Quantum principal component analysis without eigenvector recovery

2026-05-27

AI Summary

This work addresses the computational inefficiency, sensitivity to small eigenvalue gaps, and redundant steps of classical principal component analysis (PCA) in high-dimensional and quantum settings, particularly when only principal subspace scores are required. The authors propose a measurement-based soft PCA framework that reformulates PCA as a calibrated measurement task without explicitly recovering eigenvectors. By replacing hard top-k projection with an entropy-regularized Fermi–Dirac filter, the method casts PCA for the first time as a quantum measurement problem. It requires only a single fixed quantum circuit and threshold calibration to accommodate varying rank budgets or variance retention targets, while supporting coherent centering of quantum data. The approach enjoys dimension-independent sample complexity $O(\eta^{-2})$, enabling efficient estimation—with additive accuracy $\eta$—of normalized score rank, retained variance, spectral energy distribution, and post-selected filtered states.

Abstract

Principal component analysis (PCA) is traditionally implemented through a covariance or kernel matrix, leading-eigenvector extraction, and hard rank-$k$ projection. These steps can be computationally costly in high-dimensional and quantum-data settings, sensitive to small eigengaps, and unnecessary when downstream tasks only require principal-subspace scores. Such score-based objectives are important in applications such as anomaly detection, spectral-energy profiling, and other postselection tasks. To address these needs, we introduce a measurement-based soft PCA framework replacing the hard top-$k$ projector with an entropy-regularized Fermi--Dirac filter. This filter is the unique optimizer of an entropy-regularized variational formulation of PCA and converges to the classical PCA projector in the zero-temperature limit. This filter has a direct interpretation as a quantum measurement, which naturally suggests a quantum approach. For centered covariance operators represented by quantum feature states, a single fixed circuit, together with threshold calibration, accesses all optimal filters for different rank budgets or retained-variance levels without rank-dependent circuit updates or eigenvector recovery. For new inputs, the same calibrated quantum circuit yields soft principal subspace scores, spectral energy profiles, and postselected filtered states. The required centering of both training and test data is performed coherently inside the quantum protocol, which is particularly important for quantum data where no classical feature vectors or centered Gram matrix are directly available. By reframing PCA as a calibrated measurement task, this framework bypasses the need for iterative eigenvector extraction and achieves a dimension-independent sample complexity $O(η^{-2})$ for normalized fractional-rank or retained variance scoring at additive accuracy $η$.

Problem

• quantum PCA
• eigenvector recovery
• principal subspace scores
• high-dimensional data
• quantum data

Innovation

• quantum PCA
• Fermi–Dirac filter
• entropy-regularized variational formulation
• measurement-based framework
• dimension-independent sample complexity