Classical Random Walk Research Articles

Transition from quantum to classical random walk distributions with spin-orbit modes

Transition from quantum to classical random walk distributions with spin-orbit modes

The Global Mean First-Passage Time for Degree-Dependent Random Walks in a Class of Fractal Scale-Free Trees

Fractal scale-free structures are widely observed across a range of natural and synthetic systems, such as biological networks, internet architectures, and social networks, providing broad applications in the management of complex systems and the facilitation of dynamic processes. The global mean first-passage time (GMFPT) for random walks on the underlying networks has attracted significant attention as it serves as an important quantitative indicator that can be used in many different fields, such as reaction kinetics, network transport, random search, pathway elaboration, etc. In this study, we first introduce two degree-dependent random walk strategies where the transition probability is depended on the degree of neighbors. Then, we evaluate analytically the GMFPT of two degree-dependent random walk strategies on fractal scale-free tree structures by exploring the connection between first-passage times in degree-dependent random walk strategies and biased random walks on the weighted network. The exact results of the GMFPT for the two degree-dependent random walk strategies are presented and are compared with the GMFPT of the classical unbiased random walk strategy. Our work not only presents a way to evaluate the GMFPT for degree-dependent biased random walk strategies on general networks but also offers valuable insights to enrich the controlling of chemical reactions, network transport, random search, and pathway elaboration.

Branching random walks with regularly varying perturbations

We consider a modification of classical branching random walk, where we add i.i.d. perturbations to the positions of the particles in each generation. In this model, which was introduced and studied by Bandyopadhyay and Ghosh (2023), perturbations take the form 1/θ log X/E where θ is a positive parameter, X has an arbitrary distribution μ and E is exponential with parameter 1, independent of X. Working under finite mean assumption for μ, they proved almost sure convergence of the rightmost position to a constant limit, and identified the weak centered asymptotics when θ does not exceed certain critical parameter θ_0 . This paper complements their work by providing weak centered asymptotics for the case when θ > θ_0 and presenting the results to μ with regularly varying tails. We prove almost sure convergence of the rightmost position and identify the appropriate centering for the weak convergence, which is of form αn + c log n, with constants α, c depending on the ratio of θ and θ_0. We describe the limiting distribution and provide explicitly the constants appearing in the centering.

Subdiffusive dynamics in photonic random walks probed with classical light.

The random walk of photons in a tight-binding lattice is known to exhibit diffusive motion similar to classical random walks under decoherence, clearly illustrating the quantum-to-classical transition. In this study, we reveal that the random walk of intense classical light under dephasing dynamics can disentangle quantum and ensemble averaging, making it possible to observe subdiffusive walker dynamics, i.e., a behavior very distinct from both a classical and a quantum walker. These findings are demonstrated through proposing photonic random walk in synthetic temporal lattices, based on pulse dynamics in coupled fiber loops.

Absolute Zeta functions and periodicity of quantum walks on cycles

The quantum walk is a quantum counterpart of the classical random walk. On the other hand, absolute zeta functions can be considered as zeta functions over $\mathbf{F}_1$. This study presents a connection between quantum walks and absolute zeta functions. In this paper, we focus on Hadamard walks and $3$-state Grover walks on cycle graphs. The Hadamard walks and the Grover walks are typical models of the quantum walks. We consider the periods and zeta functions of such quantum walks. Moreover, we derive the explicit forms of the absolute zeta functions of corresponding zeta functions. Also, it is shown that our zeta functions of quantum walks are absolute automorphic forms.

Continuous Time Randon Walks with Resetting in a Bounded Chain

The model of classical random walks with Poissonian resetting in a one-dimensional lattice is analyzed in detail in its general version. A special emphasis is made on the resetting effects that emerge due to the variety of arbitrary initial and boundary conditions. A quantum analog of the model is also discussed.

Statistical mechanics of stochastic quantum control: d-adic Rényi circuits.

