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The time-dependent variational Monte Carlo 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 a k {\displaystyle a_{k}} are time-dependent variational parameters, X {\displaystyle X} denotes a many-body configuration and O k {\displaystyle O_{k}} are time-independent operators that define the specific ansatz. The time evolution of the parameters a k {\displaystyle a_{k}} 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 H {\displaystyle {\mathcal {H}}} is the Hamiltonian of the system, ⟨ A B ⟩ t c = ⟨ A B ⟩ t − ⟨ A ⟩ t ⟨ B ⟩ t {\displaystyle \langle AB\rangle _{t}^{c}=\langle AB\rangle _{t}-\langle A\rangle _{t}\langle B\rangle _{t}} are connected averages, and the quantum expectation values are taken over the time-dependent variational wave function, i.e., ⟨ ⋯ ⟩ t ≡ ⟨ Ψ | ⋯ | Ψ ⟩ {\displaystyle \langle \cdots \rangle _{t}\equiv \langle \Psi |\cdots |\Psi \rangle }.
In analogy with the Variational Monte Carlo approach and following the Monte Carlo method for evaluating integrals, we can interpret | Ψ | 2 ∫ | Ψ | 2 d X {\displaystyle {\frac {|\Psi |^{2}}{\int |\Psi |^{2}\,dX}}} as a probability distribution function over the multi-dimensional space spanned by the many-body configurations X {\displaystyle X}. The Metropolis–Hastings algorithm is then used to sample exactly from this probability distribution and, at each time t {\displaystyle t} , the quantities entering the equation of motion are evaluated as statistical averages over the sampled configurations. The trajectories a {\displaystyle a} of the variational parameters are then found upon numerical integration of the associated differential equation.