  A density matrix describes the statistical state of a quantum system. It is a
powerful formalism to represent both the quantum and classical uncertainty of
quantum systems and to express different statistical operations such as
measurement, system combination and expectations as linear algebra operations.
This paper explores how density matrices can be used as a building block for
machine learning models exploiting their ability to straightforwardly combine
linear algebra and probability. One of the main results of the paper is to show
that density matrices coupled with random Fourier features could approximate
arbitrary probability distributions over $\mathbb{R}^n$. Based on this finding
the paper builds different models for density estimation, classification and
regression. These models are differentiable, so it is possible to integrate
them with other differentiable components, such as deep learning architectures
and to learn their parameters using gradient-based optimization. In addition,
the paper presents optimization-less training strategies based on estimation
and model averaging. The models are evaluated in benchmark tasks and the
results are reported and discussed.
