Time-dependent variational Monte Carlo

The time-dependent variational Monte Carlo (t-VMC) method is a quantum Monte Carlo approach to study the dynamics of closed, non-relativistic quantum systems in the context of the quantum many-body problem. It is an extension of the variational Monte Carlo method, in which a time-dependent pure quantum state is encoded by some variational wave function, generally parametrized as 
where the complex-valued formula_2 are time-dependent variational parameters, formula_3 denotes a many-body configuration and formula_4 are time-independent operators that define the specific ansatz. The time evolution of the parameters formula_2 can be found upon imposing a variational principle to the wave function. In particular one can show that the optimal parameters for the evolution satisfy at each time the equation of motion
where formula_7 is the Hamiltonian of the system, formula_8 are connected averages, and the quantum expectation values are taken over the time-dependent variational wave function, i.e., formula_9.
In analogy with the Variational Monte Carlo approach and following the Monte Carlo method for evaluating integrals, we can interpret formula_10
as a probability distribution function over the multi-dimensional space spanned by the many-body configurations formula_3. The Metropolis–Hastings algorithm is then used to sample exactly from this probability distribution and, at each time formula_12, the quantities entering the equation of motion are evaluated as statistical averages over the sampled configurations. The trajectories formula_13 of the variational parameters are then found upon numerical integration of the associated differential equation.