Given access to accurate solutions of the many-electron Schr\"odinger
equation, nearly all chemistry could be derived from first principles. Exact
wavefunctions of interesting chemical systems are out of reach because they are
NP-hard to compute in general, but approximations can be found using
polynomially-scaling algorithms. The key challenge for many of these algorithms
is the choice of wavefunction approximation, or Ansatz, which must trade off
between efficiency and accuracy. Neural networks have shown impressive power as
accurate practical function approximators and promise as a compact wavefunction
Ansatz for spin systems, but problems in electronic structure require
wavefunctions that obey Fermi-Dirac statistics. Here we introduce a novel deep
learning architecture, the Fermionic Neural Network, as a powerful wavefunction
Ansatz for many-electron systems. The Fermionic Neural Network is able to
achieve accuracy beyond other variational quantum Monte Carlo Ans\"atze on a
variety of atoms and small molecules. Using no data other than atomic positions
and charges, we predict the dissociation curves of the nitrogen molecule and
hydrogen chain, two challenging strongly-correlated systems, to significantly
higher accuracy than the coupled cluster method, widely considered the most
accurate scalable method for quantum chemistry at equilibrium geometry. This
demonstrates that deep neural networks can improve the accuracy of variational
quantum Monte Carlo to the point where it outperforms other ab-initio quantum
chemistry methods, opening the possibility of accurate direct optimisation of
wavefunctions for previously intractable molecules and solids.