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 Theta~(kn) uniformly random classical examples (Haviv and Regev, CCC'15). Our main tool is an improvement of Chang's lemma for 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 ({Q^2}/{log Q} * log|C|) classical membership queries. This improves the previous-best simulation result (Servedio-Gortler, SICOMP'04) by a log Q-factor.