Walk/Zeta Correspondence for quantum and correlated random walks (original) (raw)

The Fourier and Grover walks on the two-dimensional lattice and torus

arXiv: Quantum Physics, 2018

In this paper, we consider discrete-time quantum walks with moving shift (MS) and flip-flop shift (FF) on two-dimensional lattice mathbbZ2\mathbb{Z}^2mathbbZ2 and torus piN2=(mathbbZ/N)2\pi_N^2=(\mathbb{Z}/N)^2piN2=(mathbbZ/N)2. Weak limit theorems for the Grover walks on mathbbZ2\mathbb{Z}^2mathbbZ2 with MS and FF were given by Watabe et al. and Higuchi et al., respectively. The existence of localization of the Grover walks on mathbbZ2\mathbb{Z}^2mathbbZ2 with MS and FF was shown by Inui et al. and Higuchi et al., respectively. Non-existence of localization of the Fourier walk with MS on mathbbZ2\mathbb{Z}^2mathbbZ2 was proved by Komatsu and Tate. Here our simple argument gave non-existence of localization of the Fourier walk with both MS and FF. Moreover we calculate eigenvalues and the corresponding eigenvectors of the (k1,k2)(k_1,k_2)(k1,k2)-space of the Fourier walks on piN2\pi_N^2piN2 with MS and FF for some special initial conditions. The probability distributions are also obtained. Finally, we compute amplitudes of the Grover and Fourier walks on pi22\pi_2^2pi22.

Recurrence properties of unbiased coined quantum walks on infinite d-dimensional lattices

Physical Review A, 2008

The Pólya number characterizes the recurrence of a random walk. We apply the generalization of this concept to quantum walks [M.Štefaňák, I. Jex and T. Kiss, Phys. Rev. Lett. 100, 020501 (2008)] which is based on a specific measurement scheme. The Pólya number of a quantum walk depends in general on the choice of the coin and the initial coin state, in contrast to classical random walks where the lattice dimension uniquely determines it. We analyze several examples to depict the variety of possible recurrence properties. First, we show that for the class of quantum walks driven by tensor-product coins, the Pólya number is independent of the initial conditions and the actual coin operators, thus resembling the property of the classical walks. We provide an estimation of the Pólya number for this class of quantum walks. Second, we examine the 2-D Grover walk, which exhibits localisation and thus is recurrent, except for a particular initial state for which the walk is transient. We generalize the Grover walk to show that one can construct in arbitrary dimensions a quantum walk which is recurrent. This is in great contrast with classical walks which are recurrent only for the dimensions d = 1, 2. Finally, we analyze the recurrence of the 2-D Fourier walk. This quantum walk is recurrent except for a two-dimensional subspace of the initial states. We provide an estimation of the Pólya number in its dependence on the initial state.

On Some Questions of C. Ampadu Associated with the Quantum Random Walk

Applied Mathematics, 2014

We review (not exhaustively) the quantum random walk on the line in various settings, and propose some questions that we believe have not been tackled in the literature. In a sense, this article invites the readers (beginner, intermediate, or advanced), to explore the beautiful area of quantum random walks.

Spectral and asymptotic properties of Grover walks on crystal lattices

Journal of Functional Analysis, 2014

We propose a twisted Szegedy walk for estimating the limit behavior of a discrete-time quantum walk on a crystal lattice, an infinite abelian covering graph, whose notion was introduced by [14]. First, we show that the spectrum of the twisted Szegedy walk on the quotient graph can be expressed by mapping the spectrum of a twisted random walk onto the unit circle. Secondly, we show that the spatial Fourier transform of the twisted Szegedy walk on a finite graph with appropriate parameters becomes the Grover walk on its infinite abelian covering graph. Finally, as an application, we show that if the Betti number of the quotient graph is strictly greater than one, then localization is ensured with some appropriated initial state. We also compute the limit density function for the Grover walk on Z d with flip flop shift, which implies the coexistence of linear spreading and localization. We partially obtain the abstractive shape of the limit density function: the support is within the d-dimensional sphere of radius 1/ √ d, and 2 d singular points reside on the sphere's surface.

Quantum Statistics of Random Walks

Journal of Modern Physics, 2018

The paper dealt with quantum canonical ensembles by random walks, where state transitions are triggered by the connections between labels, not by elements, which are transferred. The balance conditions of such walks lead to emission rates of the labels. The labels with emission rates definitely lower than 1 are like modes. For labels with emission rates very close to 1, the quantum numbers are concentrated around a mean value. As an application I consider the role of the zero label in a quantum gas in equilibrium.

Quantum Random Walk via Classical Random Walk With

2010

In recent years quantum random walks have garnered much interest among quantum information researchers. Part of the reason is the prospect that many hard problems can be solved efficiently by employing algorithms based on quantum random walks, in the same way that classical random walks have played a central role in many hugely successful randomized algorithms. In this paper we introduce a new representation for the quantum random walks via the classical random walk with internal states. This new representation allows for a systematic approach to finding closed form expressions for the n-step distributions for a variety of quantum random walk models, and lends itself naturally to large deviation analysis. As an example, we show how to use the new representation to arrive at the same closed form expression for the Hadamard quantum random walk on a line, previously obtained by others. We assert the proposed method works in the most general settings.

Random walk in generalized quantum theory

Physical Review D, 2005

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 certain condition of "strong positivity", the most general Markovian, translationally invariant "decoherence functional" with nearest neighbor transitions.

An analytic solution for one-dimensional quantum walks

arXiv (Cornell University), 2007

The first general analytic solutions for the one-dimensional walk in position and momentum space are derived. These solutions reveal, among other things, new symmetry features of quantum walk probability densities and further insight into the behaviour of their moments. The analytic expressions for the quantum walk probability distributions provide a means of modelling quantum phenomena that is analogous to that provided by random walks in the classical domain.