  Density estimation is a central task in statistics and machine learning. This
problem aims to determine the underlying probability density function that best
aligns with an observed data set. Some of its applications include statistical
inference, unsupervised learning, and anomaly detection. Despite its relevance,
few works have explored the application of quantum computing to density
estimation. In this article, we present a novel quantum-classical density
matrix density estimation model, called Q-DEMDE, based on the expected values
of density matrices and a novel quantum embedding called quantum Fourier
features. The method uses quantum hardware to build probability distributions
of training data via mixed quantum states. As a core subroutine, we propose a
new algorithm to estimate the expected value of a mixed density matrix from its
spectral decomposition on a quantum computer. In addition, we present an
application of the method for quantum-classical anomaly detection. We evaluated
the density estimation model with quantum random and quantum adaptive Fourier
features on different data sets on a quantum simulator and a real quantum
computer. An important result of this work is to show that it is possible to
perform density estimation and anomaly detection with high performance on
present-day quantum computers.
