  Given a quantum circuit, a quantum computer can sample the output
distribution exponentially faster in the number of bits than classical
computers. A similar exponential separation has yet to be established in
generative models through quantum sample learning: given samples from an
n-qubit computation, can we learn the underlying quantum distribution using
models with training parameters that scale polynomial in n under a fixed
training time? We study four kinds of generative models: Deep Boltzmann machine
(DBM), Generative Adversarial Networks (GANs), Long Short-Term Memory (LSTM)
and Autoregressive GAN, on learning quantum data set generated by deep random
circuits. We demonstrate the leading performance of LSTM in learning quantum
samples, and thus the autoregressive structure present in the underlying
quantum distribution from random quantum circuits. Both numerical experiments
and a theoretical proof in the case of the DBM show exponentially growing
complexity of learning-agent parameters required for achieving a fixed accuracy
as n increases. Finally, we establish a connection between learnability and the
complexity of generative models by benchmarking learnability against different
sets of samples drawn from probability distributions of variable degrees of
complexities in their quantum and classical representations.
