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    New Developments in Quantum Algorithms

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    In this survey, we describe two recent developments in quantum algorithms. The first new development is a quantum algorithm for evaluating a Boolean formula consisting of AND and OR gates of size N in time O(\sqrt{N}). This provides quantum speedups for any problem that can be expressed via Boolean formulas. This result can be also extended to span problems, a generalization of Boolean formulas. This provides an optimal quantum algorithm for any Boolean function in the black-box query model. The second new development is a quantum algorithm for solving systems of linear equations. In contrast with traditional algorithms that run in time O(N^{2.37...}) where N is the size of the system, the quantum algorithm runs in time O(\log^c N). It outputs a quantum state describing the solution of the system.Comment: 11 pages, 1 figure, to appear as an invited survey talk at MFCS'201

    Two new results about quantum exact learning

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    We present two new results about exact learning by quantum computers. First, we show how to exactly learn a kk-Fourier-sparse nn-bit Boolean function from O(k1.5(logk)2)O(k^{1.5}(\log k)^2) uniform quantum examples for that function. This improves over the bound of Θ~(kn)\widetilde{\Theta}(kn) uniformly random classical examples (Haviv and Regev, CCC'15). Our main tool is an improvement of Chang's lemma for the special case of sparse functions. Second, we show that if a concept class C\mathcal{C} can be exactly learned using QQ quantum membership queries, then it can also be learned using O(Q2logQlogC)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 logQ\log Q-factor.Comment: v3: 21 pages. Small corrections and clarification
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