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 $η$.

翻译：主成分分析（PCA）传统上通过协方差或核矩阵、主导特征向量提取以及硬秩-$k$投影来实现。这些步骤在高维和量子数据场景中计算成本高昂，对小特征间隙敏感，且当下游任务仅需主成分子空间得分时非必要。此类基于得分的目标在异常检测、光谱能量分析及其他后选择任务中具有重要应用。为满足这些需求，我们提出一种基于测量的软PCA框架，以熵正则化的费米-狄拉克滤波器替代硬性top-$k$投影器。该滤波器是PCA熵正则化变分形式的唯一优化器，并在零温极限下收敛至经典PCA投影器。此滤波器可直接解释为量子测量，自然引导出量子方法。对于由量子特征态表示的中心化协方差算子，单个固定电路配合阈值校准即可访问不同秩预算或保留方差水平下的所有最优滤波器，无需依赖秩的电路更新或特征向量恢复。对于新输入，同一校准后的量子电路可生成软主成分子空间得分、光谱能量分布及后选择滤波态。训练和测试数据所需的中心化操作在量子协议内相干完成，这对无法直接获取经典特征向量或中心化格拉姆矩阵的量子数据尤为重要。通过将PCA重新定义为校准测量任务，本框架绕过了迭代特征向量提取的需求，并在归一化分数秩或保留方差评分中实现了与维度无关的样本复杂度 $O(η^{-2})$（加性精度 $η$）。