  We train convolutional neural networks to predict whether or not a set of
measurements is informationally complete to uniquely reconstruct any given
quantum state with no prior information. In addition, we perform fidelity
benchmarking based on this measurement set without explicitly carrying out
state tomography. The networks are trained to recognize the fidelity and a
reliable measure for informational completeness. By gradually accumulating
measurements and data, these trained convolutional networks can efficiently
establish a compressive quantum-state characterization scheme by accelerating
runtime computation and greatly reducing systematic drifts in experiments. We
confirm the potential of this machine-learning approach by presenting
experimental results for both spatial-mode and multiphoton systems of large
dimensions. These predictions are further shown to improve when the networks
are trained with additional bootstrapped training sets from real experimental
data. Using a realistic beam-profile displacement error model for
Hermite-Gaussian sources, we further demonstrate numerically that the
orders-of-magnitude reduction in certification time with trained networks
greatly increases the computation yield of a large-scale quantum processor
using these sources, before state fidelity deteriorates significantly.