The dynamics of quantum information in many-body systems with large onsite Hilbert space dimension admits an enlightening description in terms of effective statistical mechanics models. Motivated by this fact, we reveal a connection between three separate models: the classically chaotic d-adic Rényi map with stochastic control, a quantum analog of this map for qudits, and a Potts model on a random graph. The classical model and its quantum analog share a transition between chaotic and controlled phases, driven by a randomly applied control map that attempts to order the system. In the quantum model, the control map necessitates measurements that concurrently drive a phase transition in the entanglement content of the late-time steady state. To explore the interplay of the control and entanglement transitions, we derive an effective Potts model from the quantum model and use it to probe information-theoretic quantities that witness both transitions. The entanglement transition is found to be in the bond-percolation universality class, consistent with other measurement-induced phase transitions, while the control transition is governed by a classical random walk. These two phase transitions can be made to coincide by varying a parameter in the model, producing a picture consistent with behavior observed in previous small-size numerical studies of the quantum model.

Disease gene prioritization with quantum walks.

Disease gene prioritization methods assign scores to genes or proteins according to their likely relevance for a given disease based on a provided set of seed genes. This scoring can be used to find new biologically relevant genes or proteins for many diseases. Although methods based on classical random walks have proven to yield competitive results, quantum walk methods have not been explored to this end. We propose a new algorithm for disease gene prioritization based on continuous-time quantum walks using the adjacency matrix of a protein-protein interaction (PPI) network. We demonstrate the success of our proposed quantum walk method by comparing it to several well-known gene prioritization methods on three disease sets, across seven different PPI networks. In order to compare these methods, we use cross-validation and examine the mean reciprocal ranks of recall and average precision values. We further validate our method by performing an enrichment analysis of the predicted genes for coronary artery disease. The data and code for the methods can be accessed at https://github.com/markgolds/qdgp.

One-dimensional continuous-time quantum Markov chains: qubit probabilities and measures

Quantum Markov chains (QMCs) are positive maps on a trace-class space describing open quantum dynamics on graphs. Such objects have a statistical resemblance with classical random walks, while at the same time they allow for internal (quantum) degrees of freedom. In this work we study continuous-time QMCs on the integer line, half-line and finite segments, so that we are able to obtain exact probability calculations in terms of the associated matrix-valued orthogonal polynomials and measures. The methods employed here are applicable to a wide range of settings, but we will restrict ourselves to classes of examples for which the Lindblad generators are induced by a single positive map, and such that the Stieltjes transforms of the measures and their inverses can be calculated explicitly.

The role of Allee effects for Gaussian and Lévy dispersals in an environmental niche.

In the study of biological populations, the Allee effect detects a critical density below which the population is severely endangered and at risk of extinction. This effect supersedes the classical logistic model, in which low densities are favorable due to lack of competition, and includes situations related to deficit of genetic pools, inbreeding depression, mate limitations, unavailability of collaborative strategies due to lack of conspecifics, etc. The goal of this paper is to provide a detailed mathematical analysis of the Allee effect. After recalling the ordinary differential equation related to the Allee effect, we will consider the situation of a diffusive population. The dispersal of this population is quite general and can include the classical Brownian motion, as well as a Lévy flight pattern, and also a "mixed" situation in which some individuals perform classical random walks and others adopt Lévy flights (which is also a case observed in nature). We study the existence and nonexistence of stationary solutions, which are an indication of the survival chance of a population at the equilibrium. We also analyze the associated evolution problem, in view of monotonicity in time of the total population, energy consideration, and long-time asymptotics. Furthermore, we also consider the case of an "inverse" Allee effect, in which low density populations may access additional benefits.

Instability in the quantum restart problem.

Repeatedly monitored quantum walks with a rate 1/τ yield discrete-time trajectories which are inherently random. With these paths the first-hitting time with sharp restart is studied. We find an instability in the optimal mean hitting time, which is not found in the corresponding classical random-walk process. This instability implies that a small change in parameters can lead to a rather large change of the optimal restart time. We show that the optimal restart time versus τ, as a control parameter, exhibits sets of staircases and plunges. The plunges, are due to the mentioned instability, which in turn is related to the quantum oscillations of the first-hitting time probability, in the absence of restarts. Furthermore, we prove that there are only two patterns of staircase structures, dependent on the parity of the distance between the target and the source in units of lattice constant. The global minimum of the hitting time is controlled not only by the restart time, as in classical problems, but also by the sampling time τ. We provide numerical evidence that this global minimum occurs for the τ minimizing the mean hitting time, given restarts taking place after each measurement. Last, we numerically show that the instability found in this work is relatively robust against stochastic perturbations in the sampling time τ.

