Classical Random Walk Research Articles

Simulation of quantum walks on a circle with polar molecules via optimal control.

Quantum walks are the quantum counterpart of classical random walks and have various applications in quantum information science. Polar molecules have rich internal energy structure and long coherence time and thus are considered as a promising candidate for quantum information processing. In this paper, we propose a theoretical scheme for implementing discrete-time quantum walks on a circle with dipole-dipole coupled SrO molecules. The states of the walker and the coin are encoded in the pendular states of polar molecules induced by an external electric field. We design the optimal microwave pulses for implementing quantum walks on a four-node circle and a three-node circle by multi-target optimal control theory. To reduce the accumulation of decoherence and improve the fidelity, we successfully realize a step of quantum walk with only one optimal pulse. Moreover, we also encode the walker into a three-level molecular qutrit and a four-level molecular ququart and design the corresponding optimal pulses for quantum walks, which can reduce the number of molecules used. It is found that all the quantum walks on a circle in our scheme can be achieved via optimal control fields with high fidelities. Our results could shed some light on the implementation of discrete-time quantum walks and high-dimensional quantum information processing with polar molecules.

Quantum walk mixing is faster than classical on periodic lattices

The quantum mixing time is a critical factor affecting the efficiency of quantum sampling and algorithm performance. It refers to the minimum time required for a quantum walk to approach its limiting distribution closely and has implications across the areas of quantum computation. This work focuses on the continuous time quantum walk mixing on a regular graph, evolving according to the unitary map U=eiĀt, where the Hamiltonian Ā is the normalized adjacency matrix of the graph. In [Physical Review A 76, 042306 (2007).], Richter previously showed that this walk mixes in time O(ndlog(d)log(1/ϵ)) with O(log(d)log(1/ϵ)) intermediate measurements when the graph is the d−dimensional periodic lattice Zn×Zn×⋯×Zn. We extend this analysis to the periodic lattice L=Zn1×Zn2×⋯×Znd, relaxing the assumption that ni are identical. We provide two quantum walks on periodic lattices that achieve faster mixing compared to classical random walks: 1. A coordinate-wise quantum walk that mixes in O((∑i=1dni)log(d/ϵ)) time with O(dlog(d/ϵ)) measurements. 2. A continuous-time quantum walk with O(log(1/ϵ)) measurements that conjecturally mixes in O(∑i=1dni(log(n1))2log(1/ϵ)) time. Our results demonstrate a quadratic speedup over the classical mixing time O(dn12log(d/ϵ)) on the generalized periodic lattice L. We have provided analytical evidence and numerical simulations to support the conjectured faster mixing time of the continuous-time quantum walk algorithm. Making progress towards proving the general conjecture that quantum walks on regular graphs mix in O(δ−1/2log(N)log(1/ϵ)) time, where δ is the spectral gap and N is the number of vertices.

Quantum walks in the normalized Laplacian

We provide a characterization of perfect state transfer in a quantum walk whose Hamiltonian is given by the normalized Laplacian. We discuss a connection between classical random walks and quantum walks only present in this model, and we also rule out several trees as candidates to host perfect state transfer.

Learning attribute and homophily measures through random walks

We investigate the statistical learning of nodal attribute functionals in homophily networks using random walks. Attributes can be discrete or continuous. A generalization of various existing canonical models, based on preferential attachment is studied (model class mathscr {P}), where new nodes form connections dependent on both their attribute values and popularity as measured by degree. An associated model class mathscr {U} is described, which is amenable to theoretical analysis and gives access to asymptotics of a host of functionals of interest. Settings where asymptotics for model class mathscr {U} transfer over to model class mathscr {P} through the phenomenon of resolvability are analyzed. For the statistical learning, we consider several canonical attribute agnostic sampling schemes such as Metropolis-Hasting random walk, versions of node2vec (Grover and Leskovec, 2016) that incorporate both classical random walk and non-backtracking propensities and propose new variants which use attribute information in addition to topological information to explore the network. Estimators for learning the attribute distribution, degree distribution for an attribute type and homophily measures are proposed. The performance of such statistical learning framework is studied on both synthetic networks (model class mathscr {P}) and real world systems, and its dependence on the network topology, degree of homophily or absence thereof, (un)balanced attributes, is assessed.

