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

Question

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.

Answer 1

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).

Answer 2

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.