  Physicists dating back to Feynman have lamented the difficulties of applying
the variational principle to quantum field theories. In non-relativistic
quantum field theories, the challenge is to parameterize and optimize over the
infinitely many $n$-particle wave functions comprising the state's Fock space
representation. Here we approach this problem by introducing neural-network
quantum field states, a deep learning ansatz that enables application of the
variational principle to non-relativistic quantum field theories in the
continuum. Our ansatz uses the Deep Sets neural network architecture to
simultaneously parameterize all of the $n$-particle wave functions comprising a
quantum field state. We employ our ansatz to approximate ground states of
various field theories, including an inhomogeneous system and a system with
long-range interactions, thus demonstrating a powerful new tool for probing
quantum field theories.
