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.

翻译：密度矩阵描述了量子系统的统计状态。它是一种强大的形式化工具，既能表示量子系统的量子不确定性和经典不确定性，也能将测量、系统组合和期望等不同统计操作表示为线性代数运算。本文探讨了如何将密度矩阵作为机器学习模型的构建模块，利用其能够直观结合线性代数与概率论的特性。本文的主要成果之一在于证明，耦合了随机傅里叶特征的密度矩阵可以逼近$\mathbb{R}^n$上的任意概率分布。基于这一发现，本文构建了用于密度估计、分类和回归的不同模型。这些模型是可微的，因此能够与深度学习架构等其他可微组件集成，并通过基于梯度的优化来学习其参数。此外，本文还提出了基于估计和模型平均的无优化训练策略。在基准任务上对模型进行了评估，并报告和讨论了相关结果。