  Modern day quantum simulators can prepare a wide variety of quantum states
but the accurate estimation of observables from tomographic measurement data
often poses a challenge. We tackle this problem by developing a quantum state
tomography scheme which relies on approximating the probability distribution
over the outcomes of an informationally complete measurement in a variational
manifold represented by a convolutional neural network. We show an excellent
representability of prototypical ground- and steady states with this ansatz
using a number of variational parameters that scales polynomially in system
size. This compressed representation allows us to reconstruct states with high
classical fidelities outperforming standard methods such as maximum likelihood
estimation. Furthermore, it achieves a reduction of the estimation error of
observables by up to an order of magnitude compared to their direct estimation
from experimental data.
