Filtered Spectral Projection for Quantum Principal Component Analysis

2026-03-13

AI Summary

Traditional quantum principal component analysis (qPCA) explicitly estimates eigenvalues and eigenvectors, yet many applications only require projecting data onto the principal spectral subspace. This work proposes the Filtered Spectral Projection Algorithm (FSPA), which forgoes explicit eigenvalue estimation and instead centers on direct spectral projection to preserve dominant spectral structure while amplifying initial state overlap. FSPA remains robust in scenarios with small spectral gaps or near-degeneracies without requiring artificial symmetry-breaking perturbations. Leveraging the equivalence among amplitude encoding, density matrices, and covariance matrices, along with eigenvalue interlacing bounds, the method demonstrates stable projection quality and downstream task performance on benchmark datasets such as Breast Cancer Wisconsin and handwritten digits, indicating that spectral projection alone suffices for most qPCA applications.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

Quantum principal component analysis (qPCA) is commonly formulated as the extraction of eigenvalues and eigenvectors of a covariance-encoded density operator. Yet in many qPCA settings, the practical objective is simpler: projecting data onto the dominant spectral subspace. In this work, we introduce a projection-first framework, the Filtered Spectral Projection Algorithm (FSPA), which bypasses explicit eigenvalue estimation while preserving the essential spectral structure. FSPA amplifies any nonzero warm-start overlap with the leading principal subspace and remains robust in small-gap and near-degenerate regimes without inducing artificial symmetry breaking in the absence of bias. To connect this approach to classical datasets, we show that for amplitude-encoded centered data, the ensemble density matrix $ρ=\sum_i p_i|ψ_i\rangle\langleψ_i|$ coincides with the covariance matrix. For uncentered data, $ρ$ corresponds to PCA without centering, and we derive eigenvalue interlacing bounds quantifying the deviation from standard PCA. We further show that ensembles of quantum states admit an equivalent centered covariance interpretation. Numerical demonstrations on benchmark datasets, including Breast Cancer Wisconsin and handwritten Digits, show that downstream performance remains stable whenever projection quality is preserved. These results suggest that, in a broad class of qPCA settings, spectral projection is the essential primitive, and explicit eigenvalue estimation is often unnecessary.

Problem

• quantum principal component analysis
• spectral projection
• eigenvalue estimation
• density operator
• principal subspace

Innovation

• Filtered Spectral Projection
• quantum PCA
• spectral subspace projection
• eigenvalue-free algorithm
• amplitude-encoded data