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Modern Physics Letters B

Quantum walks on N-cycles with an absorbing wall

Qi Han, Ning Bai, Huan Wang, Shuai Wang, Lijie Gou

Published Jul 5, 2023• 0 citations
Publisher

Abstract

In this paper, we analyze quantum walks on cycles with an absorbing wall. We set the absorbing wall on cycles with N vertices (where N is an even number), and divide [Formula: see text] into two parts, [Formula: see text] and [Formula: see text]. Due to the periodicity of the cycles, the condition [Formula: see text] (or [Formula: see text]) is applied to [Formula: see text] and [Formula: see text], then the transmission probability [Formula: see text] and reflection probability [Formula: see text] at the absorbing wall [Formula: see text] at time t are obtained. Furthermore, we show that over time, the absorbing wall absorbs less and less.

DOI
10.1142/s0217984923501385

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In this paper, we analyze quantum walks on cycles with an absorbing wall. We set the absorbing wall on cycles with N vertices (where N is an even number), and divide [Formula: see text] into two parts, [Formula: see text] and [Formula: see text]. Due to the periodicity of the cycles, the condition [Formula: see text] (or [Formula: see text]) is applied to [Formula: see text] and [Formula: see text], then the transmission probability [Formula: see text] and reflection probability [Formula: see text] at the absorbing wall [Formula: see text] at time t are obtained. Furthermore, we show that over time, the absorbing wall absorbs less and less.
Quantum walks are interesting models with potential applications to quantum algorithms and physical processes such as photosynthesis. In this paper, we study two models of one-dimensional quantum walks, namely, quantum walks with a moving absorbing wall and quantum walks with one stationary and one moving absorbing wall. For the former, we calculate numerically the survival probability, the rate of change of average position, and the rate of change of standard deviation of the particle's position in the long time limit for different wall velocities. Moreover, we also study the asymptotic behavior and the dependence of the survival probability on the initial particle's state. While for the latter, we compute the absorption probability of the right stationary wall for different velocities and initial positions of the left wall boundary. The results for these two models are compared with those obtained for the classical model. The difference between the results obtained for the quantum and classical models can be attributed to the difference in the probability distributions.
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 probability computations. In this paper, we shift our sights from the local phenomena of absorption probabilities to the global behavior of finite absorbing quantum walks in one dimension. We conduct our analysis by approximating the eigenbasis of the associated absorbing quantum walk operator matrix Qn{Q}_{n}Qn​, where nnn is the lattice size. The conditional probability distributions of these finite absorbing quantum walks exhibit distinct behavior at various timescales, namely wavelike reflections for times t=O(n)t=O(n)t=O(n), fractional quantum revivals for t=O(n2)t=O({n}^{2})t=O(n2), and stability for t=O(n3)t=O({n}^{3})t=O(n3). At the end of this paper, we demonstrate the existence of fractional quantum revivals in other sufficiently regular quantum walk systems.
In this paper, we successfully obtain an explicit expression of the limit distribution [Formula: see text] of three-state quantum walks on cycles, the total variation distance between [Formula: see text] and the average probability [Formula: see text], and lower bound on the difference between two eigenvalues, among others. Based on the above conclusions, we finally get the mixing time [Formula: see text] of the quantum walk of Grover coin on the N-cycle. [Formula: see text] is the time required to characterize [Formula: see text] approaching [Formula: see text]. Our results show that the average probability of a three-state quantum walk on a cycle can approach its limit distribution faster than that of a two-state quantum walk, which might be of significance to quantum computation.
The quantum walk is a unitary analogue to the discrete random walk, and its properties have been increasingly studied since the turn of the millennium. In comparison with the classical random walk, the quantum walk exhibits linear spreading and initial condition dependent asymmetries. As noted early on in the conjecture and subsequent calculation of absorption probabilities in the one dimensional Hadamard walk, the interaction of the quantum walk with an absorbing boundary is fundamentally divergent from classical case. Here, we will survey absorption probabilities for a more general collection of one dimensional quantum walks and extend the method to consider d-dimensional walks in the presence of d-1 dimensional absorbing walls. However, these results are concerned only with local behavior at the boundary in the form of absorption probabilities. The main results of this thesis are concerned with the global behavior of finite quantum walks, which can be described by linear spreading in the short term, modal phenomena in the mid term, and stable distributions in the exceedingly long term. These theorems will be rigorously proved in the one-dimensional case and extrapolated to higher dimensional quantum walks. To this end we introduce QWSim, a new and robust computational engine for displaying finite two dimensional quantum walks.
Abstract. The adjacency matrix of a graph [Formula: see text] is the Hamiltonian for a continuous-time quantum walk on the vertices of [Formula: see text]. Although the entries of the adjacency matrix are integers, its eigenvalues are generally irrational and, because of this, the behavior of the walk is typically not periodic. In this paper, we develop a theory to exactly study any quantum walk generated by an integral Hamiltonian, and we put emphasis on those with irrational eigenvalues—what we call irrational quantum walks. As a result, we provide exact methods to compute the average of the mixing matrices, and to decide whether pretty good (or almost perfect) state transfer occurs in a given graph. We also use our methods to study geometric properties of beautiful curves arising from entries of the quantum walk matrix and discuss possible applications of these results. Throughout the paper, we emphasize the interplay between different fields of mathematics applied to the study of quantum walks.