  Owing to their great expressivity and versatility, neural networks have
gained attention for simulating large two-dimensional quantum many-body
systems. However, their expressivity comes with the cost of a challenging
optimization due to the in general rugged and complicated loss landscape. Here,
we present a hybrid optimization scheme for neural quantum states (NQS) that
involves a data-driven pretraining with numerical or experimental data and a
second, Hamiltonian-driven optimization stage. By using both projective
measurements from the computational basis as well as expectation values from
other measurement configurations such as spin-spin correlations, our
pretraining gives access to the sign structure of the state, yielding improved
and faster convergence that is robust w.r.t. experimental imperfections and
limited datasets. We apply the hybrid scheme to the ground state search for the
2D transverse field Ising model and the 2D dipolar XY model on $6\times 6$ and
$10\times 10$ square lattices with a patched transformer wave function, using
numerical and experimental data from a programmable Rydberg quantum simulator
[Chen et al., Nature 616 (2023)], with snapshots of the quantum system obtained
from the different measurement configurations, and show that the information
from the second basis highly improves the performance. Our work paves the way
for a reliable and efficient optimization of neural quantum states.
