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 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.

Figures (5)

• Figure 1: Schematic illustration of FSPA. Repeated application of $\rho$ amplifies components proportionally to $\lambda_j^k$, exponentially suppressing subdominant eigenspaces. Renormalization removes magnitude dependence and yields projection onto the dominant eigenspace.
• Figure 2: Empirical validation of the gap-dependent oracle complexity of FSPA. The total number of oracle applications required to reach 99.99% fidelity is plotted against the theoretical scaling variable $1/\log(\lambda_1/\lambda_2)$. Linear regression confirms proportional scaling, consistent with the predicted complexity $\mathcal{O}\!\left(\frac{\log(1/\epsilon)}{\log(\lambda_1/\lambda_2)}\right)$. Classical power iteration is shown for comparison.
• Figure 3: Eigenvector instability versus subspace stability on the Breast Cancer Wisconsin datasetpedregosa2011scikit. The real-data covariance matrix is constructed from standardized diagnostic features. Small perturbations strongly rotate individual leading eigenvectors, while dominant-subspace fidelity remains stable. This illustrates why subspace-level metrics are the appropriate object of study in near-degenerate regimes.
• Figure 4: Uniform spectral rescaling at fixed gap. Lloyd-style qPCA collapses below a resolution threshold, while FSPA remains stable under global eigenvalue downscaling.
• Figure 5: Algorithmic regime map versus spectral gap. Phase-estimation-based methods show a sharp threshold behavior tied to finite resolution; FSPA degrades smoothly as the gap shrinks, consistent with gap-limited amplification.

Theorems & Definitions (4)

• Proposition 1: Eigenvalue Magnitude Invariance
• Proposition 2: Gap-Dependent Bias Amplification
• Proposition 3: Subspace Convergence
• Theorem 1: Oracle Complexity of FSPA