# Can the Metropolis-Hastings algorithm be generalized to quantum systems?

The Metropolis-Hastings algorithm is an efficient way of simulating classical ensembles using the Monte Carlo method. Is there a generalization of this algorithm to quantum systems? What I DON'T have in mind is Wick rotation to a classical Euclidean system.

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Why don't you want to discuss methods that rely on a Wick rotated classical system? – wsc Oct 11 '11 at 12:53
@wsc: I understand exactly what he wants, and it is something interesting and new, and might not exist. Metropolis Monte-Carlo produces an essentially arbitrary local probability distribution on many variables efficiently starting from coin-flips. He wants a sampling method which produces an arbitrarily entangled quantum state using only quantum information source, which is similarly efficient. The definition of quantum computation is sort of like this, but you want to specify an arbitarily entangled state, and how to get to it. This is asking for a Quantum computation monte-carlo analog. – Ron Maimon Nov 1 '11 at 2:31

It is called quantum monte carlo.

However, there is unresolved problem which does not allow to "compute everything": wavefunction of fermions should be antisymmetric, thus it changes its sign. Which is a big problem for quantum monte carlo. For bosonic systems it "just works".

UPD Both main QMC methods, variational and diffusion, are not just Wick rotation to a classical system. Variational MC is "just" a variational method with integrals computed using MC. No rotations, nothing. For trial functions there is a standard choice: Slater-Jastrow which is a jeneralization of Hartree-Fock functions with a free parameters.

I actually had in mind diffusion MC, which might look like turning into classical system, though it is not. Imaginary time is used, but serves a different purpose: to turn time evolution in the Schrodinger equation into convergence to the stationary solution. The obtained equations which are similar to diffusion equations in multi (3M where M is the number of particles involved) dimensional space gives the solution: evolution of this fictional "system of particles" which is calulated using slightly modified Metropolis algorithm gives an approximate solution of stationary Schrodinger equation as its infinite limit.

You might take a look at the introductory level paper in Rev. Mod. Phys., 73, 33 (2001).

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This answer is exactly what the OP didn't want. It's classical Monte Carlo in imaginary time. – Ron Maimon Oct 10 '11 at 18:22
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. – Misha Oct 10 '11 at 18:28
@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. – Ron Maimon Oct 10 '11 at 21:26
@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. – Misha Oct 11 '11 at 4:36
@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. – Ron Maimon Oct 11 '11 at 15:45

Were you thinking about something like this?

http://www.nature.com/nature/journal/v471/n7336/full/nature09770.html

or arXiv:0911.3635

They called the algorithm "quantum metropolis sampling". The only downside seems to be that you would actually need a working quantum computer.

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Path integral Monte Carlo might be what you're looking for. The basic idea is to sample the partition function $$Z = {\rm tr} \ \exp\{-\beta H\}$$ where $H$ is the Hamiltonian of a single quantum particle (think of an electron in a disordered environment in the simplest case). $Z$ can be factored into P parts $$Z = \int dx <x| e^{-\beta H} |x>$$ $$= \int dx_1 dx_2...dx_P <x_1| e^{-\beta H/P} |x_2>... <x_P| e^{-\beta H/P} |x_1>$$ the last expression is isomorphic to the partition function of a classical ring polymer with P 'beads' or particles. The ring geometry comes from the trace. It can be sampled with Metropolis Monte Carlo in the same way as a classical (ring) polymer. There have been numerous applications of this method, for example to study an electron in a disordered liquid or inert gas.

For a many-particle quantum system it gets tricky because exchanges between identical particles have to be accounted for. The approach was originally proposed by Feynman in 1953 to study superfluidity in He$^4$. He had to wait a couple of years until computers were powerful enough: Ceperley & Pollock were the first to do a Monte Carlo study of liquid He II in the early 1980's.

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