Classical Random Walk Research Articles

Hitting times of quantum and classical random walks in potential spaces

The spatial search problem is an interesting and important problem in computer science and especially the area of algorithms. The objective is a marked site to be found in a finite physical space, that can be modeled as a finite lattice or a graph. Many approaches have been developed to address this problem. Classical random walks and quantum walks are efficient models that address the spatial search problem. Quantum walks is a universal model of quantum computation and can be mapped directly to quantum circuits and consequently executed on quantum computers. Quantum walks utilized for quantum search proved to achieve significantly lower hitting times than their classical counterpart, classical random walks. The evolution space for the quantum walks as well as the classical random walks is up until now a free space. In our approach, we introduce external electrical potentials to the evolution space. We study the evolution of discrete time quantum and classical random walks in such potential spaces and the probability — hitting time on finding marked sites. We considered the differences in applied potential among neighboring sites as weights for the lattice — graph. We introduce these weights to the evolution space as an operator for the discrete time quantum walk and as coin probabilities for the classical random walk. Our results show that quantum walks again, evolve faster in the evolution space with the applied potential. Quantum walks also achieve better probability — hitting time on finding the marked site in the potential space. With the introduction of electrical potentials, quantum walks evolving in potential spaces, can lead to the development of novel quantum algorithms, where input parameters can be introduced as external potentials.

Quantum vs. Classical Algorithms for Solving the Heat Equation

Quantum computers are predicted to outperform classical ones for solving partial differential equations, perhaps exponentially. Here we consider a prototypical PDE—the heat equation in a rectangular region—and compare in detail the complexities of ten classical and quantum algorithms for solving it, in the sense of approximately computing the amount of heat in a given region. We find that, for spatial dimension d ge 2, there is an at most quadratic quantum speedup in terms of the allowable error epsilon using an approach based on applying amplitude estimation to an accelerated classical random walk. However, an alternative approach based on a quantum algorithm for linear equations is never faster than the best classical algorithms.

Wall Polynomials on the Real Line: A Classical Approach to OPRL Khrushchev’s Formula

The standard proof of Khrushchev’s formula for orthogonal polynomials on the unit circle given in Khrushchev (J Approx Theory 108:161–248, 2001, J Approx Theory 116:268–342, 2002) combines ideas from continued fractions and complex analysis, depending heavily on the theory of Wall polynomials. Using operator theoretic tools instead, Khrushchev’s formula has been recently extended to the setting of orthogonal polynomials on the real line in the determinate case (Grünbaum and Velázquez in Adv Math 326:352–464, 2018). This paper develops a theory of Wall polynomials on the real line, which serves as a means to prove Khrushchev’s formula for any sequence of orthogonal polynomials on the real line. This real line version of Khrushchev’s formula is used to rederive the characterization given in Simon (J Approx Theory 126:198–217, 2004) for the weak convergence of p_n^2mathrm{d}mu , where p_n are the orthonormal polynomials with respect to a measure mu supported on a bounded subset of the real line (Theorem 8.1). The generality and simplicity of such a Khrushchev’s formula also permits the analysis of the unbounded case. Among other results, we use this tool to prove that no measure mu supported on an unbounded subset of the real line yields a weakly convergent sequence p_n^2mathrm{d}mu (Corollary 8.10), but there exist instances such that p_n^2mathrm{d}mu becomes vaguely convergent (Example 8.5 and Theorem 8.6). Some other asymptoptic results related to the convergence of p_n^2mathrm{d}mu in the unbounded case are obtained via Khrushchev’s formula (Theorems 8.3, 8.7, 8.8, Proposition 8.4, Corollary 8.9). In the bounded case, we include a simple diagrammatic proof of Khrushchev’s formula on the real line which sheds light on its graph theoretical meaning, linked to Pólya’s recurrence theory for classical random walks.

Robustness of Aharonov-Bohm cages in quantum walks

It was recently shown that Aharonov-Bohm (AB) cages exist for quantum walks (QW) on certain tilings -- such as the diamond chain or the dice (or $\mathcal{T}_3$) lattice -- for a proper choice of coins. In this article, we probe the robustness of these AB cages to various perturbations. When the cages are destroyed, we analyze the leakage mechanism and characterize the resulting dynamics. Quenched disorder typically breaks the cages and leads to an exponential decay of the wavefunction similar to Anderson localization. Dynamical disorder or repeated measurements destroy phase coherence and turn the QW into a classical random walk with diffusive behavior. Combining static and dynamical disorder in a specific way leads to subdiffusion with an anomalous exponent controlled by the quenched disorder distribution. Introducing interaction to a second walker can also break the cages and restore a ballistic motion for a "molecular" bound-state.

