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- W3099681168 abstract "In this paper we provide new quantum algorithms with polynomial speed-up for a range of problems for which no such results were known, or we improve previous algorithms. First, we consider the approximation of the frequency moments F_k of order k >= 3 in the multi-pass streaming model with updates (turnstile model). We design a P-pass quantum streaming algorithm with memory M satisfying a tradeoff of P^2 M = O~(n^{1-2/k}), whereas the best classical algorithm requires P M = Theta(n^{1-2/k}). Then, we study the problem of estimating the number m of edges and the number t of triangles given query access to an n-vertex graph. We describe optimal quantum algorithms that perform O~(sqrt{n}/m^{1/4}) and O~(sqrt{n}/t^{1/6} + m^{3/4}/sqrt{t}) queries respectively. This is a quadratic speed-up compared to the classical complexity of these problems.For this purpose we develop a new quantum paradigm that we call Quantum Chebyshev's inequality. Namely we demonstrate that, in a certain model of quantum sampling, one can approximate with relative error the mean of any random variable with a number of quantum samples that is linear in the ratio of the square root of the variance to the mean. Classically the dependence is quadratic. Our algorithm subsumes a previous result of Montanaro [Montanaro, 2015]. This new paradigm is based on a refinement of the Amplitude Estimation algorithm of Brassard et al. [Brassard et al., 2002] and of previous quantum algorithms for the mean problem. We show that this speed-up is optimal, and we identify another common model of quantum sampling where it cannot be obtained. Finally, we develop a new technique called variable-time amplitude estimation that reduces the dependence of our algorithm on the sample preparation time." @default.
- W3099681168 created "2020-11-23" @default.
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- W3099681168 date "2019-07-04" @default.
- W3099681168 modified "2023-10-16" @default.
- W3099681168 title "Quantum Chebyshev's Inequality and Applications" @default.
- W3099681168 doi "https://doi.org/10.4230/lipics.icalp.2019.69" @default.
- W3099681168 hasPublicationYear "2019" @default.
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