We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang's breakthrough quantum-inspired algorithm for recommendation systems [STOC'19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gily\'en, Su, Low, and Wiebe [STOC'19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffice to generalize all recent results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: $\ell^2$-norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.

翻译：我们提出了一种用于近低秩矩阵的受量子启发的经典算法框架，推广了由Tang在推荐系统方面的突破性量子启发算法[STOC'19]所引发的系列成果。受量子线性代数算法以及Gilyén、Su、Low和Wiebe提出的量子奇异值变换（SVT）框架[STOC'19]的启发，我们在适当的量子启发采样假设下，开发了运行时间与输入维度无关的经典SVT算法。我们的结果提供了有力证据，表明在相应的QRAM数据结构输入模型下，量子SVT并未带来指数级量子加速。由于量子SVT框架本质上概括了所有已知的量子线性代数技术，我们的结果结合先前工作中的采样引理，足以概括所有近期关于去量子化量子机器学习算法的成果。特别地，我们的经典SVT框架恢复并常能改进推荐系统、主成分分析、监督聚类、支持向量机、低秩回归和半定规划求解等问题的去量子化结果。我们还给出了低秩哈密顿量模拟和判别分析方面的额外去量子化结果。我们的改进源于识别出量子启发输入模型的核心关键特征（这也是所有先前量子启发结果的基石）：ℓ²范数采样可在与矩阵维度无关的时间内近似矩阵乘积。我们将所有主要结果归结于这一事实，使论述更加简洁、自洽且直观。