From classical to quantum walks with stochastic resetting on networks.

Quantum biology seeks to explain biological phenomena via quantum mechanisms, such as enzyme reaction rates via tunnelling and photosynthesis energy efficiency via coherent superposition of states. However, less effort has been devoted to study the role of quantum mechanisms in biological evolution. In this paper, we used transcription factor networks with two and four different phenotypes, and used classical random walks (CRW) and quantum walks (QW) to compare network search behaviour and efficiency at finding novel phenotypes between CRW and QW. In the network with two phenotypes, at temporal scales comparable to decoherence time TD, QW are as efficient as CRW at finding new phenotypes. In the case of the network with four phenotypes, the QW had a higher probability of mutating to a novel phenotype than the CRW, regardless of the number of mutational steps (i.e. 1, 2 or 3) away from the new phenotype. Before quantum decoherence, the QW probabilities become higher turning the QW effectively more efficient than CRW at finding novel phenotypes under different starting conditions. Thus, our results warrant further exploration of the QW under more realistic network scenarios (i.e. larger genotype networks) in both closed and open systems (e.g. by considering Lindblad terms).

Discrete Randomness in Discrete Time Quantum Walk: Study Via Stochastic Averaging

A quantum random walk optimization model and algorithm in network cluster server traffic control and task scheduling is proposed. In order to solve the problem of server load balancing, we research and discuss the distribution theory of energy field in quantum mechanics and apply it to data clustering. We introduce the method of random walk and illuminate what the quantum random walk is. Here, we mainly research the standard model of one-dimensional quantum random walk. For the data clustering problem of high dimensional space, we can decompose onem-dimensional quantum random walk intomone-dimensional quantum random walk. In the end of the paper, we compare the quantum random walk optimization method with GA (genetic algorithm), ACO (ant colony optimization), and SAA (simulated annealing algorithm). In the same time, we prove its validity and rationality by the experiment of analog and simulation.

We consider general quantum random walks in a d-dimensional half-space. We first obtain a path integral formula for general quantum random walks in a d-dimensional space. Our path integral formula is valid for general quantum random walks on Cayley graphs as well. Then the path integral formula is applied to obtain the scaling limit of the exit distribution, the expectation of exit time and the asymptotic behaviour of the exit probabilities, for general quantum random walks in a half-space under some conditions on amplitude functions. The conditions are shown to be satisfied by both the Hadamard and Grover quantum random walks in two-dimensional half-spaces. For the two-dimensional case, we show that the critical exponent for the scaling limit of the hitting distribution is 1 as the lattice spacing tends to zero, i.e. the natural magnitude of the hitting position is of order O(1) if the lattice spacing is set to be 1/n. We also show that the rate of convergence of the total hitting probability has lower bound n−2 and upper bound n−2+ϵ for any ϵ > 0. For a quantum random walk with a fixed starting point, we show that the probability of hitting times at the hyperplane decays faster than that of the classical random walk. In both one and two dimensions, given the event of a hit, the conditional expectation of hitting times is finite, in contrast to being infinite for the classical case. In the one-dimensional case, we also obtain an exact order of the probability distribution of the hitting time at 0.

By generalizing a quantum walk to include measurement of the path taken by the quantum walker, we show that quantum walks interpolate between a fully coherent quantum dynamics and a classical random walk, depending on how strongly the walker’s coin state is measured; i.e., the quantum walk exhibits the quintessentially quantum property of complementarity.

We analyze a Markovian random walk strategy on undirected regular networks involving power matrix functions of the type where L indicates a ‘simple’ Laplacian matrix. We refer to such walks as ‘fractional random walks’ with admissible interval . We deduce probability-generating functions (network Green’s functions) for the fractional random walk. From these analytical results we establish a generalization of Polya’s recurrence theorem for fractional random walks on d-dimensional infinite lattices: The fractional random walk is transient for dimensions (recurrent for ) of the lattice. As a consequence, for the fractional random walk is transient for all lattice dimensions and in the range for dimensions . Finally, for , Polya’s classical recurrence theorem is recovered, namely the walk is transient only for lattice dimensions . The generalization of Polya’s recurrence theorem remains valid for the class of random walks with Lévy flight asymptotics for long-range steps. We also analyze the mean first passage probabilities, mean residence times, mean first passage times and global mean first passage times (Kemeny constant) for the fractional random walk. For an infinite 1D lattice (infinite ring) we obtain for the transient regime closed form expressions for the fractional lattice Green’s function matrix containing the escape and ever passage probabilities. The ever passage probabilities (fractional lattice Green’s functions) in the transient regime fulfil Riesz potential power law decay asymptotic behavior for nodes far from the departure node. The non-locality of the fractional random walk is generated by the non-diagonality of the fractional Laplacian matrix with Lévy-type heavy tailed inverse power law decay for the probability of long-range moves. This non-local and asymptotic behavior of the fractional random walk introduces small-world properties with the emergence of Lévy flights on large (infinite) lattices.