Quantum walks on random lattices: Diffusion, localization, and the absence of parametric quantum speedup

Discrete-time quantum walks, quantum generalizations of classical random walks, provide a framework for quantum information processing, quantum algorithms and quantum simulation of condensed matter systems. The key property of quantum walks, which lies at the heart of their quantum information applications, is the possibility for a parametric quantum speed-up in propagation compared to classical random walks. In this work we study propagation of quantum walks on percolation-generated two-dimensional random lattices. In large-scale simulations of topological and trivial split-step walks, we identify distinct pre-diffusive and diffusive behaviors at different time scales. Importantly, we show that even arbitrarily weak concentrations of randomly removed lattice sites give rise to a complete breakdown of the superdiffusive quantum speed-up, reducing the motion to ordinary diffusion. By increasing the randomness, quantum walks eventually stop spreading due to Anderson localization. Near the localization threshold, we find that the quantum walks become subdiffusive. The fragility of quantum speed-up implies dramatic limitations for quantum information applications of quantum walks on random geometries and graphs.

Bandit Algorithm Driven by a Classical Random Walk and a Quantum Walk.

Quantum walks (QWs) have a property that classical random walks (RWs) do not possess-the coexistence of linear spreading and localization-and this property is utilized to implement various kinds of applications. This paper proposes RW- and QW-based algorithms for multi-armed-bandit (MAB) problems. We show that, under some settings, the QW-based model realizes higher performance than the corresponding RW-based one by associating the two operations that make MAB problems difficult-exploration and exploitation-with these two behaviors of QWs.

Measurements on an Anderson chain

We study the dynamics of a monitored single particle in a one-dimensional, Anderson-localized system. The time evolution is governed by Hamiltonian dynamics for fixed time intervals, interrupted by local, projective measurements. The competition between disorder-induced localization and measurement-induced jumps leads to interesting behaviour of readout-averaged quantities. We find that measurements at random positions delocalize the average position, similar to a classical random walk. Along each quantum trajectory, the particle remains localized, however with a modified localization length. In contrast to measurement-induced delocalization, controlled measurements can be used to introduce transport in the system and localize the particle at a chosen site. In this sense, the measurements provide a controlled environment for the particle.

Strong Shannon–McMillan–Breiman’s theorem for locally compact groups

AbstractWe prove that for a vast class of random walks on a compactly generated group, the exponential growth of convolutions of a probability density function along almost every sample path is bounded by the growth of the group. As an application, we show that the almost sure and $L^1$ convergences of the Shannon–McMillan–Breiman theorem hold for compactly supported random walks on compactly generated groups with subexponential growth.

Link Prediction with Continuous-Time Classical and Quantum Walks.

Protein-protein interaction (PPI) networks consist of the physical and/or functional interactions between the proteins of an organism, and they form the basis for the field of network medicine. Since the biophysical and high-throughput methods used to form PPI networks are expensive, time-consuming, and often contain inaccuracies, the resulting networks are usually incomplete. In order to infer missing interactions in these networks, we propose a novel class of link prediction methods based on continuous-time classical and quantum walks. In the case of quantum walks, we examine the usage of both the network adjacency and Laplacian matrices for specifying the walk dynamics. We define a score function based on the corresponding transition probabilities and perform tests on six real-world PPI datasets. Our results show that continuous-time classical random walks and quantum walks using the network adjacency matrix can successfully predict missing protein-protein interactions, with performance rivalling the state-of-the-art.

Universal Framework for Record Ages under Restart.

We propose a universal framework to compute record age statistics of a stochastic time series that undergoes random restarts. The proposed framework makes minimal assumptions on the underlying process and is furthermore suited to treat generic restart protocols going beyond the Markovian setting. After benchmarking the framework for classical random walks on the 1D lattice, we derive a universal criterion underpinning the impact of restart on the age of the nth record for generic time series with nearest-neighbor transitions. Crucially, the criterion contains a penalty of order n that puts strong constraints on restart expediting the creation of records, as compared to the simple first-passage completion. The applicability of our approach is further demonstrated on an aggregation-shattering process where we compute the typical growth rates of aggregate sizes. This unified framework paves the way to explore record statistics of time series under restart in a wide range of complex systems.

Application of Random Walks in Data Processing

A random walk is known as a process that a random walker makes consecutive steps in space at equal intervals of time and the length and direction of each step is determined independently. It is an example of Markov processes, meaning that future movements of the random walker are independent of the past. The applications of random walks are quite popular in the field of mathematics, probability and computer science. Random walk related models can be used in different areas such as prediction, recommendation algorithm to recent supervised learning and networks. It is noticeable that there are few reviews about randoms for the beginners and how random walks are used nowadays in distinctive areas. Hence, the aim of the article is to provide a brief review of classical random walks, including basic concepts and models of the algorithm and then some applications in the field of computer science for the beginners to understand the significance and future of random walks.

