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
: $\Psi(X,t) = \exp \left ( \sum_k a_k(t) O_k(X) \right )$

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
: $i \sum_\langle O_k O_\rangle_t^c \dot_=\langle O_k \mathcal\rangle_t^c,$

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

In analogy with the Variational Monte Carlo approach and following the Monte Carlo method for evaluating integrals, we can interpret $\frac$
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.

References

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* {{cite thesis
, author = G. Carleo
, title = Spectral and dynamical properties of strongly correlated systems
, url = http://www.sissa.it/cm/phdsection/past_phd_thesis/2011/Carleo.pdf
, type= PhD Thesis
, pages = 107–128
, year = 2011

Quantum mechanics
Quantum Monte Carlo