  We introduce the quantum Gaussian process state, motivated via a statistical
inference for the wave function supported by a data set of unentangled product
states. We show that this condenses down to a compact and expressive parametric
form, with a variational flexibility shown to be competitive or surpassing
established alternatives. The connections of the state to its roots as a
Bayesian inference machine as well as matrix product states, also allow for
efficient deterministic training of global states from small training data with
enhanced generalization, including on application to frustrated spin physics.