Measurement-induced quantum walks.

We investigate a tight-binding quantum walk on a graph. Repeated stroboscopic measurements of the position of the particle yield a measured "trajectory," and a combination of classical and quantum mechanical properties for the walk are observed. We explore the effects of the measurements on the spreading of the packet on a one-dimensional line, showing that except for the Zeno limit, the system converges to Gaussian statistics similarly to a classical random walk. A large deviation analysis and an Edgeworth expansion yield quantum corrections to this normal behavior. We then explore the first passage time to a target state using a generating function method, yielding properties like the quantization of the mean first return time. In particular, we study the effects of certain sampling rates that cause remarkable changes in the behavior in the system, such as divergence of the mean detection time in finite systems and decomposition of the phase space into mutually exclusive regions, an effect that mimics ergodicity breaking, whose origin here is the destructive interference in quantum mechanics. For a quantum walk on a line, we show that in our system the first detection probability decays classically like (time)^{-3/2}. This is dramatically different compared to local measurements, which yield a decay rate of (time)^{-3}, indicating that the exponents of the first passage time depend on the type of measurements used.

Return Probability of Quantum and Correlated Random Walks.

The analysis of the return probability is one of the most essential and fundamental topics in the study of classical random walks. In this paper, we study the return probability of quantum and correlated random walks in the one-dimensional integer lattice by the path counting method. We show that the return probability of both quantum and correlated random walks can be expressed in terms of the Legendre polynomial. Moreover, the generating function of the return probability can be written in terms of elliptic integrals of the first and second kinds for the quantum walk.

Origin of photo-enhanced hysteretic electrical conductance in nanostructured silicon-based heterojunction

Photo-enhanced hysteretic I–V curves have been observed under reverse bias in a p-i-n structure containing electrochemically etched nanostructured silicon (Si) sandwiched between p-Si and n-type a-Si:H layers. These curves have been found to depend on intensity of incident illumination and structural morphology of the nanostructured Si layer. The conductance in trace path is lower than that in retrace path. Charge transport mechanism in this structure has been interpreted using microscopic description of charge trapping and detrapping in the defect states present at the interface of nanocrystalline silicon core and oxide shell in the active layer. An applied voltage dependent probability distribution of trapping and detrapping has been calculated in light of classical random walk problem. The trapping/detrapping of charges leading to development/destruction of potential barriers in the path of charge flow shows an analogy with the river bed deposition/erosion. The rate of trapping has been considered to depend on the empty defect states whereas the rate of detrapping depends on the already filled defects. Moreover, the rate of both trapping and detrapping is expected to depend on the charge flow rate. All these considerations lead the I–V relations for trace and retrace paths in reverse bias fitting nicely with experimental I–V loops. The observed peaks in the voltage dependent dynamic conductance in trace and retrace paths have been explained as a consequence of development and destruction of two barriers in the active layer for electrons and holes separately. Best fit values of the fitting parameters indicates that the trace path is dominated by holes whereas the retrace path is dominated by electronic transport. The difference in mobility of electron and hole leads to different trapping and detrapping rates in the two paths resulting in the observed hysteresis.

Influence of coin symmetry on infinite hitting times in quantum walks

Classical random walks on finite graphs have an underrated property: a walk from any vertex can reach every other vertex in finite time, provided they are connected. Discrete-time quantum walks on finite connected graphs however, can have infinite hitting times. This phenomenon is related to graph symmetry, as previously characterized by the group of direction-preserving graph automorphisms that trivially affect the coin Hilbert space. If a graph is symmetric enough (in a particular sense) then the associated quantum walk unitary will contain eigenvectors that do not overlap a set of target vertices, for any coin flip operator. These eigenvectors span the Infinite Hitting Time (IHT) subspace. Quantum states in the IHT subspace never reach the target vertices, leading to infinite hitting times. However, this is not the whole story: the graph of the 3D cube does not satisfy this symmetry constraint, yet quantum walks on this graph with certain symmetric coins can exhibit infinite hitting times. We study the effect of coin symmetry by analyzing the group of coin-permutation symmetries (CPS): graph automorphisms that act nontrivially on the coin Hilbert space but leave the coin operator invariant. Unitaries using highly symmetric coins with large CPS groups, such as the permutation-invariant Grover coin, are associated with higher probabilities of never arriving, as a result of their larger IHT subspaces.

