  We study two problems in high-dimensional robust statistics: \emph{robust
mean estimation} and \emph{outlier detection}. In robust mean estimation the
goal is to estimate the mean $\mu$ of a distribution on $\mathbb{R}^d$ given
$n$ independent samples, an $\varepsilon$-fraction of which have been corrupted
by a malicious adversary. In outlier detection the goal is to assign an
\emph{outlier score} to each element of a data set such that elements more
likely to be outliers are assigned higher scores. Our algorithms for both
problems are based on a new outlier scoring method we call QUE-scoring based on
\emph{quantum entropy regularization}. For robust mean estimation, this yields
the first algorithm with optimal error rates and nearly-linear running time
$\widetilde{O}(nd)$ in all parameters, improving on the previous fastest
running time $\widetilde{O}(\min(nd/\varepsilon^6, nd^2))$. For outlier
detection, we evaluate the performance of QUE-scoring via extensive experiments
on synthetic and real data, and demonstrate that it often performs better than
previously proposed algorithms. Code for these experiments is available at
https://github.com/twistedcubic/que-outlier-detection .
