  Suppose we have a weak learning algorithm $\mathcal{A}$ for a Boolean-valued
problem: $\mathcal{A}$ produces hypotheses whose bias $\gamma$ is small, only
slightly better than random guessing (this could, for instance, be due to
implementing $\mathcal{A}$ on a noisy device), can we boost the performance of
$\mathcal{A}$ so that $\mathcal{A}$'s output is correct on $2/3$ of the inputs?
  Boosting is a technique that converts a weak and inaccurate machine learning
algorithm into a strong accurate learning algorithm. The AdaBoost algorithm by
Freund and Schapire (for which they were awarded the G\"odel prize in 2003) is
one of the widely used boosting algorithms, with many applications in theory
and practice. Suppose we have a $\gamma$-weak learner for a Boolean concept
class $C$ that takes time $R(C)$, then the time complexity of AdaBoost scales
as $VC(C)\cdot poly(R(C), 1/\gamma)$, where $VC(C)$ is the $VC$-dimension of
$C$. In this paper, we show how quantum techniques can improve the time
complexity of classical AdaBoost. To this end, suppose we have a $\gamma$-weak
quantum learner for a Boolean concept class $C$ that takes time $Q(C)$, we
introduce a quantum boosting algorithm whose complexity scales as
$\sqrt{VC(C)}\cdot poly(Q(C),1/\gamma);$ thereby achieving a quadratic quantum
improvement over classical AdaBoost in terms of $VC(C)$.
