Two new results about quantum exact learning

TL;DR

This paper introduces two significant results in quantum exact learning: an improved quantum sample complexity for learning Fourier-sparse Boolean functions and a more efficient classical simulation of quantum membership query algorithms.

Contribution

It provides a tighter quantum sample complexity bound for Fourier-sparse functions and enhances classical simulation bounds for quantum learning algorithms.

Findings

Quantum algorithms learn $k$-Fourier-sparse functions with $O(k^{1.5}(\log k)^2)$ examples.

Classical queries needed are $O(rac{Q^2}{\log Q}\log|\mathcal{C}|)$ for simulating quantum queries.

Potential improvement of bounds via an extended Chang's lemma.

Abstract

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 \emph{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 $O\left(\frac{Q^2}{\log Q}\log|\mathcal{C}|\right)$ \emph{classical} membership queries. This improves the previous-best simulation result (Servedio and Gortler, SICOMP'04) by a $\log Q$-factor.