Timeline for Can the Metropolis-Hastings algorithm be generalized to quantum systems?
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| Oct 12, 2011 at 3:48 | history | edited | Misha | CC BY-SA 3.0 |
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| Oct 11, 2011 at 15:53 | comment | added | Ron Maimon | @wsc: I am not confused, I understand all of this perfectly well. He is talking about the monte-carlo on the imaginary time continuation of the quantum mechanical paths. It is exactly the same as any other Monte-Carlo, it's Wick rotated. It's quantum field theory in 0+1 dimensions, so it turns into a monte-carlo on 1d paths. It's not what the OP wanted. The OP wanted a quantum computer algorithm which will allow you to reproduce a given quantum state by a sequence of quantum operations, like Metropolis allows you to reproduce a given classical probability distribution by some moves. | |
| Oct 11, 2011 at 15:45 | comment | added | Ron Maimon | @Misha: -1--- you are making no sense: yes, a classical deterministic system is defined by a point in phase space, but a classical probabilistic system is defined by a probability distribution, which is of the same exponential size as quantum mechanics. The probability distributions reproduces the ground state properties. | |
| Oct 11, 2011 at 12:45 | comment | added | wsc | @Misha, I think you're trying to talk about Diffusion (sometimes called "Green Function") Monte Carlo. This is an essentially variational approach and only one flavor of QMC (Ron's confusion is understandable). If you want this to be a good answer you should flesh out what DMC is exactly, and why it's different from path integral or stochastic series expansion approaches (which are also called Quantum Monte Carlo). | |
| Oct 11, 2011 at 4:36 | comment | added | Misha | @Ron Maimon No, I mean that classical system is defined either (it depends on what you call classical, but OP was not precise enough there) by point in N dimensional space where N is the number of variables or distribution in 3 dimensional space. While quantum system is a distribution in 3M dimensional space where M is the number of particles. And QMC has no "time that becomes spatial dimension". "Time" there is somewhat fictional variable to let the (fictional) diffusion approach stationary solution (of the quantum problem). Only its infinite limit has a meaning. | |
| Oct 10, 2011 at 21:26 | comment | added | Ron Maimon | @Misha--- you mean that the imaginary time quantum system is one dimension more, because of the fact that time becomes a spatial dimension. Yes, we know, and yes it is just classical MC in imaginary time. | |
| Oct 10, 2011 at 18:28 | comment | added | Misha | Then he could better formulate what he wanted. It is NOT just classical MC in imaginary time, by the way. The dimensionality of the classical and quantum problems is quite different. | |
| Oct 10, 2011 at 18:22 | comment | added | Ron Maimon | This answer is exactly what the OP didn't want. It's classical Monte Carlo in imaginary time. | |
| Oct 10, 2011 at 16:05 | history | answered | Misha | CC BY-SA 3.0 |