Question

The Wiener process (say, on $\mathbb{R}$) can be thought of as a scaling limit of a classical, discrete random walk. On the other hand, one can define and study quantum random walks, when the underlying stochastic process is governed by a unitary transform + measurement (for an excellent introduction, see http://arxiv.org/abs/quant-ph/0303081).

My question is - do quantum random walks have a reasonable continuous limit, something which would give a quantum analogue of the Wiener process?

Answer 1

I believe that The theory of quantum sochastic processes of Hudson and Parthasarathy, (see the original article) provides the necessary generalization to the continuous limit and also to a more general quantum evolution semigroups.

Answer 2

In section III.B of the survey paper you cite, it describes continuous quantum walks, which are I think are a natural analogue of the Wiener process. These are basically Hamiltonian evolution when the Hamiltonian is something like the adjacency matrix (or Laplacian) of a graph.

On the relationship between continuous- and discrete-time quantum walk has some recent developments with fascinating applications to simulating Hamiltonians on quantum computers.