Asymptotic behavior of quantum walks with spatio-temporal coin fluctuations

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 nondegenerate perturbation theory and yields an explicit expression for the covariance matrix of the asymptotic Gaussian in terms of the randomness parameters.

Decoherence models for discrete-time quantum walks and their application to neutral atom experiments

“…More recently, the asymptotic spatial distributions in the presence of translationally invariant decoherence mechanisms have been analytically derived by means of a generalized group velocity operator [8]. With a similar formalism based on perturbation theory, the spatial distribution in the presence of both decoherence and spatial disorder has been analytically studied [9]. Decohered quantum walks have also been considered for algorithmic applications, namely for searching unstructured databases [10].…”

Quantum walks in external gauge fields

“…Proof. (1)⇒(2): According to (21), P αβ is given as the product T * α T * β T α T β and hence clearly invariant under conjugation by S γ if we assume (1). At the same time, P αβ is a localized operator on which the translation system S γ acts as S γ P αβ S * γ φ x = P αβ (x−γ)φ x , hence P αβ (x) cannot depend on x.…”

“…In this manuscript the systems under consideration are single particles with spin which evolve in discrete time on a lattice. Such systems are called quantum walks [1,3,4,22,28]. The minimal coupling mechanism for turning on an electromagnetic field does not carry over to such systems directly, because discrete translations have no generators P µ .…”

Transport properties of anyons in random topological environments

“…[28], the quantum walk is diffusive in the presence of both temporal and spatial disorders; in other words, localization does not occur in a spatially disordered system if temporal randomness is also present. The effect of T a a is that it multiplies the contribution from each path by the configuration average over (−1) 1 s <s n m s m s τ (s ,s ) .…”