Related papers: Return probability of quantum and correlated rando…
We analyze final-time dependent discrete-time quantum walks in one dimension. We compute asymptotics of the return probability of the quantum walk by a path counting approach. Moreover, we discuss a relation between the quantum walk and the…
Polya showed in his 1921 paper that the generating function of the return probability for a two-dimensional random walk can be written in terms of an elliptic integral. In this paper we present a similar expression for a one-dimensional…
A return probability of random walks is one of the interesting subjects. As it is well known, the return probability strongly depends on the structure of the space where the random waker moves. On the other hand, the return probability of…
Interplay between quantum interference and classical randomness can enhance performance of various quantum information tasks. In the present paper we analyze recurrence phenomena in the discrete-time quantum stochastic walk on a line, which…
Associated to a random walk on $\mathbb{Z}$ and a positive integer $n$, there is a return probability of the random walk returning to the origin after $n$ steps. An interesting question is when the set of return probabilities uniquely…
The study of random walks has increasingly been popular across diverse disciplines such as statistics, mathematics, quantum physics, where they are used to model paths consisting of successive random steps in a mathematical space. A…
There has recently been considerable interest in quantum walks in connection with quantum computing. The walk can be considered as a quantum version of the so-called correlated random walk. We clarify a strong structural similarity between…
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…
The Polya number of a classical random walk on a regular lattice is known to depend solely on the dimension of the lattice. For one and two dimensions it equals one, meaning unit probability to return to the origin. This result is extremely…
Quantum walks are known to have nontrivial interactions with absorbing boundaries. In particular it has been shown that an absorbing boundary in the one dimensional quantum walk partially reflects information, as observed by absorption…
This paper is concerned with the continuous-time quantum walk on Z, Z^d, and infinite homogeneous trees. By using the generating function method, we compute the limit of the average probability distribution for the general isotropic walk on…
We consider a discrete random walk on a diagonal lattice in two and three dimensions and obtain explicit solutions of absorption probabilities and probabilities of return in several domains. In three dimensions we consider both the cube and…
One can view quantum mechanics as a generalization of classical probability theory that provides for pairwise interference among alternatives. Adopting this perspective, we ``quantize'' the classical random walk by finding, subject to a…
In this paper, we study discrete-time quantum walks on one-dimensional lattices. We find that the coherent dynamics depends on the initial states and coin parameters. For infinite size of lattice, we derive an explicit expression for the…
Spherically symmetric random walks in arbitrary dimension $D$ can be described in terms of Gegenbauer (ultraspherical) polynomials. For example, Legendre polynomials can be used to represent the special case of two-dimensional spherically…
We study quantum walk on a ladder with combination of conventional and split-step protocols. The two components of the walk resulting from periodic boundary conditions can be made to have three kinds of probability distributions. Two of…
A random walk is known as a random process which describes a path including a succession of random steps in the mathematical space. It has increasingly been popular in various disciplines such as mathematics and computer science.…
The position density of a "particle" performing a continuous-time quantum walk on the integer lattice, viewed on length scales inversely proportional to the time t, converges (as t tends to infinity) to a probability distribution that…
Classical and quantum walks on some finite paths are introduced. It is shown that these walks have explicit solutions given in terms of exceptional Krawtchouk polynomials and their properties are explored. In particular, fractional revival…
We analyze the recurrence probability (P\'olya number) for d-dimensional unbiased quantum walks. A sufficient condition for a quantum walk to be recurrent is derived. As a by-product we find a simple criterion for localisation of quantum…