Quantum walk mixing is faster than classical on periodic lattices

Optimal computation with non-unitary quantum walks

Krawtchouk's polynomials occur classically as orthogonal polynomials with respect to the binomial distribution. They may be also expressed in the form of matrices, that emerge as arrays of the values that the polynomials take. The algebraic properties of these matrices provide a very interesting and accessible example in the approach to probability theory known as quantum probability. First it is noted how the Krawtchouk matrices are connected to the classical symmetric Bernoulli random walk. And we show how to derive Krawtchouk matrices in the quantum probability context via tensor powers of the elementary Hadamard matrix. Then connections with the classical situation are shown by calculating expectation values in the quantum case.

We present an overview of two models of quantum random walk. In the first model, the discrete quantum random walk, we present the explicit solution for the recurring amplitude of the quantum random walk on a one-dimensional lattice. We also introduce a new method of solving the problem of random walk in the most general case and use it to derive the hitting amplitude for quantum random walk on the hypercube. The second is a special model based on a local interaction between neighboring spin-1/2 particles on a one-dimensional lattice. We present explicit results for the relevant quantities and obtain an upper bound on the speed of convergence to limiting probability distribution.

We study the diffusive transport of Markovian random walks on arbitrary networks with stochastic resetting to multiple nodes. We deduce analytical expressions for the stationary occupation probability and for the mean and global first passage times. This general approach allows us to characterize the effect of resetting on the capacity of random walk strategies to reach a particular target or to explore the network. Our formalism holds for ergodic random walks and can be implemented from the spectral properties of the random walk without resetting, providing a tool to analyze the efficiency of search strategies with resetting to multiple nodes. We apply the methods developed here to the dynamics with two reset nodes and derive analytical results for normal random walks and Lévy flights on rings. We also explore the effect of resetting to multiple nodes on a comb graph, Lévy flights that visit specific locations in a continuous space, and the Google random walk strategy on regular networks.

We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. The number of steps of the quantum walk is quadratically smaller than the classical hitting time of any reversible random walk P on the graph.Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolation between the walk P and the absorbing walk P′, whose marked states are absorbing. Then our quantum walk is simply the quantum analogue of the interpolation. Contrary to previous approaches, our results remain valid when the random walk P is not state-transitive, and in the presence of multiple marked vertices.As a consequence we make a progress on an open problem related to the spatial search on the 2D-grid.

For network graphs, numerous graph features are intimately linked to eigenvalues of the Laplacian matrix, such as connectivity and diameter. Thus, it is very important to solve eigenvalues of the Laplacian matrix for graphs. Similarly, for higher-order networks, eigenvalues of combinatorial Laplacian matrices are also important for invariants of graphs. However, for large-scale networks, it is difficult to calculate eigenvalues of the Laplacian matrix directly because it is either very difficult to obtain the whole network structure or requires a lot of computing resources. Therefore, this article makes the following contributions. Firstly, this paper proposes a random walk approach for estimating the bounds of the greatest eigenvalues of Laplacian matrices for large-scale networks. Considering the relationship between the spectral moments of the adjacency matrix and the closed paths in the network, we utilize the relationship between the adjacency matrix and the Laplacian matrix to establish the relationship between the Laplacian matrix and the closed paths. Then, we employ equiprobable random walks to sample the large graph to obtain the small graph. Through algebraic topology knowledge, we obtain the bounds of the largest eigenvalue of the Laplacian matrix of the large graph by using Laplacian spectral moments of the small graph. Secondly, for high-order networks, this paper proposes a method based on random walks and graph transformations. The graph transformation we propose mainly converts graphs with second-order simplices into ordinary weighted graphs, thereby transforming the problem of solving the spectral moments of the second-order combined Laplacian matrix into solving the spectral moments of the adjacency matrix. Then, we use the aforementioned random walk method to solve bounds of the greatest eigenvalue of the second-order combinatorial Laplacian matrix. Finally, by comparing the proposed method with existing algorithms in synthetic and real networks, its accuracy and superiority are demonstrated.

We present a detailed circuit implementation of Szegedy's quantization of the Metropolis-Hastings walk. This quantum walk is usually defined with respect to an oracle. We find that a direct implementation of this oracle requires costly arithmetic operations. We thus reformulate the quantum walk, circumventing its implementation altogether by closely following the classical Metropolis-Hastings walk. We also present heuristic quantum algorithms that use the quantum walk in the context of discrete optimization problems and numerically study their performances. Our numerical results indicate polynomial quantum speedups in heuristic settings.

Let be an abstract space and a denumerable (finite or infinite) alphabet. Suppose that is a family of functions such that for all we have and a family of transformations . The pair ((Sa)a, (pa)a) is termed an iterated function system with place dependent probabilities. Such systems can be thought as generalisations of random dynamical systems. As a matter of fact, suppose we start from a given ; we pick then randomly, with probability pa (x), the transformation Sa and evolve to Sa (x). We are interested in the behaviour of the system when the iteration continues indefinitely.Random walks of the above type are omnipresent in both classical and quantum Physics. To give a small sample of occurrences we mention: random walks on the affine group, random walks on Penrose lattices, random walks on partially directed lattices, evolution of density matrices induced by repeated quantum measurements, quantum channels, quantum random walks, etc.In this article, we review some basic properties of such systems and provide with a pathfinder in the extensive bibliography (both on mathematical and physical sides) where the main results have been originally published.