Quantum walks as thermalisations, with application to fullerene graphs

We consider to what extent quantum walks can constitute models of thermalisation, analogously to how classical random walks can be models for classical thermalisation. In a quantum walk over a graph, a walker moves in a superposition of node positions via a unitary time evolution. We show a quantum walk can be interpreted as an equilibration of a kind investigated in the literature on thermalisation in unitarily evolving quantum systems. This connection implies that recent results concerning the equilibration of observables can be applied to analyse the node position statistics of quantum walks. We illustrate this in the case of a family of graphs known as fullerenes. We find that a bound from Short et al., implying that certain expectation values will at most times be close to their time-averaged value, applies tightly to the node position probabilities. Nevertheless, the node position statistics do not thermalise in the standard sense. In particular, quantum walks over fullerene graphs constitute a counter-example to the hypothesis that subsystems equilibrate to the Gibbs state. We also exploit the bridge created to show how quantum walks can be used to probe the universality of the eigenstate thermalisation hypothesis relation. We find that whilst in C60 with a single walker, the eigenstate thermalisation hypothesis relation does not hold for node position projectors, it does hold for the average position, enforced by a symmetry of the Hamiltonian. The findings suggest a unified study of quantum walks and quantum self-thermalisations is natural and feasible.

Photonic random walks with traps.

Random walks (RW) behave very differently for classical and quantum particles. Here we unveil a ubiquitous distinctive behavior of random walks of a photon in a one-dimensional lattice in the presence of a finite number of traps, at which the photon can be destroyed and the walk terminates. While for a classical random walk, the photon is unavoidably destroyed by the traps. For a quantum walk, the photon can remain alive, and the walk continues forever. Such an intriguing behavior is illustrated by considering photonic random walks in synthetic mesh lattices with controllable decoherence, which enables the switch from quantum to classical random walks.

From unbiased to maximal-entropy random walks on hypergraphs.

Random walks have been intensively studied on regular and complex networks, which are used to represent pairwise interactions. Nonetheless, recent works have demonstrated that many real-world processes are better captured by higher-order relationships, which are naturally represented by hypergraphs. Here we study random walks on hypergraphs. Due to the higher-order nature of these mathematical objects, one can define more than one type of walks. In particular, we study the unbiased and the maximal entropy random walk on hypergraphs with two types of steps, emphasizing their similarities and differences. We characterize these dynamic processes by examining their stationary distributions and associated hitting times. To illustrate our findings, we present a toy example and conduct extensive analyses of artificial and real hypergraphs, providing insights into both their structural and dynamical properties. We hope that our findings motivate further research extending the analysis to different classes of random walks as well as to practical applications.

An approach to the Second Quantum Revolution for secondary school students: the case of the random walk algorithm

The teaching of quantum technologies has now become a leading topic and is at the heart of numerous international programs (e.g., quantum flagship, National Quantum Initiative, UK national quantum technology program) with the aim of widening the workforce and preparing the next generations of experts. In the present contribution, we present an approach for teaching the Second Quantum Revolution to secondary school students that we developed in recent years. The approach aims to emphasize the ongoing revolution as first of all a cultural revolution and its intrinsic interdisciplinary character. As an emblematic case of some aspects of the approach, we present an activity on the classical and quantum random walk algorithms whose main aims are to recontextualize some basic quantum concepts (quantum state, state manipulation/evolution, measurement, and entanglement) in the case of the algorithm and to reflect on the main differences between the classical and quantum case in terms of logic behind its functioning as well as from an epistemological perspective.

Search algorithm on strongly regular graph by lackadaisical quantum walk

Quantum walk is a widely used method in designing quantum algorithms. In this article, we consider the lackadaisical discrete-time quantum walk (DTQW) on strongly regular graphs (SRG). When there is a single marked vertex in a SRG, we prove that lackadaisical DTQW can find the marked vertex with asymptotic success probability 1, with a quadratic speedup compared to classical random walk. This paper deals with any parameter family of SRG and argues that, by adding self-loops with proper weights, the asymptotic success probability can reach 1. The running time and asymptotic success probability matches the result of continuous-time quantum walk, and improves the result of DTQW.

