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 $k \geq 3$ in the multi-pass streaming model with updates (turnstile model). We design a $P$-pass quantum streaming algorithm with space memory $M$ satisfying a tradeoff of $P^2 M = \tilde{O}(n^{1-2/k})$, 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 $\tilde{O}(\sqrt{n}/m^{1/4})$ and $\tilde{O}(\sqrt{n}/t^{1/6} + m^{3/4}/\sqrt{t})$ 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 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 dependency is quadratic. Our result is optimal and subsumes a previous result of Montanaro [Mon15]. This new paradigm is based on a refinement of the Amplitude Estimation algorithm of Brassard et al. [BHMT02], and of previous quantum algorithms for the mean estimation problem. For our applications, we also adapt the variable-time amplitude amplification technique of Ambainis [Amb10] into a variable-time amplitude estimation algorithm, improving a recent result of Chakraborty, Gily\'en and Jeffery [CGJ18].