Quantum canonical correlation analysis algorithm

Abstract

Canonical correlation analysis (CCA) is a fundamental technique used to analyze data correlation in various fields, including video and medical data analysis. In this paper, we propose a quantum canonical correlation analysis (QCCA) algorithm. First, we introduce a combined density matrix representation method that transforms CCA into generalized eigenvalue decomposition. Moreover, to address the challenge of performing generalized eigenvalue decomposition in high-dimensional scenarios, we propose a quantum method for extracting the canonical principal axes. In this method, two sets of variables are transformed into a reduced density matrix, so that the product of variable matrices can be accelerated by phase estimation and controlled rotation. Complexity analysis shows that the QCCA algorithm achieves exponential acceleration in variable dimensions n, p and variable size m compared to classical algorithms. The QCCA algorithm serves as a foundation for the subsequent development of quantum algorithms for classification, regression, and other machine learning tasks.