Quantum-Stable Robust Principal Component Analysis: Theory and Evidence from NISQ Regimes

Abstract

Robust Principal Component Analysis (RPCA) is a fundamental technique for extracting low-rank structures from data corrupted by sparse anomalies and noise. While classical RPCA and its stable variants have been widely studied, their computational cost grows prohibitively with data dimensionality. Motivated by emerging trends in quantum machine learning, this paper introduces Quantum-Stable RPCA, the first quantum algorithm for robust low-rank and sparse decomposition under noisy intermediate-scale quantum (NISQ) constraints. Our approach integrates Quantum Singular Value Thresholding (QSVT) for low-rank recovery with Quantum Sparse Approximation (QSA) for anomaly detection, while explicitly modeling gate, decoherence, and measurement errors inherent in quantum hardware. We establish six theoretical results covering recovery guarantees, identifiability, robustness to approximate structure, quantum noise resilience, convergence of alternating minimization, and generalization bounds, thereby extending classical RPCA theory to the quantum regime. Extensive Qiskit-based simulations confirm that Quantum-Stable RPCA delivers significant runtime improvements over classical solvers while maintaining competitive reconstruction accuracy, even under realistic NISQ noise levels. This work provides a blueprint for quantum-enhanced robust learning, bridging machine learning and signal processing paradigms.

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