  There is currently a large interest in understanding the potential advantages
quantum devices can offer for probabilistic modelling. In this work we
investigate, within two different oracle models, the probably approximately
correct (PAC) learnability of quantum circuit Born machines, i.e., the output
distributions of local quantum circuits. We first show a negative result,
namely, that the output distributions of super-logarithmic depth Clifford
circuits are not sample-efficiently learnable in the statistical query model,
i.e., when given query access to empirical expectation values of bounded
functions over the sample space. This immediately implies the hardness, for
both quantum and classical algorithms, of learning from statistical queries the
output distributions of local quantum circuits using any gate set which
includes the Clifford group. As many practical generative modelling algorithms
use statistical queries -- including those for training quantum circuit Born
machines -- our result is broadly applicable and strongly limits the
possibility of a meaningful quantum advantage for learning the output
distributions of local quantum circuits. As a positive result, we show that in
a more powerful oracle model, namely when directly given access to samples, the
output distributions of local Clifford circuits are computationally efficiently
PAC learnable by a classical learner. Our results are equally applicable to the
problems of learning an algorithm for generating samples from the target
distribution (generative modelling) and learning an algorithm for evaluating
its probabilities (density modelling). They provide the first rigorous insights
into the learnability of output distributions of local quantum circuits from
the probabilistic modelling perspective.
