Two new results about quantum exact learning

We present two new results about exact learning by quantum computers. First, we show how to exactly learn a $k$ Fourier-sparse $n$ -bit Boolean function from $O(k^{1.5}(\log <!--\; \; -->k{)}^2)$ uniform quantum examples for that function. This improves over the bound of $\stackrel{~<!--\; ~\; -->}{\Theta }(kn)$ uniformly random \emph{classical} examples (Haviv and Regev, CCC'15). Second, we show that if a concept class $\mathcal{C}$ can be exactly learned using $Q$ quantum membership queries, then it can also be learned using $O(\frac{Q^2}{\log <!--\; \; -->Q}\log <!--\; \; -->|\mathcal{C}|)$ \emph{classical} membership queries. This improves the previous-best simulation result (Servedio and Gortler, SICOMP'04) by a $\log <!--\; \; -->Q$ -factor.