提出一种更优的量子谱投影算法，直接聚焦主子空间投影，无需估算特征值。
Filtered Spectral Projection for Quantum Principal Component Analysis
- 先投影后处理，跳过特征值估计，保留核心谱结构。
- 在小间隙和近简并情况下仍稳定，复杂度达理论最优。
- 适合量子机器学习、化学模拟等需高效谱投影的场景。
量子主成分分析（qPCA）通常被定义为提取协方差编码密度算符的本征值与本征向量，但在许多实际场景中，目标仅是投影到主导谱子空间。本文提出一种投影优先框架——滤波谱投影算法（FSPA），避免显式本征值估计，同时保持关键谱结构。FSPA能放大非零初始重叠与主子空间的信号，在小本征值间隙和近简并情形下仍具鲁棒性，且无偏时不会引入人工对称性破缺。理论证明其预言复杂度为 $\mathcal{O}((\log(1/ε)+\log(1/|a_1|^2))/\log(λ_1/λ_2))$，并给出匹配下界，表明其为最优投影原语。针对退化谱推导收敛率，给出电路资源分析：仅需 $n+\mathcal{O}(1)$ 量子比特开销，与系统维度无关。扩展至阈值谱投影（Threshold-FSPA），当阈值位于本征值之间时，收敛于 $\mathcal{O}(\log(1/ε))$ 次调用。在密度矩阵指数访问模型下，相比经典方法实现指数级样本复杂度优势。对于经典数据集，证明幅度编码中心化数据对应的混合态密度矩阵 $ρ=\sum_i p_i|ψ_i\rangle\langleψ_i|$ 等于协方差矩阵。在化学密度矩阵、噪声电路输出、乳腺癌威斯康星、手写数字及1–4量子比特可扩展性测试中验证理论。最小Qiskit实现确认幅值不变性、信号增强及无虚假对称性破缺。结果确立FSPA为最优且可部署的量子谱投影原语。
原文摘要 · Abstract (English)
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 $\mathcal{O}((\log(1/ε)+\log(1/|a_1|^2))/\log(λ_1/λ_2))$,which is tight by a matching lower bound, establishing it as an\emph{optimal} projection primitive. We derive a convergence rate for degenerate spectra, give a circuit resource analysis with $n+\mathcal{O}(1)$ qubit overhead independent of system dimension, and extend the method to threshold spectral projection, Threshold-FSPA, which converges in $\mathcal{O}(\log(1/ε))$ 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 $ρ=\sum_i p_i|ψ_i\rangle\langleψ_i|$ 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.
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