  Recently, deep neural networks have proven capable of predicting some output
properties of relevant random quantum circuits, indicating a strategy to
emulate quantum computers alternative to direct simulation methods such as,
e.g., tensor-network methods. However, the reach of this alternative strategy
is not yet clear. Here we investigate if and to what extent neural networks can
learn to predict the output expectation values of circuits often employed in
variational quantum algorithms, namely, circuits formed by layers of CNOT gates
alternated with random single-qubit rotations. On the one hand, we find that
the computational cost of supervised learning scales exponentially with the
inter-layer variance of the random angles. This allows entering a regime where
quantum computers can easily outperform classical neural networks. On the other
hand, circuits featuring only inter-qubit angle variations are easily emulated.
In fact, thanks to a suitable scalable design, neural networks accurately
predict the output of larger and deeper circuits than those used for training,
even reaching circuit sizes which turn out to be intractable for the most
common simulation libraries, considering both state-vector and tensor-network
algorithms. We provide a repository of testing data in this regime, to be used
for future benchmarking of quantum devices and novel classical algorithms.
