Abstract
In this article we give several new results on the complexity of algorithms that learn Boolean functions from quantum queries and quantum examples.
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Hunziker et al.[Quantum Information Processing, to appear] conjectured that for any class C of Boolean functions, the number of quantum black-box queries which are required to exactly identify an unknown function from C is \(O(\frac{\log |C|}{\sqrt{{\hat{\gamma}}^{C}}})\), where \(\hat{\gamma}^{C}\) is a combinatorial parameter of the class C. We essentially resolve this conjecture in the affirmative by giving a quantum algorithm that, for any class C, identifies any unknown function from C using \(O(\frac{\log |C| \log \log |C|}{\sqrt{{\hat{\gamma}}^{C}}})\) quantum black-box queries.
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We consider a range of natural problems intermediate between the exact learning problem (in which the learner must obtain all bits of information about the black-box function) and the usual problem of computing a predicate (in which the learner must obtain only one bit of information about the black-box function). We give positive and negative results on when the quantum and classical query complexities of these intermediate problems are polynomially related to each other.
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Finally, we improve the known lower bounds on the number of quantum examples (as opposed to quantum black-box queries) required for ɛ, Δ-PAC learning any concept class of Vapnik-Chervonenkis dimension d over the domain \(\{0,1\}^n\) from \(\Omega({\frac d n})\) to \(\Omega(\frac{1}{\epsilon}\log \frac{1}{\delta}+d+\frac{\sqrt{d}}{\epsilon})\). This new lower bound comes closer to matching known upper bounds for classical PAC learning.
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Pacs: 03.67.Lx, 89.80.+h, 02.70.-c
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Atici, A., Servedio, R.A. Improved Bounds on Quantum Learning Algorithms. Quantum Inf Process 4, 355–386 (2005). https://doi.org/10.1007/s11128-005-0001-2
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DOI: https://doi.org/10.1007/s11128-005-0001-2