Asymptotic behavior of quantum walks with spatio-temporal coin fluctuations

TL;DR

This paper analytically studies the long-time behavior of decoherent quantum walks with spatio-temporal coin fluctuations, showing they converge to classical Gaussian diffusion under mild conditions.

Contribution

It provides the first analytical proof that certain decoherent quantum walks exhibit classical diffusive behavior, with explicit covariance matrix expressions.

Findings

Position distribution converges to a Gaussian in the long-time limit.

Quantum walks with local random coin operations behave classically asymptotically.

Explicit covariance matrix derived from randomness parameters.

Abstract

Quantum walks subject to decoherence generically suffer the loss of their genuine quantum feature, a quadratically faster spreading compared to classical random walks. This intuitive statement has been verified analytically for certain models and is also supported by numerical studies of a variety of examples. In this paper we analyze the long-time behavior of a particular class of decoherent quantum walks, which, to the best of our knowledge, was only studied at the level of numerical simulations before. We consider a local coin operation which is randomly and independently chosen for each time step and each lattice site and prove that, under rather mild conditions, this leads to classical behavior: With the same scaling as needed for a classical diffusion the position distribution converges to a Gaussian, which is independent of the initial state. Our method is based on non-degenerate perturbation theory and yields an explicit expression for the covariance matrix of the asymptotic Gaussian in terms of the randomness parameters.