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Zhongguo kexue. Wulixue Lixue Tianwenxue

Quantum algorithm for angle-based anomaly detection

Hongmiao Rao, Chaohua Yu, Yingpei Wu, Dexi Liu, Xiping Liu

Published Nov 6, 2024• 0 citations
Publisher

Abstract

Anomaly detection is a prominent task in machine learning and data mining, which plays a key role in various domains such as financial fraud detection, network intrusion detection, and healthcare. Angle-based outlier detection (ABOD) algorithm stands out as one of the most common approaches for anomaly detection, the core step of which involves computing the angular variance among distance vectors from each point to others, and then marking those points with angular variance lower than a threshold as outliers. The time complexity of classical ABOD algorithm scales as mathcalOM³) with respect to the number of samples M, making it hard to efficiently handle large-scale datasets. To this end, a quantum ABOD algorithm is proposed, in which quantum amplitude estimation is harnessed to compute angular variance in quantum parallel, and quantum amplitude amplification is further applied to search out the outliers. Compared with its classical counterpart, our quantum ABOD algorithm has time complexity Ołeft[\sqrt{M}\log(M)\right] with respect to M, achieving a polynomial speedup with respect to M.

DOI
10.1360/sspma-2024-0471

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Anomaly detection is a prominent task in machine learning and data mining, which plays a key role in various domains such as financial fraud detection, network intrusion detection, and healthcare. Angle-based outlier detection (ABOD) algorithm stands out as one of the most common approaches for anomaly detection, the core step of which involves computing the angular variance among distance vectors from each point to others, and then marking those points with angular variance lower than a threshold as outliers. The time complexity of classical ABOD algorithm scales as mathcalOM³) with respect to the number of samples M, making it hard to efficiently handle large-scale datasets. To this end, a quantum ABOD algorithm is proposed, in which quantum amplitude estimation is harnessed to compute angular variance in quantum parallel, and quantum amplitude amplification is further applied to search out the outliers. Compared with its classical counterpart, our quantum ABOD algorithm has time complexity Ołeft[\sqrt{M}\log(M)\right] with respect to M, achieving a polynomial speedup with respect to M.
Anomaly detection plays a crucial role in identifying rare and critical events, anomalies, or outliers that are not part of the normal data distribution. With the large volume of modern data and the widespread use of sensors and IoT devices, datasets have evolved into high-dimensional structures, posing significant challenges to anomaly detection tasks. In this work, we investigate to employ a hybrid quantum-classical algorithm to help handle such challenges. It is mainly based on Local Outlier Factor (LOF) algorithm to detect anomalies. However, we enhance distance estimation and k-nearest neighbor search via quantum algorithms. The proof-of-principle numerical results demonstrate the performance of the proposed method is competitive to previous algorithm.
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.
Abstract Anomaly detection of sequences is a hot topic in data mining. Anomaly detection using piecewise aggregate approximation in the amplitude domain (called ADPAAD) is one of the widely used methods in anomaly detection of sequences. The core step in the classical algorithm for performing ADPAAD is to construct an approximate representation of the subsequence, where the elements of each subsequence are divided into several subsections according to the amplitude domain, and then the average of the subsections is computed. It is computationally expensive when processing large‐scale sequences. In this paper, a quantum algorithm for ADPAAD is proposed, which can divide the subsequence elements and compute the average in parallel. The quantum algorithm can achieve polynomial speedups on the number of subsequences and the length of subsequences over its classical counterpart.
The density peak anomaly detection algorithm based on KNN, one of the most frequently utilized classical algorithms, is widely applied in communication fields, such as network fault detection, network traffic detection, user behavior detection, and network intrusion detection. In this algorithm, the local density value and KNN distance of all data points must be calculated, which faces enormous computational cost and computational delay challenges when the data set is processed as the network scale increases. To address this trouble, we introduce quantum computing into the algorithm and propose a quantum version of the density peak anomaly detection algorithm based on KNN. The proposed quantum algorithm first utilizes the inner product estimate algorithm and the quantum minimum search algorithm to accelerate the determination of the K nearest neighbor set of each data point and the calculation of KNN distance. Then, it performs amplitude estimation to concurrently obtain the local density of all data points. Our quantum algorithm is shown to be exponentially faster than its classical counterpart in the dimensionality of the data points and polynomially faster than the classical algorithm in the number of data points.
Anomaly detection, the task of identifying rare and unusual events within a dataset, is a critical component in numerous applications, including fraud detection, cybersecurity, and industrial monitoring. Traditional anomaly detection methods often rely on classical machine learning algorithms which can be sensitive to noise and may struggle with complex, high-dimensional data. This paper investigates the application of quantum machine learning (QML) algorithms to enhance anomaly detection performance. We propose a framework where anomaly detection problems are formulated as quantum machine learning tasks, leveraging quantum algorithms for data analysis and pattern recognition. The core claim of this work is that QML can significantly improve the precision and efficiency of anomaly detection compared to conventional techniques. Specifically, we explore the potential of quantum support vector machines (QSVMs) and quantum clustering algorithms for anomaly identification. The theoretical underpinnings of these approaches are outlined, and we present a conceptual model demonstrating the transformation of a classical anomaly detection problem into a quantum equivalent. This work lays the groundwork for future research in applying QML to real-world anomaly detection challenges.
This work explores the potential of Quantum Matrix Multiplication (QMM) to accelerate several computational tasks, demonstrating substantial speedups. We present three distinct applications showcasing QMM’s versatility and efficiency. We introduce a novel Hybrid-Quantum Angle Based Outlier Detection (H-QABOD) algorithm, leveraging QMM to efficiently identify outliers in numerical datasets. We evaluate H-QABOD’s performance on both synthetic and real-world datasets, benchmarking it against its classical counterpart. Furthermore, we investigate QMM’s application within quantum linear algebra, specifically for multitrace estimation and Frobenius norm computation.