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 $widetilde{Theta}(kn)$ uniformly random classical examples (Haviv and Regev, CCC15). Our main tool is an improvement of Changs lemma for the special case of sparse functions. 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 $Oleft(frac{Q^2}{log Q}log|mathcal{C}|right)$ classical membership queries. This improves the previous-best simulation result (Servedio and Gortler, SICOMP04) by a $log Q$-factor.