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 Θ(^e kn) uniformly random classical examples (Haviv and Regev, CCC'15). Additionally, we provide a possible direction to improve our O^e(k^1.5) 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 (_log^Q2_Q log |C| ) classical membership queries. This improves the previous-best simulation result (Servedio and Gortler, SICOMP'04) by a log Q-factor.