[evanb]

If you want to study a quantum mechanical system in equilibrium at inverse temperature β, the interesting quantity is the partition function Z = tr exp[-β H]. This can be converted into a path integral Z = ∫ dφ exp[-S[φ]] which can be importance-sampled via the Metropolis-Hastings algorithm [mh] via Markov-chain Monte Carlo.

This approach is commonly used in lattice field theory [lft], where the Hamiltonian H is that of a discretized spacetime (or the problem is formulated in terms of the action S to begin with).

Real-time problems in quantum mechanics involve exp[i t H] which brings a horrible complication called the sign problem [sign]. The one-sentence summary is that exp[-β H] is positive-definite but exp[i t H] is not and it's not clear how to incorporate a complex Boltzmann weight as a probability for MCMC.

mh: https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_al...

lft: https://en.wikipedia.org/wiki/Lattice_field_theory

sign: https://en.wikipedia.org/wiki/Numerical_sign_problem