  The widespread use of machine learning has raised the question of quantum
supremacy for supervised learning as compared to quantum computational
advantage. In fact, a recent work shows that computational and learning
advantage are, in general, not equivalent, i.e., the additional information
provided by a training set can reduce the hardness of some problems. This paper
investigates under which conditions they are found to be equivalent or, at
least, highly related. The existence of efficient algorithms to generate
training sets emerges as the cornerstone of such conditions. These results are
applied to prove that there is a quantum speed-up for some learning tasks based
on the prime factorization problem, assuming the classical intractability of
this problem.
