Classical Random Walk Research Articles

Imprinting the quantum statistics of photons on free electrons.

The interaction between free electrons and light stands at the base of both classical and quantum physics, with applications in free-electron acceleration, radiation sources, and electron microscopy. Yet to this day, all experiments involving free-electron–light interactions are fully explained by describing the light as a classical wave. We observed quantum statistics effects of photons on free-electron–light interactions. We demonstrate interactions that pass continuously from Poissonian to super-Poissonian and up to thermal statistics, revealing a transition from quantum walk to classical random walk on the free-electron energy ladder. The electron walker serves as the probe in nondestructive quantum detection, measuring the second-order photon-correlation g(2)(0) and higher-orders g(n)(0). Unlike conventional quantum-optical detectors, the electron can perform both quantum weak measurements and projective measurements by evolving into an entangled joint state with the photons. These findings inspire hitherto inaccessible concepts in quantum optics, including free-electron–based ultrafast quantum tomography of light.

Generalized quantum-classical correspondence for random walks on graphs

We introduce a minimal set of physically motivated postulates that the Hamiltonian H of a continuous-time quantum walk should satisfy in order to properly represent the quantum counterpart of the classical random walk on a given graph. We found that these conditions are satisfied by infinitely many quantum Hamiltonians, which provide novel degrees of freedom for quantum enhanced protocols, In particular, the on-site energies, i.e. the diagonal elements of H, and the phases of the off-diagonal elements are unconstrained on the quantum side. The diagonal elements represent a potential energy landscape for the quantum walk, and may be controlled by the interaction with a classical scalar field, whereas, for regular lattices in generic dimension, the off-diagonal phases of H may be tuned by the interaction with a classical gauge field residing on the edges, e.g., the electro-magnetic vector potential for a charged walker.

Variations of the elephant random walk

AbstractIn the classical simple random walk the steps are independent, that is, the walker has no memory. In contrast, in the elephant random walk, which was introduced by Schütz and Trimper [19] in 2004, the next step always depends on the whole path so far. Our main aim is to prove analogous results when the elephant has only a restricted memory, for example remembering only the most remote step(s), the most recent step(s), or both. We also extend the models to cover more general step sizes.

Quantum Speed-Up in Collisional Battery Charging.

We present a collision model for the charging of a quantum battery by identical nonequilibrium qubit units. When the units are prepared in a mixture of energy eigenstates, the energy gain in the battery can be described by a classical random walk, where both average energy and variance grow linearly with time. Conversely, when the qubits contain quantum coherence, interference effects buildup in the battery and lead to a faster spreading of the energy distribution, reminiscent of a quantum random walk. This can be exploited for faster and more efficient charging of a battery initialized in the ground state. Specifically, we show that coherent protocols can yield higher charging power than any possible incoherent strategy, demonstrating a quantum speed-up at the level of a single battery. Finally, we characterize the amount of extractable work from the battery through the notion of ergotropy.

Investigating the quench dynamics of the bound states in a spin-orbital-coupling system using a trapped ion

The quantum walk (QW), as the quantum analog of classical random walk, provides a feasible platform to study the topological phenomenon and non-equilibrium dynamics. Here, we propose a novel scheme to realize the quantum walk with a single trapped ion where the Fock states provides the walk space and zero phonon state $\left|n=0\right\rangle $ serves as its natural boundary. Thus, our scheme offers the unique opportunity to investigate the dynamics of the bound states of the corresponding topological systems. Particularly, the quench dynamics of the bound states can be extensively studied by tuning the bulk parameters and the local boundary operator, which are experimentally accessible. Our proposal not only offers a new approach to exploring the character of the bound states of the topological systems, but also offers a way to determine different phases through the dynamical processes.

Grover/Zeta Correspondence based on the Konno–Sato theorem

Recently the Ihara zeta function for the finite graph was extended to infinite one by Clair and Chinta et al. In this paper, we obtain the same expressions by a different approach from their analytical method. Our new approach is to take a suitable limit of a sequence of finite graphs via the Konno-Sato theorem. This theorem is related to explicit formulas of characteristic polynomials for the evolution matrix of the Grover walk. The walk is one of the most well-investigated quantum walks which are quantum counterpart of classical random walks. We call the relation between the Grover walk and the zeta function based on the Konno-Sato theorem "Grover/Zeta Correspondence" here.

