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 $O(polylogN)$, 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.
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 A\ensuremath∈RN\ifmmode×\else\texttimes\fiN with time complexity 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 is used for identifying data that deviate from ``normal'' data patterns. Its usage on classical data finds diverse applications in many important areas such as finance, fraud detection, medical diagnoses, data cleaning, and surveillance. With the advent of quantum technologies, anomaly detection of quantum data, in the form of quantum states, may become an important component of quantum applications. Machine-learning algorithms are playing pivotal roles in anomaly detection using classical data. Two widely used algorithms are the kernel principal component analysis and the one-class support vector machine. We find corresponding quantum algorithms to detect anomalies in quantum states. We show that these two quantum algorithms can be performed using resources that are logarithmic in the dimensionality of quantum states. For pure quantum states, these resources can also be logarithmic in the number of quantum states used for training the machine-learning algorithm. This makes these algorithms potentially applicable to big quantum data applications.
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.
A core task in quantum anomaly detection is to compute an anomaly score that quantifies how strongly a test quantum state deviates from a given quantum dataset assumed to be normal. Classically, principal component analysis (PCA) for centered data computes the anomaly score by evaluating the test sample relative to the subspace spanned by the selected leading eigenvectors. However, for quantum data that lack a standard centering, explicitly recovering principal eigenvectors, constructing full Gram matrices, or loading quantum-random-access-memory-style data can be more costly than estimating the anomaly score itself. To avoid these costs, we propose Quantum Spectral Anomaly Detection (QSPADE), which computes PCA-like anomaly scores directly from the spectrum of the average state of the normal dataset. By replacing hard PCA rank selection with a smooth, temperature-controlled spectral threshold, QSPADE makes near-threshold spectral components contribute partially to the anomaly score. This makes the score vary continuously rather than jump when a borderline component is included or excluded, and makes it less sensitive to noise or arbitrary hard cutoffs near the threshold. In the zero-temperature limit, QSPADE recovers the hard-projector PCA score. The proposed measurement-based quantum detector can be calibrated with a sample complexity independent of the data dimension. Numerical simulations show that QSPADE behaves like kernel-PCA on encoded classical data and detects changes across a transverse-field Ising transition without predefined order parameters. Consequently, QSPADE gives an efficient framework for both quantum-kernel anomaly detection on encoded classical data and the monitoring of quantum-native systems where diagnostic observables are unknown.
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.
Quantum computing, with its potential to enhance various machine learning tasks, allows significant advancements in kernel calculation and model precision. Utilizing the one-class Support Vector Machine alongside a quantum kernel, known for its classically challenging representational capacity, notable improvements in average precision compared to classical counterparts were observed in previous studies. Conventional calculations of these kernels, however, present a quadratic time complexity concerning data size, posing challenges in practical applications. To mitigate this, we explore two distinct approaches: utilizing randomized measurements to evaluate the quantum kernel and implementing the variable subsampling ensemble method, both targeting linear time complexity. Experimental results demonstrate a substantial reduction in training and inference times by up to 95\% and 25\% respectively, employing these methods. Although unstable, the average precision of randomized measurements discernibly surpasses that of the classical Radial Basis Function kernel, suggesting a promising direction for further research in scalable, efficient quantum computing applications in machine learning.
This paper evaluates the performance of two quantum machine learning algorithms: Quantum Support Vector Machine (QSVM) and Quantum Principal Component Analysis (Q-PCA). The scalability of QSVM is analyzed across increasing dataset sizes and feature dimensions, revealing faster-than-linear growth in kernel evaluation and SVM training times. However, this computational overhead is accompanied by a notable decline in classification accuracy, highlighting challenges in applying QSVM to larger or more complex datasets. In contrast, Q-PCA demonstrates near-linear scalability with dataset size and qubit count, leveraging the independence of quantum circuit execution and structured operations to maintain efficiency. A QNN-based implementation of Q-PCA further validates its ability to align data with principal components, maximize variance, and enable effective dimensionality reduction. These findings emphasize the potential of Q-PCA for efficient quantum-enhanced data analysis while underscoring the need for optimization in QSVM and further exploration of Q-PCA’s capabilities for real-world quantum computing applications.