# Quantum Anomaly Detection with Density Estimation and Multivariate Gaussian Distribution

## TL;DR

This paper introduces quantum algorithms for anomaly detection using density estimation and Gaussian models, achieving exponential improvements in resource complexity over classical methods, and includes quantum procedures for determinant estimation and quantum kernel methods.

## Contribution

The paper presents novel quantum algorithms for anomaly detection and related tasks with significantly reduced resource complexity compared to classical algorithms.

## Key findings

- Quantum algorithms have logarithmic resource complexity in data dimensions.
- Efficient quantum determinant estimation with polylogarithmic time complexity.
- Extension to quantum kernel PCA and quantum one-class SVM for classical data.

## Abstract

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 $\mathcal{A}\in\mathcal{R}^{N\times N}$ with time complexity $O(poly\log N)$ 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.