# Filtered Spectral Projection for Quantum Principal Component Analysis

arXiv: https://arxiv.org/abs/2603.13441 · HTML: https://mlsanity.com/paper/2603.13441/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 17 references that are in the map.
- First posted on arXiv 13 Mar 2026.

## Builds on (most cited first, 2 of 2)

- [Quantum principal component analysis only achieves an exponential speedup because of its state preparation assumptions](https://mlsanity.com/paper/1811.00414/) (2018)
- [Covariance matrix preparation for quantum principal component analysis](https://mlsanity.com/paper/2204.03495/) (2022)

## Built on by (most cited first, 1 of 1)

- [Deep Learning for Semen Analysis in Male Infertility: Computer Vision, Multimodal Fusion, and Clinical Translation](https://mlsanity.com/paper/2607.05311/) (2026)

## Abstract (arXiv)

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 O((log(1/ε)+log(1/|a1|2))/log(λ1/λ2)),which is tight by a matching lower bound, establishing it as anoptimal projection primitive. We derive a convergence rate for degenerate spectra, give a circuit resource analysis with n+O(1) qubit overhead independent of system dimension, and extend the method to threshold spectral projection, Threshold-FSPA, which converges in O(log(1/ε)) 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 ρ=sumi pi|psiiranglelanglepsii| 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.