Two new results about quantum exact learning

Srinivasan Arunachalam1, Sourav Chakraborty2, Troy Lee3, Manaswi Paraashar2, and Ronald de Wolf4

1IBM T. J. Watson Research Center
2Indian Statistical Institute, Kolkata, India
3Centre for Quantum Software and Information, University of Technology Sydney, Australia
4QuSoft, CWI and University of Amsterdam, the Netherlands

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

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[2] Seyed Shakib Vedaie, Archismita Dalal, Eduardo J. Páez, and Barry C. Sanders, "Framework for learning and control in the classical and quantum domains", Annals of Physics 458, 169471 (2023).

[3] Anurag Anshu and Srinivasan Arunachalam, "A survey on the complexity of learning quantum states", Nature Reviews Physics 6 1, 59 (2023).

[4] Dylan Herman, Rudy Raymond, Muyuan Li, Nicolas Robles, Antonio Mezzacapo, and Marco Pistoia, "Expressivity of Variational Quantum Machine Learning on the Boolean Cube", IEEE Transactions on Quantum Engineering 4, 1 (2023).

[5] András Gilyén and Tongyang Li, "Distributional property testing in a quantum world", arXiv:1902.00814, (2019).

[6] Srinivasan Arunachalam and Reevu Maity, "Quantum Boosting", arXiv:2002.05056, (2020).

[7] Srinivasan Arunachalam, Alex B. Grilo, and Aarthi Sundaram, "Quantum hardness of learning shallow classical circuits", arXiv:1903.02840, (2019).

[8] Matthias C. Caro, "Quantum learning Boolean linear functions w.r.t. product distributions", Quantum Information Processing 19 6, 172 (2020).

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