  Supervised machine learning is emerging as a powerful computational tool to
predict the properties of complex quantum systems at a limited computational
cost. In this article, we quantify how accurately deep neural networks can
learn the properties of disordered quantum systems as a function of the system
size. We implement a scalable convolutional network that can address arbitrary
system sizes. This network is compared with a recently introduced extensive
convolutional architecture [K. Mills et al., Chem. Sci. 10, 4129 (2019)] and
with conventional dense networks with all-to-all connectivity. The networks are
trained to predict the exact ground-state energies of various disordered
systems, namely a continuous-space single-particle Hamiltonian for cold-atoms
in speckle disorder, and different setups of a quantum Ising chain with random
couplings, including one with only short-range interactions and one augmented
with a long-range term. In all testbeds we consider, the scalable network
retains high accuracy as the system size increases. Furthermore, we demonstrate
that the network scalability enables a transfer-learning protocol, whereby a
pre-training performed on small systems drastically accelerates the learning of
large-system properties, allowing reaching high accuracy with small training
sets. In fact, with the scalable network one can even extrapolate to sizes
larger than those included in the training set, accurately reproducing the
results of state-of-the-art quantum Monte Carlo simulations.
