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(k1.5(log k)2) uniform quantum examples for that function. This improves over the bound of Θ(e kn) uniformly random classical examples (Haviv and Regev, CCC'15). Additionally, we provide a possible direction to improve our Oe(k1.5) upper bound by proving an improvement of Chang's lemma for k-Fourier-sparse Boolean functions. Second, we show that if a concept class C can be exactly learned using Q quantum membership queries, then it can also be learned using O (logQ2Q log |C| ) classical membership queries. This improves the previous-best simulation result (Servedio and Gortler, SICOMP'04) by a log Q-factor.