Quantum-assisted anomaly detection with multivariate Gaussian distribution

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.