  The Expectation-Maximization (EM) algorithm is a fundamental tool in
unsupervised machine learning. It is often used as an efficient way to solve
Maximum Likelihood (ML) estimation problems, especially for models with latent
variables. It is also the algorithm of choice to fit mixture models: generative
models that represent unlabelled points originating from $k$ different
processes, as samples from $k$ multivariate distributions. In this work we
define and use a quantum version of EM to fit a Gaussian Mixture Model. Given
quantum access to a dataset of $n$ vectors of dimension $d$, our algorithm has
convergence and precision guarantees similar to the classical algorithm, but
the runtime is only polylogarithmic in the number of elements in the training
set, and is polynomial in other parameters - as the dimension of the feature
space, and the number of components in the mixture. We generalize further the
algorithm in two directions. First, we show how to fit any mixture model of
probability distributions in the exponential family. Then, we show how to use
this algorithm to compute the Maximum a Posteriori (MAP) estimate of a mixture
model: the Bayesian approach to likelihood estimation problems. We discuss the
performance of the algorithm on a dataset that is expected to be classified
successfully by this algorithm, arguing that on those cases we can give strong
guarantees on the runtime.