Induced on-demand revival in coined quantum walks on infinite d -dimensional lattices

The study of recurrences and revivals in quantum systems has attracted a great deal of interest because of its importance in the control of quantum systems and its potential use in developing new technologies. In this work, we introduce a protocol to induce full-state revivals in a huge class of quantum walks on a $d$-dimensional lattice governed by a $c$-dimensional coin system. The protocol requires two repeated interventions in the coin degree of freedom. We also present a characterization of the walks that admits such a protocol. Moreover, we modify the quantity known as P\'olya number, typically used in the study of recurrences in classical random walks and quantum walks, to create a witness of the first revival of the walk.

Modeling Random Exit Selection in Intercity Expressway Traffic with Quantum Walk

In intercity expressway traffic, the multiplicity of available routes leads to randomness in exit selection. Random exit selection by drivers is hard to observe, and thus it is a challenge to model intercity expressway traffic sufficiently. In this paper, we developed a Random Quantum Traffic Model (RQTM), which modeled the stochastic traffic fluctuation caused by random exit selection and the residual regularity fluctuation with the quantum walk and autoregressive moving average model (ARMA), respectively. The RQTM considered the random exit selection of a driver as a quantum stochastic process with a dynamic probability function. A quantum walk was applied to update the probability function, which simulated when and where a driver will leave the expressway. We validated our model with hourly traffic data from seven exits from the Nanjing–Changzhou expressway in eastern China. For the seven exits, the coefficients of determination of the RQTM ranged from 0.5 to 0.85. Compared with the classical random walk and the ARMA model, the coefficients of determination were increased by 21.28% to 104.98%, and the relative mean square error decreased by 11.61% to 32.92%. We conclude that the RQTM provides new potential for modeling traffic dynamics with consideration of unobservable random driver decision making.

Quantum walk on a comb with infinite teeth

We study continuous time quantum walk on a comb with infinite teeth and show that the return probability to the starting point decays with time t as t −1. We analyse the diffusion along the spine and into the teeth and show that the walk can escape into the teeth with a finite probability and goes to infinity along the spine with a finite probability. The walk along the spine and into the teeth behaves qualitatively as a quantum walk on a line. This behaviour is quite different from that of classical random walk on the comb.

Electric‐Circuit Simulation of Quantum Fast Hitting with Exponential Speedup

AbstractThe ultimate goal of developing quantum algorithms and constructing quantum computers is to achieve faster information processing than using current classical computers. Quantum walks are powerful kernels in quantum computing protocols and possess strong capabilities in speeding up various simulation and optimization tasks. One striking example is provided by quantum walkers evolving on unbalanced trees, which demonstrate faster hitting performances than classical random walk. However, direct experimental construction of unbalanced trees to prove quantum advantage with exponential speedup remains a great challenge due to the highly complex arrangements of the structure. This study attempts to simulate quantum algorithm by classical circuit. Inspired by the quantum algorithm, the classical circuit networks are designed and fabricated with unbalanced tree structures. It is then demonstrated, both theoretically and experimentally, that the quantum algorithm for the fast hitting problem can be simulated in the structure. It is shown that the hitting efficiency of electric signals in the circuit networks with unbalanced tree structures is exponentially faster than the corresponding cases of classical random walks. Because classical circuit networks possess good scalability and stability, the results open up a scalable new path toward quantum speedup in complex problems.

An improved algorithm for computing hitting probabilities of quantum walks

Quantum walks are the quantum corresponding version of classical random walks, providing polynomial and even exponential acceleration for some classical difficult problems. In recent years, there have been numerous researches on quantum walks, among which the hitting probabilities problem is another crucial problem. In 2021, Guan et al. have proposed an HHL-based algorithm to calculate the hitting probabilities of quantum walks, which can effectively transform the original problem into an inverse matrix problem, then achieve an acceleration effect on the corresponding classical algorithm under some specific conditions. However, we propose a new algorithm consisting of an improved quantum matrix inversion algorithm with complexity Oκ2log1.5(κ/ε)polylog(n), where κ is the condition number of the matrix, n+1 is the number of positions of the one-dimensional quantum walks, and ε is the expected accuracy of the output state. Compared to the HHL-based algorithm, our improved algorithm achieves an exponential improvement in the dependence on accuracy.

