Quantum algorithm for the covariance matrix preparation and its application

Abstract Performing the eigendecomposition of the covariance matrix of the dataset is of great significance in the field of machine learning. However, classical operations will become time-consuming when involving large data sets. In this paper, in order to address this problem, we design an efficient quantum algorithm to prepare the covariance matrix state by means of quantum amplitude estimation. After that, we research on its application in principal component analysis and Mahalanobis distance calculation. Specifically, we obtain the transformation matrix for quantum principal component analysis based on the singular value estimation algorithm and the amplitude amplification algorithm. Furthermore, we invoke the quantum matrix inversion algorithm to calculate the Mahalanobis distance. The final complexity analysis shows that our proposed algorithms can achieve speedup compared to their classical counterparts under certain conditions.