Quantum algorithms for anomaly detection using amplitude estimation

TL;DR

This paper introduces a new quantum anomaly detection algorithm based on amplitude estimation, achieving exponential speedup on training data size, and extends the approach to kernel PCA-based detection.

Contribution

The paper corrects previous assumptions about quantum anomaly detection algorithms and proposes a novel quantum algorithm with exponential speedup on data size.

Findings

Achieves exponential speedup on the number of training data points M.

Identifies issues with previous quantum ADDE algorithms.

Extends quantum approach to kernel PCA-based anomaly detection.

Abstract

Anomaly detection plays a critical role in fraud detection, health care, intrusion detection, military surveillance, etc. Anomaly detection algorithm based on density estimation (called ADDE algorithm) is one of widely used algorithms. Liang et al. proposed a quantum version of the ADDE algorithm [Phys. Rev. A 99, 052310 (2019)] and it is believed that the algorithm has exponential speedups on both the number and the dimension of training data point over the classical algorithm. In this paper, we find that Liang et al.'s algorithm doesn't actually execute. Then we propose a new quantum ADDE algorithm based on amplitude estimation. It is shown that our algorithm can achieves exponential speedup on the number $M$ of training data points compared with the classical counterpart. Besides, the idea of our algorithm can be applied to optimize the anomaly detection algorithm based on kernel principal component analysis (called ADKPCA algorithm). Different from the quantum ADKPCA proposed by Liu et al. [Phys. Rev. A 97, 042315 (2018)], compared with the classical counterpart, which offer exponential speedup on the dimension $d$ of data points, our algorithm achieves exponential speedup on $M$.