting times, transition probabilities, limiting distributions, transport velocity, and entropy across diverse graph models.Examples include star graphs, threshold networks, trees, ultrametric spaces, and small-world networks.
  • Hitting times: For continuous quantum walks, hitting times may be infinite or finite depending on the measurement rate when measurements occur at Poisson-distributed random times.This definition incorporates measurements into the hitting-time analysis.
  • Randomness: Quantum evolution itself is deterministic, whereas randomness in quantum walks arises from decoherence or measurement processes involving walkers or coins.This distinction explains why quantum walks can serve as quantum counterparts of stochastic algorithms.

2.6 How are continuous and discrete quantum walks connected?

The review addresses how discrete and continuous quantum walks can be related despite their different mathematical forms. It describes limit-process constructions in both directions and notes equivalent formulations of discrete walks.

  • Motivation: Transforming between discrete and continuous quantum walks is nontrivial, despite the classical connection between discrete and continuous random walks.The issue also reflects the differing roles of the coin degree of freedom in quantum-walk models.
  • Strauch’s contribution: Strauch constructs a discrete walk whose unitary evolution approaches a continuous walk through a limit process.The construction uses amplitudes on a discrete lattice and does not require a coin degree of freedom.
  • Childs’s contribution: Childs provides an ϵ-approximation framework that simulates a continuous quantum walk as a limit of discrete quantum walks.The procedure enlarges the Hilbert space, builds a modified isometry, applies discrete-walk steps, and projects onto a chosen basis.
  • Algorithms: The review notes a continuous-time quantum-walk algorithm for distinctness, a problem previously solved with a discrete quantum-walk algorithm.The continuous-time framework also includes a notion of query complexity.
  • Discrete-walk formulations: Coined and scattering discrete quantum walks admit a general framework for unitary equivalence.This provides another formulation-level connection among quantum-walk models.

2.7 Are quantum walks really quantum?

The section examines whether quantum-walk behavior is genuinely quantum or can be reproduced by classical wave interference. It concludes that classical setups can reproduce some single-walker statistics, while specifically quantum resources matter for complementarity, multiple walkers or coins, and entanglement.

  • Classical reproduction: Classical electromagnetic-wave implementations can reproduce some one-walker quantum-walk statistics, including variance enhancement over classical random walks.This motivates distinguishing statistical resemblance from genuinely quantum behavior.
  • Genuinely quantum properties: Quantum mechanical descriptions remain important for indivisibility and complementarity, properties identified as unavailable for exact classical reproduction.Complementarity is also presented as a resource for testing quantum-computer realizations and computation.
  • Quantum-walk statistics: The quadratic increase in walker variance is attributed to quantum evolution through Markovian and interference components of the evolution equation.The equation can be separated into these two contributions.
  • Multiple systems and entanglement: Multiple walkers or coins enable the detection, quantification, and use of quantum-mechanical properties such as entanglement.The review treats entanglement both as an outcome of quantum walks and as a resource for constructing new walks.
  • Entanglement studies: Research on entanglement in quantum walks includes analytical limits, multipartite measures, directional correlations, decoherence effects, and experimental violations of classical limits.Reported studies span lines, trees, cycles, lattices, and multi-particle settings.

2.8 Experimental proposals and realizations of quantum walks

The review surveys experimental proposals and realizations of discrete and continuous quantum walks using optical, atomic, solid-state, and superconducting platforms. It also covers experiments probing multiparticle statistics and correlations.

  • Discrete-time quantum walks: Discrete quantum walks have been proposed or implemented with classical optical devices, single photons, optical lattices, quantum dots, and parametric down-conversion.These proposals span both classical optical realizations and single-photon implementations.
  • Continuous-time quantum walks: Continuous-time quantum walks have been proposed using Skyrmion-burst remnants and implemented on optical chips with arrays of waveguides.The optical-chip architecture implements a two-photon continuous quantum walk.
  • Other proposed platforms: Additional platforms include superconducting circuit quantum electrodynamics, Bose–Einstein condensates, quantum dots, and ion traps.These proposals target walks on circles, multistep walks, and related implementations.
  • Multiparticle experiments: Experiments have studied two-boson dynamics, particle statistics, correlations in interacting and non-interacting walks, and quantum correlations in photons.The reported work uses lattice, coined-walk, and parametric-down-conversion settings.