Nonclassical oscillations in pre- and post-selected quantum walks

Quantum walks are counterparts of classical random walks. They spread faster, which can be exploited in information processing tasks, and constitute a versatile simulation platform for many quantum systems. Yet, some of their properties can be emulated with classical light. This rises a question: which aspects of the model are truly nonclassical? We address it by carrying out a photonic experiment based on a pre- and post-selection paradox. The paradox implies that if somebody could choose to ask, either if the particle is at position x = 0 at even time steps, or at position x = d (d > 1) at odd time steps, the answer would be positive, no matter the question asked. Therefore, the particle seems to undergo long distance oscillations despite the fact that the model allows to jump one position at a time. We translate this paradox into a Bell-like inequality and experimentally confirm its violation up to eight standard deviations.

A Gaussian approximation theorem for Lévy processes

Without higher moment assumptions, this note establishes the decay of the Kolmogorov distance in a central limit theorem for Lévy processes. This theorem can be viewed as a continuous-time extension of the classical random walk result by Friedman et al. (1966).

Current Trends in Random Walks on Random Lattices

In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset of the real space. In this case we target the virtual first passage (or escape) time. Thus, unlike standard random walk problems, rather than crossing the boundary, we deal with the walker’s escape location arbitrarily distant from the boundary. In this paper, we give a short historical background on random walk, discuss various directions in the development of random walk theory, and survey most of our results obtained in the last 25–30 years, including the very recent ones dated 2020–21. Among different applications of such random walks, we discuss stock markets, stochastic networks, games, and queueing.

Remarks on the speeds of a class of random walks on the integers

In recent years, there has been an interest in deriving certain important probabilistic results as consequences of deterministic ones; see for instance [2] and [1]. In this work, we continue on this path by deducing a well known equivalence between the speed of random walks on the integers and the growth of the size of their ranges. This result is an immediate consequence of the Kesten-Spitzer-Whitman theorem, and by appearances is probabilistic in nature, but we will show that it follows easily from an elementary deterministic result. We also investigate the common property of recurrent random walks of having speed zero, and show by example that this property need not be shared by deterministic sequences. However, if we consider the inter-arrival times (times at which the sequence is equal to 0) then we find a sufficient deterministic condition for a sequence to have zero speed, and show that this can be used to derive several probabilistic results.

Quantum walks on a programmable two-dimensional 62-qubit superconducting processor.

Quantum walks are the quantum mechanical analog of classical random walks and an extremely powerful tool in quantum simulations, quantum search algorithms, and even for universal quantum computing. In our work, we have designed and fabricated an 8-by-8 two-dimensional square superconducting qubit array composed of 62 functional qubits. We used this device to demonstrate high-fidelity single- and two-particle quantum walks. Furthermore, with the high programmability of the quantum processor, we implemented a Mach-Zehnder interferometer where the quantum walker coherently traverses in two paths before interfering and exiting. By tuning the disorders on the evolution paths, we observed interference fringes with single and double walkers. Our work is a milestone in the field, bringing future larger-scale quantum applications closer to realization for noisy intermediate-scale quantum processors.

A Little Bit of Classical Magic to Achieve (Super-)Quantum Speedup

We introduce nebit, a classical bit with a signed probability distribution. We study its properties and basic transformations that can be applied to it. Then, we introduce a simple dynamical model – a classical random walk supplemented with nebits. We show that such a model exhibits some counterintuitive non-classical properties and that it can achieve or even exceed the speedup of Grover’s quantum search algorithm. The proposed classical dynamics never reveals negativity of nebits and thus we do not need any operational interpretation of negative probabilities. We argue that nebits can be useful as a measure of non-classicality as well as a tool to find new quantum algorithms.

