Covariance Matrix Preparation for Quantum Principal Component Analysis

Principal component analysis (PCA) is a dimensionality reduction method in data that involves diagonalizing the covariance matrix of dataset. Recently, quantum algorithms have been formulated for PCA based on density matrix. These assume can be encoded matrix, but concrete protocol this encoding has lacking. Our work aims to address gap. Assuming amplitude data, with given by ensemble {pi,|ψi⟩}, then one easily prepare average ρ¯=∑ipi|ψi⟩⟨ψi|. We first show ρ¯ precisely whenever dataset centered. For datasets, we exploit global phase symmetry argue there always exists centered consistent ρ¯, and hence interpreted as This provides simple means preparing arbitrary datasets or classical datasets. uncentered our so-called “PCA without centering,” which interpret symmetrized closely corresponds standard PCA, derive equations inequalities bound deviation spectrum obtained from PCA. numerically illustrate Modified National Institute Standards Technology (MNIST) handwritten digit also natural meaningful, implement molecular ground-state datasets.1 MoreReceived 14 April 2022Revised 5 July 2022Accepted 16 August 2022DOI:https://doi.org/10.1103/PRXQuantum.3.030334Published American Physical Society under terms Creative Commons Attribution 4.0 International license. Further distribution must maintain attribution author(s) published article's title, journal citation, DOI.Published SocietyPhysics Subject Headings (PhySH)Research AreasMachine learningQuantum algorithmsQuantum computationQuantum Information