  In this study, we give an extension of Montanaro's arXiv/archive:1504.06987
quantum Monte Carlo method, tailored for computing expected values of random
variables that exhibit infinite variance. This addresses a challenge in
analyzing heavy-tailed distributions, which are commonly encountered in various
scientific and engineering fields. Our quantum algorithm efficiently estimates
means for variables with a finite $(1+\delta)^{\text{th}}$ moment, where
$\delta$ lies between 0 and 1. It provides a quadratic speedup over the
classical Monte Carlo method in both the accuracy parameter $\epsilon$ and the
specified moment of the distribution. We establish both classical and quantum
lower bounds, showcasing the near-optimal efficiency of our algorithm among
quantum methods. Our work focuses not on creating new algorithms, but on
analyzing the execution of existing algorithms with available additional
information about the random variable. Additionally, we categorize these
scenarios and demonstrate a hierarchy in the types of supplementary information
that can be provided.