Localization of two dimensional quantum walks defined by generalized Grover coins

Localization phenomena of quantum walks makes the propagation dynamics of a walker strikingly different from that corresponding to classical random walks. In this paper, we study the localization phenomena of four-state discrete-time quantum walks on two-dimensional lattices with coin operators as one-parameter orthogonal matrices that are also permutative, a combinatorial structure of the Grover matrix. We show that the proposed walks localize at its initial position for canonical initial coin states when the coin belongs to classes which contain the Grover matrix that we consider in this paper, however, the localization phenomena depends on the coin parameter when the class of parametric coins does not contain the Grover matrix.

The Application of Graph Embedding Based on Random Walk

In the historical process of scientific development, computers have a lofty position, and in recent years, graph embedding algorithms and models are one of the most popular subjects. A large number of similar data structures are indistinguishable by humans, but graph embedding can quickly compare and analyze these data structures. Existing research on random walk-based graph embedding methods is very rich. In order to summarize and classify the status quo of the more mature classical models and compare and integrate them, many different classical models are discussed in this paper. Based on different models, the problems solved, algorithm ideas, strategies, advantages, and disadvantages of the models are discussed in detail, and the application performance of some models is evaluated. DeepWalk model, Node2Vec model, HARP model are three graph embedding models based on the classical random walk model. Calculations for different data can occur by generating different node sequences. The three most important models in attribute random walk models are TriDNR model, GraphRNA model and FEATHER model. The model that only targets the information data in the shallow network is no longer suitable for the rapidly developing network. Attribute random walk models can handle data in deeper networks. At the end of this paper, the full text is summarized and the future prospect of this field is made.

Directivity of Quantum Walk via Its Random Walk Replica

Quantum walks (QWs) exhibit different properties compared with classical random walks (RWs), most notably by linear spreading and localization. In the meantime, random walks that replicate quantum walks, which we refer to as quantum-walk-replicating random walks (QWRWs), have been studied in the literature where the eventual properties of QWRW coincide with those of QWs. However, we consider that the unique attributes of QWRWs have not been fully utilized in the former studies to obtain deeper or new insights into QWs. In this paper, we highlight the directivity of one-dimensional discrete quantum walks via QWRWs. By exploiting the fact that QWRW allows trajectories of individual walkers to be considered, we first discuss the determination of future directions of QWRWs, through which the effect of linear spreading and localization is manifested in another way. Furthermore, the transition probabilities of QWRWs can also be visualized and show a highly complex shape, representing QWs in a novel way. Moreover, we discuss the first return time to the origin between RWs and QWs, which is made possible via the notion of QWRWs. We observe that the first return time statistics of QWs are quite different from RWs, caused by both the linear spreading and localization properties of QWs.

The influence of a transport process on the epidemic threshold

By generating transient encounters between individuals beyond their immediate social environment, transport can have a profound impact on the spreading of an epidemic. In this work, we consider epidemic dynamics in the presence of the transport process that gives rise to a multiplex network model. In addition to a static layer, the (multiplex) epidemic network consists of a second dynamic layer in which any two individuals are connected for the time they occupy the same site during a random walk they perform on a separate transport network. We develop a mean-field description of the stochastic network model and study the influence the transport process has on the epidemic threshold. We show that any transport process generally lowers the epidemic threshold because of the additional connections it generates. In contrast, considering also random walks of fractional order that in some sense are a more realistic model of human mobility, we find that these non-local transport dynamics raise the epidemic threshold in comparison to a classical local random walk. We also test our model on a realistic transport network (the Munich U-Bahn network), and carefully compare mean-field solutions with stochastic trajectories in a range of scenarios.

Quadratic Speedup for Spatial Search by Continuous-Time Quantum Walk.

