量子稳定鲁棒主成分分析：理论与NISQ体制下的证据
Quantum-Stable Robust Principal Component Analysis: Theory and Evidence from NISQ Regimes
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中文总结 AI 辅助
本文提出首个NISQ约束下的量子稳定RPCA算法，结合QSVT与QSA，建立六项理论保证，并通过Qiskit模拟验证其显著加速与竞争性精度。
中文摘要 AI 辅助
鲁棒主成分分析（RPCA）是从被稀疏异常和噪声破坏的数据中提取低秩结构的基本技术。虽然经典RPCA及其稳定变体已被广泛研究，但其计算成本随数据维度增长而变得难以承受。受量子机器学习新兴趋势的推动，本文提出了量子稳定RPCA，这是首个在含噪中等规模量子（NISQ）约束下进行鲁棒低秩和稀疏分解的量子算法。我们的方法将用于低秩恢复的量子奇异值阈值化（QSVT）与用于异常检测的量子稀疏逼近（QSA）相结合，同时显式建模量子硬件中固有的门误差、退相干误差和测量误差。我们建立了六项理论结果，涵盖恢复保证、可辨识性、对近似结构的鲁棒性、量子噪声韧性、交替最小化的收敛性以及泛化界，从而将经典RPCA理论扩展到量子领域。基于Qiskit的大量模拟证实，即使在现实NISQ噪声水平下，量子稳定RPCA相比经典求解器也能实现显著的运行时间改进，同时保持有竞争力的重建精度。这项工作为量子增强鲁棒学习提供了蓝图，桥接了机器学习和信号处理范式。
英文摘要
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.
发表机构
- Institute of Advancing Intelligence (IAI), TCG CREST（advancing intelligence 研究所 (IAI)，TCG CREST）
机构由 AI 辅助整理，请以论文原文为准。
