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

: $\backslash Psi(X,t)\; =\; \backslash exp\; \backslash left\; (\; \backslash sum\_k\; a\_k(t)\; O\_k(X)\; \backslash right\; )$

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

: $i\; \backslash sum\_\{k^\{\backslash prime\}\}\backslash langle\; O\_k\; O\_\{k^\{\backslash prime\}\}\backslash rangle\_t^c\; \backslash dot\{a\}\_\{k^\{\backslash prime\}\}=\backslash langle\; O\_k\; \backslash mathcal\{H\}\backslash rangle\_t^c,$

where $\backslash mathcal\{H\}$ is the Hamiltonian of the system, $\backslash langle\; AB\; \backslash rangle\_t^c=\backslash langle\; AB\backslash rangle\_t-\backslash langle\; A\backslash rangle\_t\backslash langle\; B\backslash rangle\_t$ are connected averages, and the quantum expectation values are taken over the time-dependent variational wave function, i.e., $\backslash langle\backslash cdots\backslash rangle\_t\; \backslash equiv\backslash langle\backslash Psi(t)|\backslash cdots|\backslash Psi(t)\backslash rangle$.

In analogy with the Variational Monte Carlo approach and following the Monte Carlo method for evaluating integrals, we can interpret $\backslash frac\{\; |\; \backslash Psi(X,t)\; |\; ^2\; \}\; \{\; \backslash int\; |\; \backslash Psi(X,t)\; |\; ^2\; \backslash ,\; dX\; \}$

as a probability distribution function over the multi-dimensional space spanned by the many-body configurations $X$. The Metropolis–Hastings algorithm is then used to sample exactly from this probability distribution and, at each time $t$, the quantities entering the equation of motion are evaluated as statistical averages over the sampled configurations. The trajectories $a(t)$ of the variational parameters are then found upon numerical integration of the associated differential equation.