Random Time-Dependent Quantum Walks

TL;DR

This paper analyzes the long-term behavior of random quantum walks on lattices, revealing drift, diffusion, and large deviation properties, with both averaged and specific examples, extending understanding of quantum stochastic processes.

Contribution

It introduces a comprehensive analysis of time-dependent quantum walks with random unitary updates, including drift, diffusion, and deviation principles, and generalizes to Markovian updates.

Findings

Averaged distribution shows linear drift and diffusive behavior with a computed diffusion matrix.

A moderate deviation principle is established for the averaged distribution.

In a specific example, the distribution exhibits deterministic drift and a random diffusion matrix.

Abstract

We consider the discrete time unitary dynamics given by a quantum walk on the lattice $\Z^d$ performed by a quantum 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 $\Z^d$ when the sequence of unitary updates is given by an i.i.d. sequence of random matrices. When averaged over the randomness, this distribution is shown to display a drift proportional to the time and its centered counterpart is shown to display a diffusive behavior with a diffusion matrix we compute. A moderate deviation principle is 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. A generalization to unitary updates distributed according to a Markov process is also provided. An example of i.i.d. random updates for which the analysis of the distribution can be performed without averaging is worked out. The distribution also displays a deterministic drift proportional to time and its centered counterpart gives rise to a random diffusion matrix whose law we compute. A large deviation principle is shown to hold for this example. We finally show that, in general, the expectation of the random diffusion matrix equals the diffusion matrix of the averaged distribution.