Principal component analysis (PCA) is an important dimensionality reduction method in machine learning and data analysis. Recently, the quantum version of PCA has been established to diagonalize quantum states. Although these quantum algorithms promise quantum advantages, they require substantial resources beyond the reach of state-of-the-art quantum technologies. This work aims to reduce resource requirements and improve the efficiency of quantum PCA. Assuming that the quantum state is accessed through a purified quantum query model and a sampling model, we propose quantum algorithms that use minimal resource requirements for ancillary qubits to reveal properties of eigenvectors and eigenvalues of a state. In particular, we show that estimating eigenvalue λ with error ε and success probability larger than λ(1−η) requests a query complexity O~(ϵ−1) and a sample complexity O~(ϵ−2η−1) , respectively. To our knowledge, our result is the first quantum speedup that achieves asymptotic linear scaling in 1/ϵ for quantum PCA. As applications, we discuss estimating the minimum relative entropy of entanglement of bipartite pure-states and performing quantum state discrimination tasks. We show that quantum speedups are maintained when the pure state has a low Schmidt number and states of discrimination have a low rank. This study opens up a new quantum PCA method for high-dimensional quantum data analysis and discusses its application in quantum information processing tasks.

中文翻译：

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资源高效的量子主成分分析


主成分分析（PCA）是机器学习和数据分析中重要的降维方法。最近，PCA 的量子版本已经建立，用于对角化量子态。尽管这些量子算法有望带来量子优势，但它们需要大量资源，超出了最先进的量子技术的范围。这项工作旨在减少资源需求并提高量子 PCA 的效率。假设通过纯化的量子查询模型和采样模型访问量子态，我们提出了量子算法，该算法使用辅助量子位的最小资源需求来揭示状态的特征向量和特征值的属性。特别地，我们表明，以误差 ε 和成功概率大于 λ(1−η) 来估计特征值 λ 分别需要查询复杂度 O~(ϵ−1) 和样本复杂度 O~(ϵ−2η−1)。据我们所知，我们的结果是第一个在量子 PCA 中实现 1/ϵ 渐近线性缩放的量子加速。作为应用，我们讨论估计二分纯态纠缠的最小相对熵并执行量子态辨别任务。我们证明，当纯态具有较低的施密特数并且歧视态具有较低的等级时，量子加速得以维持。该研究为高维量子数据分析开辟了一种新的量子PCA方法，并讨论了其在量子信息处理任务中的应用。