  [New and updated results were published in Nature Chemistry,
doi:10.1038/s41557-020-0544-y.] The electronic Schr\"odinger equation describes
fundamental properties of molecules and materials, but can only be solved
analytically for the hydrogen atom. The numerically exact full
configuration-interaction method is exponentially expensive in the number of
electrons. Quantum Monte Carlo is a possible way out: it scales well to large
molecules, can be parallelized, and its accuracy has, as yet, only been limited
by the flexibility of the used wave function ansatz. Here we propose PauliNet,
a deep-learning wave function ansatz that achieves nearly exact solutions of
the electronic Schr\"odinger equation. PauliNet has a multireference
Hartree-Fock solution built in as a baseline, incorporates the physics of valid
wave functions, and is trained using variational quantum Monte Carlo (VMC).
PauliNet outperforms comparable state-of-the-art VMC ansatzes for atoms,
diatomic molecules and a strongly-correlated hydrogen chain by a margin and is
yet computationally efficient. We anticipate that thanks to the favourable
scaling with system size, this method may become a new leading method for
highly accurate electronic-strucutre calculations on medium-sized molecular
systems.
