  We revisit the problem of characterising the complexity of Quantum PAC
learning, as introduced by Bshouty and Jackson [SIAM J. Comput. 1998, 28,
1136-1153]. Several quantum advantages have been demonstrated in this setting,
however, none are generic: they apply to particular concept classes and
typically only work when the distribution that generates the data is known. In
the general case, it was recently shown by Arunachalam and de Wolf [JMLR, 19
(2018) 1-36] that quantum PAC learners can only achieve constant factor
advantages over classical PAC learners.
  We show that with a natural extension of the definition of quantum PAC
learning used by Arunachalam and de Wolf, we can achieve a generic advantage in
quantum learning. To be precise, for any concept class $\mathcal{C}$ of VC
dimension $d$, we show there is an $(\epsilon, \delta)$-quantum PAC learner
with sample complexity \[ O\left(\frac{1}{\sqrt{\epsilon}}\left[d+
\log(\frac{1}{\delta})\right]\log^9(1/\epsilon)\right). \] Up to
polylogarithmic factors, this is a square root improvement over the classical
learning sample complexity. We show the tightness of our result by proving an
$\Omega(d/\sqrt{\epsilon})$ lower bound that matches our upper bound up to
polylogarithmic factors.
