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arXiv:2201.10006 · Physical Review A

Quantum density estimation with density matrices: Application to quantum anomaly detection

Diego H. Useche, Oscar A. Bustos-Brinez, Joseph A. Gallego-Mejia, Fabio Augusto Gonzalez

Published Mar 19, 2024• 9 citations
arXivHTMLPDF

Abstract

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 dataset. 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 quantum-classical density-matrix density estimation model, called Q-DEMDE, based on the expected values of density matrices and a 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 an 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 datasets 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.

DOI
10.1103/physreva.109.032418

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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 dataset. 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 quantum-classical density-matrix density estimation model, called Q-DEMDE, based on the expected values of density matrices and a 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 an 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 datasets 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.
Density estimation is a fundamental task in statistics and machine learning that aims to estimate, from a set of samples, the probability density function of the distribution that generated them. There are different methods for addressing this problem but recently deep-neural density estimation methods have emerged as a powerful alternative. This paper presents a novel method for neural density estimation based on density matrices and adaptive Fourier features. Density matrices are commonly used in quantum mechanics to represent the quantum state of a physical system. In this work, they are used to estimate probability densities using an operation called quantum measurement. The proposed method can be trained without optimization using an averaging operation over the samples of the training dataset. It can also be integrated with deep learning architectures and trained using gradient descent. The performance of the proposed method was evaluated on a range of synthetic and real datasets and compared with fast kernel density estimation and state-of-the-art neural density estimation methods. The results demonstrate that the proposed method achieves competitive performance while being faster and more efficient than existing methods.
Abstract 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 Rn\mathbb {R}^{n}Rn 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.
Density estimation is a fundamental task in statistics and machine learning applications. Kernel density estimation is a powerful tool for non-parametric density estimation in low dimensions; however, its performance is poor in higher dimensions. Moreover, its prediction complexity scale linearly with more training data points. This paper presents a method for neural density estimation that can be seen as a type of kernel density estimation, but without the high prediction computational complexity. The method is based on density matrices, a formalism used in quantum mechanics, and adaptive Fourier features. The method can be trained without optimization, but it could be also integrated with deep learning architectures and trained using gradient descent. Thus, it could be seen as a form of neural density estimation method. The method was evaluated in different synthetic and real datasets, and its performance compared against state-of-the-art neural density estimation methods, obtaining competitive results.
We study quantum anomaly detection with density estimation and multivariate Gaussian distribution. Both algorithms are constructed using the standard gate-based model of quantum computing. Compared with the corresponding classical algorithms, the resource complexities of our quantum algorithm are logarithmic in the dimensionality of quantum states and the number of training quantum states. We also present a quantum procedure for efficiently estimating the determinant of any Hermitian operators $\mathcal{A}\ensuremath{\in}{\mathcal{R}}^{N\ifmmode\times\else\texttimes\fi{}N}$ with time complexity O(polylogN)O(polylogN)O(polylogN), which forms an important subroutine in our quantum anomaly detection with multivariate Gaussian distribution. Finally, our results also include the modified quantum kernel principal component analysis (PCA) and the quantum one-class support vector machine (SVM) for detecting classical data.
Anomaly detection with multivariate Gaussian distribution, which we refer to as Gassian anomaly detection (GAD), is a prominent task in data mining and machine learning. The core task of GAD is to obtain the mean value vector and the covariance matrix that characterize the probability density function of an unknown multivariate Gaussian distribution used to detect anomalies, which could be time-consuming when addressing a large dataset. Recently, several quantum algorithms have been proposed for GAD with substantial speedup over the classical GAD. However, they all require quantum phase estimation as key subroutines so that their quantum ciruits have long depth and are unfavorable in the noisy intermediate-scale and early fault-tolerant quantum eras. In this paper, we propose a quantum algorithm for GAD biult on arithmetic-free black-box quantum state preparation (AFQSP), which significantly shortens the quantum circuit depth and reduces the burden of quantum hardware. Specifically, we take advantage of AFQSP to estimate the magnitude of every mean value and that of every covariance matrix element in the classical form, and develop a technique referred to as Hadamard sign test to further reveal their signs, so that anomaly detection of any data point can be done immediately on a classical computer at little cost. It is shown that our quantum algorithm for GAD achieves exponential speedup over the classical GAD when handling low-dimensional datasets with well-conditioned data matrices, and is also time competitive compared to the prior quantum algorithms for GAD. Moreover, our algorithm releases the requirements of input data being quantum, mean centered, or feature correlated in the prior quantum algorithms for GAD, meaning that our algorithm is more practical on input data. Our work highlights the role of AFQSP in bringing quantum machine learning closer to reality.
In an era where data underpins decision-making across science, politics, and economics, ensuring high data quality is of paramount importance. Conventional computing algorithms for enhancing data quality, including anomaly detection, demand substantial computational resources, lengthy processing times, and extensive training datasets. This work aims to explore the potential advantages of quantum computing for enhancing data quality, with a particular focus on detection. We begin by examining quantum techniques that could replace key subroutines in conventional anomaly detection frameworks to mitigate their computational intensity. We then provide practical demonstrations of quantum-based anomaly detection methods, highlighting their capabilities. We present a technical implementation for detecting volatility regime changes in stock market data using quantum reservoir computing, which is a special type of quantum machine learning model. The experimental results indicate that quantum-based embeddings are a competitive alternative to classical ones in this particular example. Finally, we identify unresolved challenges and limitations in applying quantum computing to data quality tasks. Our findings open up new avenues for innovative research and commercial applications that aim to advance data quality through quantum technologies.