  In classical statistical learning theory, one of the most well studied
problems is that of binary classification. The information-theoretic sample
complexity of this task is tightly characterized by the Vapnik-Chervonenkis
(VC) dimension. A quantum analog of this task, with training data given as a
quantum state has also been intensely studied and is now known to have the same
sample complexity as its classical counterpart.
  We propose a novel quantum version of the classical binary classification
task by considering maps with classical input and quantum output and
corresponding classical-quantum training data. We discuss learning strategies
for the agnostic and for the realizable case and study their performance to
obtain sample complexity upper bounds. Moreover, we provide sample complexity
lower bounds which show that our upper bounds are essentially tight for pure
output states. In particular, we see that the sample complexity is the same as
in the classical binary classification task w.r.t. its dependence on accuracy,
confidence and the VC-dimension.
