Quantum Non-Identical Mean Estimation: Efficient Algorithms and Fundamental Limits

TL;DR

This paper explores quantum algorithms for mean estimation with non-identical samples, demonstrating quadratic speed-ups under certain conditions and establishing fundamental lower bounds for the problem.

Contribution

It introduces quantum mean estimators that achieve quadratic speed-up for non-identical distributions and proves fundamental lower bounds, advancing understanding of quantum query complexity.

Findings

Quantum algorithms achieve quadratic speed-up for bounded and sub-Gaussian variables.

Fundamental lower bounds show no quadratic speed-up possible in general for non-identical samples.

Techniques include reduction to Bernoulli variables and adversarial oracle simulation.

Abstract

We systematically investigate quantum algorithms and lower bounds for mean estimation given query access to non-identically distributed samples. On the one hand, we give quantum mean estimators with quadratic quantum speed-up given samples from different bounded or sub-Gaussian random variables. On the other hand, we prove that, in general, it is impossible for any quantum algorithm to achieve quadratic speed-up over the number of classical samples needed to estimate the mean $\mu$, where the samples come from different random variables with mean close to $\mu$. Technically, our quantum algorithms reduce bounded and sub-Gaussian random variables to the Bernoulli case, and use an uncomputation trick to overcome the challenge that direct amplitude estimation does not work with non-identical query access. Our quantum query lower bounds are established by simulating non-identical oracles by parallel oracles, and also by an adversarial method with non-identical oracles. Both results pave the way for proving quantum query lower bounds with non-identical oracles in general, which may be of independent interest.