Filtered Spectral Projection for Quantum Principal Component Analysis
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.