  It is believed that one of the first useful applications for a quantum
computer will be the preparation of groundstates of molecular Hamiltonians. A
crucial task involving state preparation and readout is obtaining physical
observables of such states, which are typically estimated using projective
measurements on the qubits. At present, measurement data is costly and
time-consuming to obtain on any quantum computing architecture, which has
significant consequences for the statistical errors of estimators. In this
paper, we adapt common neural network models (restricted Boltzmann machines and
recurrent neural networks) to learn complex groundstate wavefunctions for
several prototypical molecular qubit Hamiltonians from typical measurement
data. By relating the accuracy $\varepsilon$ of the reconstructed groundstate
energy to the number of measurements, we find that using a neural network model
provides a robust improvement over using single-copy measurement outcomes alone
to reconstruct observables. This enhancement yields an asymptotic scaling near
$\varepsilon^{-1}$ for the model-based approaches, as opposed to
$\varepsilon^{-2}$ in the case of classical shadow tomography.
