Correlated Markov Quantum Walks

We consider the discrete time unitary dynamics given by a quantum walk on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}^d}$$\end{document} performed by a particle with internal degree of freedom, called coin state, according to the following iterated rule: a unitary update of the coin state takes place, followed by a shift on the lattice, conditioned on the coin state of the particle. We study the large time behavior of the quantum mechanical probability distribution of the position observable in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}^d}$$\end{document} for random updates of the coin states of the following form. The random sequences of unitary updates are given by a site-dependent function of a Markov chain in time, with the following properties: on each site, they share the same stationary Markovian distribution and, for each fixed time, they form a deterministic periodic pattern on the lattice. We prove a Feynman–Kac formula to express the characteristic function of the averaged distribution over the randomness at time n in terms of the nth power of an operator M. By analyzing the spectrum of M, we show that this distribution possesses a drift proportional to the time and its centered counterpart displays a diffusive behavior with a diffusion matrix we compute. Moderate and large deviation principles are also proven to hold for the averaged distribution and the limit of the suitably rescaled corresponding characteristic function is shown to satisfy a diffusion equation. An example of random updates for which the analysis of the distribution can be performed without averaging is worked out. The random distribution displays a deterministic drift proportional to time and its centered counterpart gives rise to a random diffusion matrix, the law of which we compute. We complete the picture by presenting an uncorrelated example.