  The potential energy surface (PES) of molecules with respect to their nuclear
positions is a primary tool in understanding chemical reactions from first
principles. However, obtaining this information is complicated by the fact that
sampling a large number of ground states over a high-dimensional PES can
require a vast number of state preparations. In this work, we propose using a
generative quantum machine learning model to prepare quantum states at
arbitrary points on the PES. The model is trained using quantum data consisting
of ground-state wavefunctions associated with different classical nuclear
coordinates. Our approach uses a classical neural network to convert the
nuclear coordinates of a molecule into quantum parameters of a variational
quantum circuit. The model is trained using a fidelity loss function to
optimize the neural network parameters. We show that gradient evaluation is
efficient and numerically demonstrate our method's ability to prepare
wavefunctions on the PES of hydrogen chains, water, and beryllium hydride. In
all cases, we find that a small number of training points are needed to achieve
very high overlap with the groundstates in practice. From a theoretical
perspective, we further prove limitations on these protocols by showing that if
we were able to learn across an avoided crossing using a small number of
samples, then we would be able to violate Grover's lower bound. Additionally,
we prove lower bounds on the amount of quantum data needed to learn a locally
optimal neural network function using arguments from quantum Fisher
information. This work further identifies that quantum chemistry can be an
important use case for quantum machine learning.