A Highly Secured Image Encryption Scheme using Quantum Walk and Chaos

The use of multimedia data sharing has drastically increased in the past few decades due to the revolutionary improvements in communication technologies such as the 4th generation (4G) and 5th generation (5G) etc. Researchers have proposed many image encryption algorithms based on the classical random walk and chaos theory for sharing an image in a secure way. Instead of the classical random walk, this paper proposes the quantum walk to achieve high image security. Classical random walk exhibits randomness due to the stochastic transitions between states, on the other hand, the quantum walk is more random and achieve randomness due to the superposition, and the interference of the wave functions. The proposed image encryption scheme is evaluated using extensive security metrics such as correlation coefficient, entropy, histogram, time complexity, number of pixels change rate and unified average intensity etc. All experimental results validate the proposed scheme, and it is concluded that the proposed scheme is highly secured, lightweight and computationally efficient. In the proposed scheme, the values of the correlation coefficient, entropy, mean square error (MSE), number of pixels change rate (NPCR), unified average change intensity (UACI) and contrast are 0.0069, 7.9970, 40.39, 99.60%, 33.47 and 10.4542 respectively.

Three dimensional image encryption algorithm based on quantum random walk and multidimensional chaos

With the development of computer network technology, people’s requirements for information security is increasing day by day. However, the classical encryption technology has the defects of small key space and easy crack. The problems of image encryption technology in protecting image information security and private content need solving urgently. As a new type of quantum key generator, quantum random walk has a large key space. Compared with the classical random walk, the computing speed and security are significantly improved. This paper presents a three-dimensional image encryption algorithm that is based on quantum random walk and involves Lorenz and Rossler multidimensional chaos. Firstly, Gaussian pyramid is used to segment the image. Secondly, the Hamming distances of several sub images are calculated by using the random sequence generated by quantum random walk and the random sequence generated by Lorenz chaotic system in multi-dimensional chaos, and then synthesized, and the Euclidean distances between the three RGB channels of the image are calculated. Finally, the sequence value obtained from the remainder of Hamming distance and Euclidean distance, as an initial value is input into the Rossler system in multi-dimensional chaos to generate a random sequence which is used as the key to XOR the RGB channel of the image so as to create an encrypted image. The corresponding decryption scheme is the inverse process of the encryption process. In addition, in terms of transmission security, this paper uses a blind watermark embedding algorithm based on DCT and SVD to embed the watermark information into the encrypted image, so that the receiver can extract the watermark and judge whether the image is damaged by the attack in the transmission process according to the integrity of the watermark information. If it is not attacked maliciously, the image will be decrypted. This operation further improves the protection of image information security.The experimental results show that the peak signal-to-noise ratio of the encrypted image is stable between 7 and 9 and the encryption effect is good, the GVD score is close to 1, the correlation of the encrypted image is uniformly distributed, and the correlation coefficient is close to 0, and the key space is 2<sup>128</sup> in size and the encrypted histogram is evenly distributed, showing a high ability to resist statistical analysis attacks.

An invariance principle for ergodic scale-free random environments

There are many classical random walk in random environment results that apply to ergodic random planar environments. We extend some of these results to random environments in which the length scale varies from place to place, so that the law of the environment is in a certain sense only translation invariant {\em modulo scaling}. For our purposes, an ``environment'' consists of an infinite random planar map embedded in $\mathbb C$, each of whose edges comes with a positive real conductance. Our main result is that under modest constraints (translation invariance modulo scaling together with the finiteness of a type of specific energy) a random walk in this kind of environment converges to Brownian motion modulo time parameterization in the quenched sense. Environments of the type considered here arise naturally in the study of random planar maps and Liouville quantum gravity. In fact, the results of this paper are used in separate works to prove that certain random planar maps (embedded in the plane via the so-called Tutte embedding) have scaling limits given by SLE-decorated Liouville quantum gravity, and also to provide a more explicit construction of Brownian motion on the Brownian map. However, the results of this paper are much more general and can be read independently of that program. One general consequence of our main result is that if a translation invariant (modulo scaling) random embedded planar map and its dual have finite energy per area, then they are close on large scales to a minimal energy embedding (the harmonic embedding). To establish Brownian motion convergence for an {\em infinite} energy embedding, it suffices to show that one can perturb it to make the energy finite.

Unstructured search by random and quantum walk

The task of finding an entry in an unsorted list of $N$ elements famously takes $O(N)$ queries to an oracle for a classical computer and $O(\sqrt{N})$ queries for a quantum computer using Grover's algorithm. Reformulated as a spatial search problem, this corresponds to searching the complete graph, or all-to-all network, for a marked vertex by querying an oracle. In this tutorial, we derive how discrete- and continuous-time (classical) random walks and quantum walks solve this problem in a thorough and pedagogical manner, providing an accessible introduction to how random and quantum walks can be used to search spatial regions. Some of the results are already known, but many are new. For large $N$, the random walks converge to the same evolution, both taking $N \ln(1/\epsilon)$ time to reach a success probability of $1-\epsilon$. In contrast, the discrete-time quantum walk asymptotically takes $\pi\sqrt{N}/2\sqrt{2}$ timesteps to reach a success probability of $1/2$, while the continuous-time quantum walk takes $\pi\sqrt{N}/2$ time to reach a success probability of $1$.

