Quantum Chebyshev’s Inequality and Applications
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 estimation 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.
Quantum algorithms
approximation algorithms
sublinear-time algorithms
Monte Carlo method
streaming algorithms
subgraph counting
Theory of computation~Quantum computation theory
69:1-69:16
Track A: Algorithms, Complexity and Games
This research was supported by the French ANR project ANR-18-CE47-0010 (QUDATA) and the QuantERA ERA-NET Cofund project QuantAlgo.
A full version of the paper is available at https://arxiv.org/abs/1807.06456.
The authors want to thank the anonymous referees for their valuable comments and suggestions which helped to improve this paper.
Yassine
Hamoudi
Yassine Hamoudi
Université de Paris, IRIF, CNRS, F-75013 Paris, France
https://orcid.org/0000-0002-3762-0612
Frédéric
Magniez
Frédéric Magniez
Université de Paris, IRIF, CNRS, F-75013 Paris, France
https://orcid.org/0000-0003-2384-9026
10.4230/LIPIcs.ICALP.2019.69
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Yassine Hamoudi and Frédéric Magniez
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