Continuous-time quantum walks provide a natural framework to tackle the fundamental problem of finding a node among a set of marked nodes in a graph, known as spatial search. Whether spatial search by continuous-time quantum walk provides a quadratic advantage over classical random walks has been an outstanding problem. Thus far, this advantage is obtained only for specific graphs or when a single node of the underlying graph is marked. In this Letter, we provide a new continuous-time quantum walk search algorithm that completely resolves this: our algorithm can find a marked node in any graph with any number of marked nodes, in a time that is quadratically faster than classical random walks. The overall algorithm is quite simple, requiring time evolution of the quantum walk Hamiltonian followed by a projective measurement. A key component of our algorithm is a purely analogue procedure to perform operations on a state of the form e^{-tH^{2}}|ψ⟩, which, for a given Hamiltonian H, only requires evolving H for time scaling as sqrt[t]. This allows us to quadratically fast-forward the dynamics of a continuous-time classical random walk. The applications of our Letter thus go beyond the realm of quantum walks and can lead to new analog quantum algorithms for preparing ground states of Hamiltonians or solving optimization problems.

Decoherence and classicalization of continuous-time quantum walks on graphs

We address decoherence and classicalization of continuous-time quantum walks (CTQWs) on graphs. In particular, we investigate three different models of decoherence and employ the quantum-classical (QC) dynamical distance as a figure of merit to assess whether, and to which extent, decoherence classicalizes the CTQW, i.e. turns it into the analogue classical process. We show that the dynamics arising from intrinsic decoherence, i.e. dephasing in the energy basis, do not fully classicalize the walker and partially preserves quantum features. On the other hand, dephasing in the position basis, as described by the Haken–Strobl master equation or by the quantum stochastic walk (QSW) model, asymptotically destroys the quantumness of the walker, making it equivalent to a classical random walk. We also investigate how fast is the classicalization process and observe a larger rate of convergence of the QC-distance to its asymptotic value for intrinsic decoherence and the QSW models, whereas in the Haken–Strobl scenario, larger values of the decoherence rate induce localization of the walker.

Long range random walks and associated geometries on groups of polynomial growth

In the context of countable groups of polynomial volume growth, we consider a large class of random walks that are allowed to take long jumps along multiple subgroups according to power law distributions. For such a random walk, we study the large time behavior of its probability of return at time $n$ in terms of the key parameters describing the driving measure and the structure of the underlying group. We obtain assorted estimates including near-diagonal two-sided estimates and the Holder continuity of the solutions of the associated discrete parabolic difference equation. In each case, these estimates involve the construction of a geometry adapted to the walk.

Random walks with memory profile of the κ-exponential: a first measurement of diffusive regimes

A new class of random walks, Elephant Random Walks (ERW) in the memory type, non-Markovian, was proposed by Schütz and Trimper characterized by diffusive regimes typical of anomalous diffusion. Following this line, we proposed a model of random walks guided by the principle of kinetic interaction (KIP), where, according to KIP, the temporal evolution of the distribution function of identical particles subject to binary collisions leads us to an ever-increasing functional with time, satisfying the statement of the irreversibility of the second law of Thermodynamics. We derived a distribution of the exponential that is discrete. In the limit , the distribution of the -exponential falls into the ordinary exponential distribution, where we found the typical Brownian diffusion. In the upper limit of anti-equilibrium ( ), we found a transition from the ordinary diffusive regime to the super-diffusive regime. Additionally, moving away from the anti-equilibrium region, we found the ERW’s diffusive behavior. Our results are yet another case of anomalous transport in complex systems, which are associated with particle diffusion for which the variance spreads non-linearly with time.

A Novel EPANET Integration for the Diffusive–Dispersive Transport of Contaminants

The EPANET model is commonly used to model hydraulic behaviour and water quality within water distribution networks. The standard version of the model solves the advective transport equation by solving a mass balance of the fundamental plug flow substance that considers the advective transport and kinetic reaction processes. Over the years, several versions of the model have been developed, which have made it possible to improve the modelling of water quality through the introduction of additional terms within the transport equation to solve the problem of dispersive transport (EPANET-AZRED) and to consider multiple interacting species in the mass flow and on the pipe walls (EPANET multi-species extension). The present study proposes a novel integration of the EPANET-DD (dynamic-dispersion) model, which enables the advective–diffusive–dispersive transport equation in dynamic flow conditions to be solved in the two-dimensional case, through the classical random walk method, implementing the diffusion and dispersion equations proposed by Romero-Gomez and Choi (2011). The model was applied to the University of Enna “KORE” laboratory network to verify its effectiveness in modelling diffusive–dispersive transport mechanisms in the presence of variable flow regimes. The results showed that the EPANET-DD model could better represent the actual data than previously developed versions of the EPANET model.