Two new results about quantum exact learning

Quantum 5, 587 (2021).

https://doi.org/10.22331/q-2021-11-24-587

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, CCC’15). Additionally, we provide a possible direction to improve our $widetilde{O}(k^{1.5})$ upper bound by proving an improvement of Chang’s lemma for $k$-Fourier-sparse Boolean functions. Second, we show that if a concept class $mathcal{C}$ 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, SICOMP’04) by a $log Q$-factor.