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arXiv:2104.09104

Time-inhomogeneous Quantum Walks with Decoherence on Discrete Infinite Spaces

Chia-Han Chou, Wei-Shih Yang

Published Apr 19, 2021• 0 citations
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Abstract

In quantum computation theory, quantum random walks have been utilized by many quantum search algorithms which provide improved performance over their classical counterparts. However, due to the importance of the quantum decoherence phenomenon, decoherent quantum walks and their applications have been studied on a wide variety of structures. Recently, a unified time-inhomogeneous coin-turning random walk with rescaled limiting distributions, Bernoulli, uniform, arcsine and semi-circle laws as parameter varies have been obtained. In this paper we study the quantum analogue of these models. We obtained a representation theorem for time-inhomogeneous quantum walk on discrete infinite state space. Additionally, the convergence of the distributions of the decoherent quantum walks are numerically estimated as an application of the representation theorem, and the convergence in distribution of the quantum analogues of Bernoulli, uniform, arcsine and semicircle laws are statistically analyzed.

DOI
10.48550/arxiv.2104.09104

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In this paper, we study the discrete-time quantum random walks on a line subject to decoherence. The convergence of the rescaled position probability distribution p(x,t)p(x,t)p(x,t) depends mainly on the spectrum of the superoperator Lkk{\mathcal{L}}_{kk}Lkk​. We show that if 1 is an eigenvalue of the superoperator with multiplicity one and there is no other eigenvalue whose modulus equals 1, then $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{P}(\frac{\ensuremath{\nu}}{\sqrt{t}},t)$ converges to a convex combination of normal distributions. In terms of position space, the rescaled probability mass function ${p}_{t}(x,t)\ensuremath{\equiv}p(\sqrt{t}x,t)$, $x\ensuremath{\in}Z/\sqrt{t}$, converges in distribution to a continuous convex combination of normal distributions. We give a necessary and sufficient condition for a U(2) decoherent quantum walk that satisfies the eigenvalue conditions. We also give a complete description of the behavior of quantum walks whose eigenvalues do not satisfy these assumptions. Specific examples such as the Hadamard walk and walks under real and complex rotations are illustrated. For the O(2) quantum random walks, an explicit formula is provided for the scaling limit of p(x,t)p(x,t)p(x,t) and their moments. We also obtain exact critical exponents for their moments at the critical point and show universality classes with respect to these critical exponents.
The evolution equation of the Space-time inhomogeneous quantum random walk in terms of quantum Bernoulli noise not only depends on the change of time, but also on the spatial position of the walker.The model is defined on the space of square summable functions.The canonical isomorphism relationship also can deduce the representation of the model in the tensor space.Mathematical induction gives the probability distribution of the quantum random walk in the general initial state and find that it has the same limit distribution as the classical random walk.
The quantum random walk has been much studied recently, largely due to its highly nonclassical behavior. In this paper, we study one possible route to classical behavior for the discrete quantum walk on the line: the presence of decoherence in the quantum ``coin'' which drives the walk. We find exact analytical expressions for the time dependence of the first two moments of position, and show that in the long-time limit the variance grows linearly with time, unlike the unitary walk. We compare this to the results of direct numerical simulation, and see how the form of the position distribution changes from the unitary to the usual classical result as we increase the strength of the decoherence.