Filtered Spectral Projection for Quantum Principal Component Analysis

TL;DR

The paper introduces FSPA, an optimal quantum spectral projection algorithm that efficiently projects onto dominant spectral subspaces without explicit eigenvalue estimation, with broad applications and validated by numerical tests.

Contribution

FSPA is a novel, optimal quantum projection method that bypasses eigenvalue estimation, offering robustness, efficiency, and applicability to various quantum and classical datasets.

Findings

FSPA achieves an optimal oracle complexity with tight bounds.

FSPA demonstrates exponential copy-complexity advantage in the density matrix exponentiation model.

Numerical tests confirm FSPA's effectiveness on diverse datasets and its implementation validity.

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.