3 Algorithms based on quantum walks and classical simulation of quantum algorithms-quantum walks

The review examines quantum-walk algorithms for search, graph traversal, and quantum-system simulation, alongside links between quantum walks and classical stochastic processes. It also emphasizes that apparent quantum-walk advantages require careful comparison with classical algorithms and implementations.

  • Algorithmic applications: Quantum-walk algorithms address unordered-list and structured search problems, including grids, hypercubes, element distinctness, and triangles.The section introduces oracle and hitting-time concepts before reviewing these algorithmic applications.
  • Search: Grover’s algorithm searches an unstructured space in O(√N) steps, compared with at least O(N) classical steps.The review notes applications including 3-SAT.
  • Grid search: Quantum-walk search performance on grids depends on dimension and coin-operator selection, with reviewed algorithms achieving dimension-dependent complexities.The cited results include logarithmic corrections and higher-dimensional variants.
  • Quantum walks from Markov chains: Szegedy’s framework quantizes a stochastic matrix and gives, for reversible symmetric chains, quantum hitting time at most the square root of the classical hitting time.The framework constructs a quantum evolution operator from a classical stochastic matrix.
  • Graph traversal: A continuous quantum walk can traverse a graph in polynomial time while the corresponding classical random walk requires exponential time, but this does not imply exponential speedup when a deterministic polynomial-time traversal exists.The review uses this result to distinguish quantum-walk versus classical-random-walk comparisons from comparisons against the best classical algorithm.
  • Quantum-system simulation: Continuous quantum walks have also been used to model energy-transfer phenomena and photosynthetic processes through a non-unitary walk on a directed graph.The framework analyzes environmental action on coherent quantum dynamics.
  • Classical simulation and quantumness: Classical physical implementations can reproduce the enhanced variance of a discrete quantum walk, leaving the quantum properties responsible for algorithmic speedup unresolved.The review identifies this as an open question about which quantum-mechanical properties and operations enhance computational capability.

4 Universality of quantum walks

Quantum-walk universality is established for both continuous- and discrete-time models by representing computation as quantum-information propagation through graph-based wires and gate structures.

  • Universality results: Universal computation has been formally proved for continuous-time and discrete-time quantum walks.The review discusses proofs by Childs, Lovett et al., and Underwood and Feder.
  • Continuous-time quantum walks: Continuous-time universality encodes algorithms as scattering-based propagation on graphs whose subgraphs represent quantum operators connected by wires.The graph structure depends on the problem, while operator subgraphs have maximum degree three.
  • Continuous-time quantum walks: The continuous-time construction uses eigenvalues and eigenvectors of computational graphs to calculate propagation through scattering processes.Finite graphs can be modeled with the same mathematical framework without significant changes.
  • Scope of the constructions: The review presents wire exchanges, gate widgets, and phase operations as logical constructions for universal computation, not implementations on actual quantum hardware.The discrete-time discussion includes Cnot, phase, and basis-changing gate widgets, while the continuous-time proposal is explicitly theoretical.
  • Discrete-time quantum walks: Discrete-time universality uses a universal gate set and quantum wires representing basis states rather than qubits.The construction maintains a close conceptual link with the continuous-time proposal.
  • Discrete-time quantum walks: A four-dimensional Grover coin transfers equal-amplitude, equal-phase inputs across even-degree vertices, enabling perfect quantum-information transfer between graph sides.The physical interpretation is a unitary transfer operation, while the shift operator is specified by its expected behavior rather than a hardware implementation.
  • Scope of the constructions: Together, the universality proofs provide alternative computational models that researchers can select according to their backgrounds and interests.The review characterizes these models as a toolbox for quantum computation.

5 Conclusions

The review synthesizes foundations, randomness, mathematical connections, entanglement, experiments, algorithms, and universality across discrete- and continuous-time quantum walks. It presents quantum walks as an established research field with continuing open problems.

  • Review scope: The review covers theoretical foundations of discrete- and continuous-time quantum walks, including their mathematical connections and the role of randomness.It also discusses the quantumness of quantum walks.
  • Review scope: The review summarizes work on discrete-time quantum walks and entanglement, experimental proposals and realizations, and algorithms based on quantum walks.The scope spans both discrete- and continuous-time algorithmic approaches.
  • Main conclusion: Computational universality is reported as a central result for both continuous- and discrete-time quantum walks.This conclusion is presented alongside the review’s broader synthesis of quantum-walk research.
  • Main conclusion: Quantum walks are characterized as a solid research field with open problems relevant to physicists, computer scientists, and engineers.The review is intended to encourage further work in quantum walks.
Loading 1201.4780v2…