Quantum analogue of Wiener process - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T10:19:46Z http://mathoverflow.net/feeds/question/15973 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/15973/quantum-analogue-of-wiener-process Quantum analogue of Wiener process Marcin Kotowski 2010-02-21T14:54:53Z 2011-06-27T16:23:23Z <p>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 <em>quantum random walks</em>, when the underlying stochastic process is governed by a unitary transform + measurement (for an excellent introduction, see <a href="http://arxiv.org/abs/quant-ph/0303081" rel="nofollow">http://arxiv.org/abs/quant-ph/0303081</a>).</p> <p>My question is - do quantum random walks have a reasonable continuous limit, something which would give a quantum analogue of the Wiener process?</p> http://mathoverflow.net/questions/15973/quantum-analogue-of-wiener-process/15976#15976 Answer by David Bar Moshe for Quantum analogue of Wiener process David Bar Moshe 2010-02-21T16:18:40Z 2010-02-21T16:18:40Z <p>I believe that The theory of <a href="http://mathworld.wolfram.com/QuantumStochasticCalculus.html" rel="nofollow">quantum sochastic processes</a> of Hudson and Parthasarathy, (see the original <a href="http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&amp;id=pdf%5F1&amp;handle=euclid.cmp/1103941122" rel="nofollow">article</a>) provides the necessary generalization to the continuous limit and also to a more general quantum evolution semigroups.</p> http://mathoverflow.net/questions/15973/quantum-analogue-of-wiener-process/68944#68944 Answer by aram for Quantum analogue of Wiener process aram 2011-06-27T16:23:23Z 2011-06-27T16:23:23Z <p>In section III.B of the survey paper you cite, it describes <em>continuous quantum walks</em>, 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.</p> <p><a href="http://arxiv.org/abs/0810.0312" rel="nofollow">On the relationship between continuous- and discrete-time quantum walk</a> has some recent developments with fascinating applications to simulating Hamiltonians on quantum computers.</p>