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Time-dependent variational Monte Carlo

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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

Ψ ( X , t ) = exp ⁡ ( ∑ k a k ( t ) O k ( X ) ) {\displaystyle \Psi (X,t)=\exp \left(\sum _{k}a_{k}(t)O_{k}(X)\right)}

where the complex-valued a k ( t ) {\displaystyle a_{k}(t)} are time-dependent variational parameters, X {\displaystyle X} denotes a many-body configuration and O k ( X ) {\displaystyle O_{k}(X)} are time-independent operators that define the specific ansatz. The time evolution of the parameters a k ( t ) {\displaystyle a_{k}(t)} 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

i ∑ k ′ ⟨ O k O k ′ ⟩ t c a ˙ k ′ = ⟨ O k H ⟩ t c , {\displaystyle i\sum _{k^{\prime }}\langle O_{k}O_{k^{\prime }}\rangle _{t}^{c}{\dot {a}}_{k^{\prime }}=\langle O_{k}{\mathcal {H}}\rangle _{t}^{c},}

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 ( t ) | ⋯ | Ψ ( t ) ⟩ {\displaystyle \langle \cdots \rangle _{t}\equiv \langle \Psi (t)|\cdots |\Psi (t)\rangle } .

In analogy with the Variational Monte Carlo approach and following the Monte Carlo method for evaluating integrals, we can interpret | Ψ ( X , t ) | 2 ∫ | Ψ ( X , t ) | 2 d X {\displaystyle {\frac {|\Psi (X,t)|^{2}}{\int |\Psi (X,t)|^{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 ( t ) {\displaystyle a(t)} of the variational parameters are then found upon numerical integration of the associated differential equation.

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