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    Improved Bounds on Quantum Learning Algorithms

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    In this article we give several new results on the complexity of algorithms that learn Boolean functions from quantum queries and quantum examples. Hunziker et al. 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(logCγ^C)O(\frac{\log |C|}{\sqrt{{\hat{\gamma}}^{C}}}), where γ^C\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(logCloglogCγ^C)O(\frac{\log |C| \log \log |C|}{\sqrt{{\hat{\gamma}}^{C}}}) quantum black-box queries. 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. Finally, we improve the known lower bounds on the number of quantum examples (as opposed to quantum black-box queries) required for (ϵ,δ)(\epsilon,\delta)-PAC learning any concept class of Vapnik-Chervonenkis dimension d over the domain {0,1}n\{0,1\}^n from Ω(dn)\Omega(\frac{d}{n}) to Ω(1ϵlog1δ+d+dϵ)\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.Comment: Minor corrections. 18 pages. To appear in Quantum Information Processing. Requires: algorithm.sty, algorithmic.sty to buil

    Improved Bounds on the Randomized and Quantum Complexity of Initial-Value Problems

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    We deal with the problem, initiated in [8], of finding randomized and quantum complexity of initial-value problems. We showed in [8] that a speed-up in both settings over the worst-case deterministic complexity is possible. In the present paper we prove, by defining new algorithms, that further improvement in upper bounds on the randomized and quantum complexity can be achieved. In the H\"older class of right-hand side functions with r continuous bounded partial derivatives, with r-th derivative being a H\"older function with exponent \rho, the \epsilon-complexity is shown to be O((1/\epsilon)^{1/(r+\rho+1/3)}) in the randomized setting, and O((1/\epsilon)^{1/(r+\rho+1/2)}) on a quantum computer (up to logarithmic factors). This is an improvement for the general problem over the results from [8]. The gap still remaining between upper and lower bounds on the complexity is further discussed for a special problem. We consider scalar autonomous problems, with the aim of computing the solution at the end point of the interval of integration. For this problem, we fill up the gap by establishing (essentially) matching upper and lower complexity bounds. We show that the complexity in this case is of order (1/\epsilon)^{1/(r+\rho+1/2)} in the randomized setting, and (1/\epsilon)^{1/(r+\rho+1)} in the quantum setting (again up to logarithmic factors).Comment: 17 pages, extended version (new section added), to appear in the Journal of Complexit
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