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arXiv:2505.02316

Quantum-assisted anomaly detection with multivariate Gaussian distribution

Chao‐Hua Yu, Rao, Hong-Miao, Yingyan Wu, Liu, De-Xi, Xiping Liu, Lin‐Chun Wan

Published May 5, 2025• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.2505.02316

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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

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, 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.

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.

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Abstract Performing the eigendecomposition of the covariance matrix of the dataset is of great significance in the field of machine learning. However, classical operations will become time-consuming when involving large data sets. In this paper, in order to address this problem, we design an efficient quantum algorithm to prepare the covariance matrix state by means of quantum amplitude estimation. After that, we research on its application in principal component analysis and Mahalanobis distance calculation. Specifically, we obtain the transformation matrix for quantum principal component analysis based on the singular value estimation algorithm and the amplitude amplification algorithm. Furthermore, we invoke the quantum matrix inversion algorithm to calculate the Mahalanobis distance. The final complexity analysis shows that our proposed algorithms can achieve speedup compared to their classical counterparts under certain conditions.