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0 CONTINUOUS-TIME QUANTUM WALK ON INTEGER LATTICES AND HOMOGENEOUS TREES

Vladislav Kargin

Published Aug 14, 2016• 4 citations
OpenAlex

Abstract

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 ℤ, and for nearest-neighbor walks on ℤ d and infinite homogeneous trees. In addition, we compute the asymptotic approximation for the probability of the return to zero at time t in all these cases.

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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 ℤ, and for nearest-neighbor walks on ℤ d and infinite homogeneous trees. In addition, we compute the asymptotic approximation for the probability of the return to zero at time t in all these cases.
ABSTRACT In this paper, we explore continuous‐time quantum walk on homogeneous tree. By applying the stratification method, we treat vertices in the same stratum of a homogeneous tree as a single vertex and obtain the probability amplitudes for different strata in finite homogeneous tree parts. In particular, when (where denotes the number of strata), we derive the probability amplitudes for each stratum of the homogeneous tree, considering the homogeneous tree's transition from a finite to infinite structure.
We consider a discrete-time quantum walk Wt given by the Grover transformation on the homogeneous tree. We reduce Wt to a quantum walk Xt on a half line with a wall at the origin. This paper presents two types of limit theorems for Xt. The first one is Xt as t→∞, which corresponds to a localization in the case of an initial qubit state. The second one is Xt⁄t as t→∞, whose limit density is given by the Konno density function [1–4]. The density appears in various situations of discrete-time cases. The corresponding similar limit theorem was proved in [5] for a continuous-time case on the homogeneous tree.
We formulate a framework for discrete-time quantum walks, motivated by classical random walks with memory. We present a specific representation of the classical walk with memory 2, on which this is based. The framework has no need for coin spaces, it imposes no constraints on the evolution operator other than unitarity, and is unifying of other approaches. As an example we construct a symmetric discrete-time quantum walk on the semi-infinite binary tree. The generating function of the amplitude at the root is computed in closed form, as a function of time and the initial level nnn in the tree, and we find the asymptotic and a full numerical solution for the amplitude. It exhibits a sharp interference peak and a power-law tail, as opposed to the exponentially decaying tail of a broadly peaked distribution of the classical symmetric random walk on a binary tree. The probability peak is orders of magnitude larger than it is for the classical walk (already at small nnn). The quantum walk shows a polynomial algorithmic speedup in nnn over the classical walk, which we conjecture to be of the order 2/32/32/3, based on strong trends in data.
We analyze several families of one and two-dimensional nearest neighbor Quantum Random Walks. Using a multivariate generating function analysis we give a simplified proof of a known phenomenon for two-chirality walks on the line, namely that the walk has linear speed rather than the diffusive behavior observed in classical random walks. We also demonstrate Airy phenomena between the regions of polynomial and exponential decay. For a three-chirality walk on the line we demonstrate similar behavior, with the addition of a bound state, in which the probability of finding the particle at the origin does not go to zero with time. For each of these walks on the line we obtain exact formulae for the leading asymptotic term of the wave function and the location probabilities. Analyzing two-dimensional walks we again find a region of polynomial decay which grows linearly with time. The limiting shape of the feasible region is, however, quite different. The limit region turns out to be an algebraic set, which we characterize as the rational image of a compact algebraic variety. We also compute the probability profile within the limit region, which is essentially a negative power of the Gaussian curvature of the same algebraic variety. We close with preliminary work concerning walks in higher dimensions.
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+ for any > 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.