  Describing the dynamics of many-electron quantum systems is crucial for
applications such as predicting electronic structures in quantum chemistry, the
properties of condensed matter systems, and the behaviors of complex materials.
However, the real-time evolution of non-equilibrium quantum electronic systems
poses a significant challenge for theoretical and computational approaches, due
to the system's exploration of a vast configuration space. This work introduces
a variational approach for fermionic time-dependent wave functions, surpassing
mean-field approximations by capturing many-body correlations. The proposed
methodology involves parameterizing the time-evolving quantum state, enabling
the approximation of the state's evolution. To account for electron
correlations, we employ time-dependent Jastrow factors and backflow
transformations. We also show that we can incorporate neural networks to
parameterize these functions. The time-dependent variational Monte Carlo
technique is employed to efficiently compute the optimal time-dependent
parameters. The approach is demonstrated in three distinct systems: the
solvable harmonic interaction model, the dynamics of a diatomic molecule in
intense laser fields, and a quenched quantum dot. In all cases, we show clear
signatures of many-body correlations in the dynamics not captured by mean-field
methods. The results showcase the ability of our variational approach to
accurately capture the time evolution of quantum states, providing insight into
the quantum dynamics of interacting electronic systems, beyond the capabilities
of mean-field.
