  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.