Lévy Gradient Descent: Augmented Random Search for Geophysical Inverse Problems

A new efficient random search algorithm is introduced for solving inversion problems in geophysical studies. The proposed algorithm is inherently a stochastic optimization method which is built on the concept of gradient descending and Levy flights. Therefore, the algorithm is referred to as the Levy gradient descent (L-GD). In which, the Levy flights is a special class of random walk which consists of many short steps along with a few large steps. Such movements are observed in a varying range of fields, including animals’ foraging patterns, fluid dynamics, transport of light and so on. Meanwhile, the Levy flights typically shows much higher speed in searching for sparsely located targets compared to the well-known Brownian walks, which make them preferable to drive random search algorithms. As shown in the paper, besides optimal solutions of the inverse problems, the L-GD algorithm could also produce estimations on the error distributions of the resultant model parameters. Following a detailed introduction of the methodology, parameter settings of the algorithm are discussed in length through statistical experiments. Subsequently, the proposed algorithm is evaluated using numeric tests and shows attracting properties of the global convergence and significant higher searching efficiency compared to commonly adopted stochastic optimization techniques in geophysical inversions. Moreover, the L-GD algorithm is applied to the inversions of gravity and seismic travel time data and has achieved the same accuracy as gradient-based optimization methods. Meanwhile, though error estimations generated by L-GD algorithm are essentially qualitative, they could still provide valuable information to help evaluating the resultant model parameters, which is of great importance for practical geophysical inversions.

Collective migrations in an epithelial–cancerous cell monolayer

Collective cell migration is extensively observed in embryo development and cancer invasion. During these processes, the interactions between cells with distinct identities and fates are of importance for boundary formation and host defense against cancer. In this paper, we explore the collective dynamics of a two-dimensional cell mixing monolayer consisting of non-tumorigenic mammary epithelial cells and breast cancer cells. We show that the epithelial–cancerous cell mixing system displays unique sorting behaviors. The epithelial cells aggregate to form scattered clusters, which perform random motion with simultaneous translation and rotation, strikingly distinct from the classical persistent random walk of individual migratory cells. The motility of cancer cells is markedly promoted by the epithelial clusters, exhibiting remarkable contact enhancement of locomotion. A discrete model based on the Johnson–Kendall–Roberts contact mechanics is proposed to identify the influence of intercellular interactions, active migration forces and cell–substrate friction forces on the collective cell dynamics. These findings could advance our understanding of many biological processes, such as cancer metastasis and tissue morphogenesis. The epithelial–cancerous cell mixing system displays unique collective dynamics. The epithelial cells aggregate to form scattered clusters, which perform random motion with simultaneous translation and rotation, strikingly distinct from the classical persistent random walk of individual migratory cells. The interactions between the two cell types drive the evolution of collective migration dynamics and give rise to a behavior akin to the contact enhancement of locomotion. This study revealed the distinctive dynamic features and the underlying regulatory mechanisms arising from epithelial–cancerous interactions.

A quantum annealing approach to ionic diffusion in solids

We have developed a framework for using quantum annealing computation to evaluate a key quantity in ionic diffusion in solids, the correlation factor. Existing methods can only calculate the correlation factor analytically in the case of physically unrealistic models, making it difficult to relate microstructural information about diffusion path networks obtainable by current ab initio techniques to macroscopic quantities such as diffusion coefficients. We have mapped the problem into a quantum spin system described by the Ising Hamiltonian. By applying our framework in combination with ab initio technique, it is possible to understand how diffusion coefficients are controlled by temperatures, pressures, atomic substitutions, and other factors. We have calculated the correlation factor in a simple case with a known exact result by a variety of computational methods, including simulated quantum annealing on the spin models, the classical random walk, the matrix description, and quantum annealing on D-Wave with hybrid solver . This comparison shows that all the evaluations give consistent results with each other, but that many of the conventional approaches require infeasible computational costs. Quantum annealing is also currently infeasible because of the cost and scarcity of qubits, but we argue that when technological advances alter this situation, quantum annealing will easily outperform all existing methods.

