Filtered Spectral Projection for Quantum Principal Component Analysis

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 goal is simpler: projection onto the dominant spectral subspace. Here we introduce a projection-first framework, the Filtered Spectral Projection Algorithm (FSPA), which bypasses explicit eigenvalue estimation while preserving the relevant spectral structure. FSPA amplifies any nonzero warm-start overlap with the leading subspace and remains robust in small-gap and near-degenerate regimes, without artificial symmetry breaking in the absence of bias. We show that FSPA achieves an oracle complexity $\mathcal{O}((\log(1/\epsilon)+\log(1/|a_1|^2))/\log(\lambda_1/\lambda_2))$,which is tight by a matching lower bound, establishing it as an\emph{optimal} projection primitive. We derive a convergence rate for degenerate spectra, give a circuit resource analysis with $n+\mathcal{O}(1)$ qubit overhead independent of system dimension, and extend the method to threshold spectral projection, Threshold-FSPA, which converges in $\mathcal{O}(\log(1/\epsilon))$ calls when the threshold lies between eigenvalues. In the density matrix exponentiation access model, FSPA gives an exponential copy-complexity advantage over classical methods. For classical datasets, we show that for amplitude-encoded centered data the ensemble density matrix $\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|$ equals the covariance matrix. Numerical tests on chemistry density matrices, noisy circuit outputs, Breast Cancer Wisconsin, handwritten Digits, and 1--4-qubit scalability confirm the theory. A minimal Qiskit implementation validates magnitude invariance, signal amplification, and no spurious symmetry breaking. These results establish FSPA as an optimal and deployable quantum spectral projection primitive.