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.

翻译：本文系统研究了在非相同分布样本查询访问下均值估计的量子算法与下界。一方面，我们针对来自不同有界或次高斯随机变量的样本，提出了具有二次量子加速的量子均值估计器。另一方面，我们证明：一般而言，当样本来自均值靠近$\mu$的不同随机变量时，任何量子算法都无法在经典样本数量需求上实现二次加速——即达到估计均值$\mu$所需的经典样本量。技术层面，我们的量子算法将非相同有界与次高斯随机变量归约为伯努利情形，并采用反计算技巧克服非相同查询访问下直接幅度估计失效的难题。量子查询下界则通过并行预言机模拟非相同预言机，以及基于非相同预言机的对抗方法建立。这两项工作为一般性非相同预言机下量子查询下界的证明开辟了道路，相关结论可能具有独立研究价值。