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 ~ ( k n ) uniformly random c l a s s i c a l examples (Haviv and Regev, CCC'15). Additionally, we provide a possible direction to improve our O ~ ( 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 ( Q 2 log ⁡ Q log ⁡ | C | ) c l a s s i c a l membership queries. This improves the previous-best simulation result (Servedio and Gortler, SICOMP'04) by a log ⁡ Q -factor.