From diffusion in compartmentalized media to non-Gaussian random walks

In this work we establish a link between two different phenomena that were studied in a large and growing number of biological, composite and soft media: the diffusion in compartmentalized environment and the non-Gaussian diffusion that exhibits linear or power-law growth of the mean square displacement joined by the exponential shape of the positional probability density. We explore a microscopic model that gives rise to transient confinement, similar to the one observed for hop-diffusion on top of a cellular membrane. The compartmentalization of the media is achieved by introducing randomly placed, identical barriers. Using this model of a heterogeneous medium we derive a general class of random walks with simple jump rules that are dictated by the geometry of the compartments. Exponential decay of positional probability density is observed and we also quantify the significant decrease of the long time diffusion constant. Our results suggest that the observed exponential decay is a general feature of the transient regime in compartmentalized media.

Discrete-time quantum walk with time-correlated noise

We investigate the dynamics of discrete-time quantum walk subject to time correlated noise. Noise is described as an unitary coin-type operator before each step, and attention is focused on the noise generated by a Gaussian Ornstein Uhlenbeck process, going beyond the usual telegraph noise, where the random variables are consist of only -1 and 1. Under the first-order approximation of BCH formula, the master equation of noisy discrete-time quantum walk is derived. The dynamics given by the master equation are in good agreement with those given by numerical simulations within a certain period of steps, which is controlled by noise parameters. Two remarker behaviors of long time noisy dynamics are observed in numerical simulations, corresponding to two opposite noise regimes: in slow noise regime, with the increase of the noise amplitude, the quantum coherence is suppressed, and the dynamics of noisy discrete-time quantum walk gradually transits to that of classical random walk. In fast noise regime, the walker is confined into few lattice sites, and the width of wave packet is much narrower compared with that in slow noise regime.

A Quantum Walk Model of Financial Options

Financial markets are often modeled using a random walk, for example in the binomial option pricing model which is a discrete version of the Black-Scholes formula. This paper presents an alternative approach to option pricing based on a quantum walk model. The quantum walk, which incorporates superposition states and allows for effects such as interference, was originally developed in physics, but has also seen application in areas such as cognitive psychology, where it is used to model dynamic decision-making processes. It is shown here that the quantum walk model captures key aspects of investor behavior, while the collapsed state captures the observed behavior of markets. The resulting option price model agrees quite closely with the classical random walk model, but helps to explain some observed anomalies. The method also has the advantage that it can be run directly on a quantum computer.

The geometry of random walk isomorphism theorems

The classical random walk isomorphism theorems relate the local times of a continuous-time random walk to the square of a Gaussian free field. A Gaussian free field is a spin system that takes values in Euclidean space, and this article generalises the classical isomorphism theorems to spin systems taking values in hyperbolic and spherical geometries. The corresponding random walks are no longer Markovian: they are the vertex-reinforced and vertex-diminished jump processes. We also investigate supersymmetric versions of these formulas. Our proofs are based on exploiting the continuous symmetries of the corresponding spin systems. The classical isomorphism theorems use the translation symmetry of Euclidean space, while in hyperbolic and spherical geometries the relevant symmetries are Lorentz boosts and rotations, respectively. These very short proofs are new even in the Euclidean case. Isomorphism theorems are useful tools, and to illustrate this we present several applications. These include simple proofs of exponential decay for spin system correlations, exact formulas for the resolvents of the joint processes of random walks together with their local times, and a new derivation of the Sabot–Tarrès formula for the limiting local time of the vertex-reinforced jump process.

From classical to quantum walks with stochastic resetting on networks.

Random walks are fundamental models of stochastic processes with applications in various fields, including physics, biology, and computer science. We study classical and quantum random walks under the influence of stochastic resetting on arbitrary networks. Based on the mathematical formalism of quantum stochastic walks, we provide a framework of classical and quantum walks whose evolution is determined by graph Laplacians. We study the influence of quantum effects on the stationary and long-time average probability distribution by interpolating between the classical and quantum regime. We compare our analytical results on stationary and long-time average probability distributions with numerical simulations on different networks, revealing differences in the way resets affect the sampling properties of classical and quantum walks.