  We define a new model of quantum learning that we call Predictive Quantum
(PQ). This is a quantum analogue of PAC, where during the testing phase the
student is only required to answer a polynomial number of testing queries.
  We demonstrate a relational concept class that is efficiently learnable in
PQ, while in any "reasonable" classical model exponential amount of training
data would be required. This is the first unconditional separation between
quantum and classical learning.
  We show that our separation is the best possible in several ways; in
particular, there is no analogous result for a functional class, as well as for
several weaker versions of quantum learning. In order to demonstrate tightness
of our separation we consider a special case of one-way communication that we
call single-input mode, where Bob receives no input. Somewhat surprisingly,
this setting becomes nontrivial when relational communication tasks are
considered. In particular, any problem with two-sided input can be transformed
into a single-input relational problem of equal classical one-way cost. We show
that the situation is different in the quantum case, where the same
transformation can make the communication complexity exponentially larger. This
happens if and only if the original problem has exponential gap between quantum
and classical one-way communication costs. We believe that these auxiliary
results might be of independent interest.
