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arXiv · 1810.00481

Two new results about quantum exact learning

Srinivasan Arunachalam·Sourav Chakraborty·Troy Lee·Manaswi Paraashar·Ronald de Wolf
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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.

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Srinivasan Arunachalam, Sourav Chakraborty, Troy Lee, Manaswi Paraashar, Ronald de Wolf. 2018-09-30. Two new results about quantum exact learning. https://doi.org/10.22331/q-2021-11-24-587

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