  Programmable quantum simulators based on Rydberg atom arrays are a
fast-emerging quantum platform, bringing together long coherence times,
high-fidelity operations, and large numbers of interacting qubits
deterministically arranged in flexible geometries. Today's Rydberg array
devices are demonstrating their utility as quantum simulators for studying
phases and phase transitions in quantum matter. In this paper, we show that
unprocessed and imperfect experimental projective measurement data can be used
to enhance in silico simulations of quantum matter, by improving the
performance of variational Monte Carlo simulations. As an example, we focus on
data spanning the disordered-to-checkerboard transition in a $16 \times 16$
square lattice array [S. Ebadi et al. Nature 595, 227 (2021)] and employ
data-enhanced variational Monte Carlo to train powerful autoregressive
wavefunction ans\"atze based on recurrent neural networks (RNNs). We observe
universal improvements in the convergence times of our simulations with this
hybrid training scheme. Notably, we also find that pre-training with
experimental data enables relatively simple RNN ans\"atze to accurately capture
phases of matter that are not learned with a purely variational training
approach. Our work highlights the promise of hybrid quantum--classical
approaches for large-scale simulation of quantum many-body systems, combining
autoregressive language models with experimental data from existing quantum
devices.
