11
$\begingroup$

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?

$\endgroup$
3
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.