量子密度估计：密度矩阵方法及其在量子异常检测中的应用
Quantum density estimation with density matrices: Application to quantum anomaly detection
AI总结：
本文提出基于密度矩阵期望值和量子傅里叶特征的量子-经典密度估计模型Q-DEMDE，并应用于异常检测，在模拟器和真实量子计算机上验证了高性能。
摘要：
密度估计是统计学和机器学习中的核心任务。该问题旨在确定与观测数据集最匹配的潜在概率密度函数。其应用包括统计推断、无监督学习和异常检测。尽管其重要性，很少有工作探索量子计算在密度估计中的应用。在本文中，我们提出了一种新颖的量子-经典密度矩阵密度估计模型，称为Q-DEMDE，基于密度矩阵的期望值和一种新颖的量子嵌入，称为量子傅里叶特征。该方法利用量子硬件通过混合量子态构建训练数据的概率分布。作为核心子程序，我们提出了一种新算法，在量子计算机上从其谱分解估计混合密度矩阵的期望值。此外，我们展示了该方法在量子-经典异常检测中的应用。我们在量子模拟器和真实量子计算机上，使用量子随机和量子自适应傅里叶特征，对不同数据集评估了密度估计模型。这项工作的一项重要结果是表明，在当今的量子计算机上，可以以高性能执行密度估计和异常检测。
英文摘要：
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.
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