Research progress of integrated optical quantum computing

<sec>Quantum computing, based on the inherent superposition and entanglement properties of quantum states, can break through the limits of classical computing power. However, under the present technical conditions, the number of qubits that can be manipulated is still limited. In addition, the preparation of high-precision quantum gates and additional quantum error correction systems requires more auxiliary bits, which leads to extra cost. Therefore, it seems to be a long-term goal to realize a universal fault-tolerant quantum computer.</sec><sec>The development of analog quantum computing is a transition path that can be used to simulate many-body physics problems. Quantum walk, as the quantum counterpart of classical random walks, is a research hotspot in analog quantum computing. Owing to the unique quantum superposition characteristics, quantum walk exhibits the ballistic transport properties of outward diffusion, so quantum walk provides acceleration in computing power for various algorithms. Based on quantum walk, different computing models are derived to deal with practical physical problems in different fields, such as biology, physics, economics, and computer science.</sec><sec>A large number of technical routes are devoted to the experiments on realizing quantum walk, including optical fiber networks, superconducting systems, nuclear magnetic resonance systems, and trapped ion atom systems. Among these routes, photons are considered as the reliable information carriers in the experiments on quantum walking due to their controllability, long coherence time. and fast speed.</sec><sec>Therefore, in this review, we focus on different quantum walk theories and experimental implementations in optical versions, such as traditional optical platforms, optical fiber platforms, and integrated optical quantum platform. In recent years, the rapid development of integrated optical quantum platforms has driven the experiments on quantum walk to move towards the stage of integration and miniaturization, and at the same time, the experimental scale and the number of qubits have gradually increased.</sec><sec>To this end, we summarize the technological progress of integrated optical quantum computing, including various integrated optical quantum experimental platforms and their applications. Secondly, we specifically discuss the experiment on quantum walk and practical applications based on integrated optical quantum platforms. Finally, we briefly describe other quantum algorithms and corresponding experimental implementations.</sec><sec>These quantum computing schemes provide computational speedups for specific physical problems. In the future, with the further development of integrated optical quantum technology, along with the increase in the number of controllable qubits and the realization of the supporting quantum error correction system, a larger-scale many-body physical system can be constructed to further expand these algorithms and move towards the field of optical quantum computing, a new stage.</sec>

Quantum Walks: Schur Functions Meet Symmetry Protected Topological Phases

This paper uncovers and exploits a link between a central object in harmonic analysis, the so-called Schur functions, and the very hot topic of symmetry protected topological phases of quantum matter. This connection is found in the setting of quantum walks, i.e. quantum analogs of classical random walks. We prove that topological indices classifying symmetry protected topological phases of quantum walks are encoded by matrix Schur functions built out of the walk. This main result of the paper reduces the calculation of these topological indices to a linear algebra problem: calculating symmetry indices of finite-dimensional unitaries obtained by evaluating such matrix Schur functions at the symmetry protected points pm 1. The Schur representation fully covers the complete set of symmetry indices for 1D quantum walks with a group of symmetries realizing any of the symmetry types of the tenfold way. The main advantage of the Schur approach is its validity in the absence of translation invariance, which allows us to go beyond standard Fourier methods, leading to the complete classification of non-translation invariant phases for typical examples.

Infinite randomness with continuously varying critical exponents in the random XYZ spin chain

We study the antiferromagnetic XYZ spin chain with quenched bond randomness, focusing on a critical line between localized Ising magnetic phases. A previous calculation using the spectrum-bifurcation renormalization group, and assuming marginal many-body localization, proposed that critical indices vary continuously. In this work we solve the low-energy physics using an unbiased numerically exact tensor network method named the "rigorous renormalization group." We find a line of fixed points consistent with infinite-randomness phenomenology, with indeed continuously varying critical exponents for average spin correlations. A self-consistent Hartree-Fock-type treatment of the $z$ couplings as interactions added to the free-fermion random XY model captures much of the important physics including the varying exponents; we provide an understanding of this as a result of local correlation induced between the mean-field couplings. We solve the problem of the locally-correlated XY spin chain with arbitrary degree of correlation and provide analytical strong-disorder renormalization group proofs of continuously varying exponents based on an associated classical random walk problem. This is also an example of a line of fixed points with continuously varying exponents in the equivalent disordered free-fermion chain. We argue that this line of fixed points also controls an extended region of the critical interacting XYZ spin chain.