Strongly interacting photonic quantum walk using single atom beamsplitters

Photonics provide an efficient way to implement quantum walks, the quantum analog of classical random walks, which demonstrate rich physics with potential applications. However, most photonic quantum walks do not involve photon interactions, which limits their potential to explore strongly correlated many-body physics of light. We propose a strongly interacting discrete-time photonic quantum walk using a network of single atom beamsplitters. We calculate output statistics of the quantum walk for the case of two photons, which reveals the strongly correlated transport of photons. Particularly, the walk can exhibit either bosonlike or fermionlike statistics which is tunable by postselecting the two-photon detection time interval. Also, the walk can sort different types of two-photon bound states into distinct pairs of output ports under certain conditions. These unique phenomena show that our quantum walk is an intriguing platform to explore strongly correlated quantum many-body states of light. Finally, we propose an experimental realization based on time-multiplexed synthetic dimensions. Published by the American Physical Society 2024

Link prediction using extended neighborhood based local random walk in multilayer social networks

One of these challenges in the analysis of social networks is the problem of link prediction. The purpose of this problem is to find links that have not yet been observed, but may exist in the future. There are many solutions for link prediction on monoplex networks. However, many real social networks model communication in multiple layers, which are known as multilayer social networks. A solution for multilayer networks involves taking into account the information of all layers to make predictions for a target layer. Among the existing solutions, local random walk has been confirmed as an efficient technique for link prediction in monoplex networks, but this technique is inefficient for link prediction in multilayer networks due to computational complexity. In order to address this issue, in this paper we propose Extended Neighborhood based Local Random Walk (ENLRW) for link prediction in multilayer networks. ENLRW is an extended version of the classical local random walk technique in which the nearest neighbors are considered based on the extended neighborhood concept. ENLRW calculates the similarity between vertices by integrating several different metrics through reliable paths that include intra-layer and inter-layer information. Besides, ENLRW considers vertex influence as a similarity metric to provide an effective reliable biased random walk. The results of the simulations show that the use of different inter-layer and intra-layer information as well as the local random walk configuration with extended neighborhood provides a trade-off between precision and complexity. Specifically, ENLRW improves the average precision by 3.1% compared to the best available state-of-the-art method.

Resolving degeneracies in Google search via quantum stochastic walks

The internet is one of the most valuable technologies invented to date. Among them, Google is the most widely used search engine. The PageRank algorithm is the backbone of Google search, ranking web pages according to relevance and recency. We employ quantum stochastic walks (QSWs) with the hope of bettering the classical PageRank (CPR) algorithm, which is based on classical continuous time random walks. We implement QSW via two schemes: only incoherence and dephasing with incoherence. PageRank using QSW with only incoherence or QSW with dephasing and incoherence best resolves degeneracies that are unresolvable via CPR and with a convergence time comparable to that for CPR, which is generally the minimum. For some networks, the two QSW schemes obtain a convergence time lower than CPR and an almost degeneracy-free ranking compared to CPR.

Quantum Walk Computing: Theory, Implementation, and Application

The classical random walk formalism plays an important role in a wide range of applications. Its quantum counterpart, the quantum walk, is proposed as an important theoretical model for quantum computing. By exploiting quantum effects such as superposition, interference, and entanglement, quantum walks and their variations have been extensively studied for achieving computing power beyond that of classical computing and have been broadly used in designing quantum algorithms for algebraic and optimization problems, graph and network analysis, and quantum Hamiltonian and biochemical process simulations. Moreover, quantum walk models have been proven capable of universal quantum computation. Unlike conventional quantum circuit models, quantum walks provide a feasible path for implementing application-specific quantum computing, particularly in the noisy intermediate-scale quantum era. Recently, remarkable progress has been achieved in implementing a wide variety of quantum walks and quantum walk applications, which demonstrates the great potential of quantum walks. In this review, we provide a thorough summary of quantum walks and quantum walk computing, including theories and characteristics, physical implementations, and applications. We also discuss the challenges facing quantum walk computing, which aims to realize a practical quantum computer in the near future.