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arXiv 1910.06151cs.DScs.LGquant-ph
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用于去量子化量子机器学习的基于采样的次线性低秩矩阵算术框架

Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning

University of Texas at Austin（德克萨斯大学奥斯汀分校）；Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）；California Institute of Technology（加州理工学院）；University of Maryland（马里兰大学）；University of Washington（华盛顿大学）

Nai-Hui Chia, András Gilyén, Tongyang Li, Han-Hsuan Lin, Ewin Tang, Chunhao Wang

备注：
77 pages, 2 figures. v2: revised to add more connection to QSVT, improve existing results. v3: revised structure, introduction rewritten for clarity. v4: minor correction to regression result
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AI总结：

本文提出基于ℓ²-范数采样的经典SVT框架，在近似低秩矩阵上实现与维度无关的矩阵运算，推广并改进多类量子机器学习算法的去量子化结果。

摘要：

我们提出了一个用于在近似低秩矩阵上运行量子启发经典算法的算法框架，推广了由 Tang 针对推荐系统的突破性量子启发算法所开启的一系列结果［STOC'19］。受量子线性代数算法以及 Gilyén、Su、Low 和 Wiebe［STOC'19］提出的量子奇异值变换（SVT）框架启发，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。我们的结果有力地表明，在相应的 QRAM 数据结构输入模型中，量子 SVT 并不会带来指数级量子加速。由于量子 SVT 框架基本上概括了所有已知的量子线性代数技术，因此我们的结果结合先前工作中的采样引理，足以推广近期所有关于量子机器学习算法去量子化的结果。特别是，我们的经典 SVT 框架恢复并通常改进了推荐系统、主成分分析、监督聚类、支持向量机、低秩回归以及半定规划求解方面的去量子化结果。我们还给出了关于低秩 Hamiltonian 模拟和判别分析的额外去量子化结果。我们的改进来自识别出量子启发输入模型中作为所有先前量子启发结果核心的关键特征：ℓ²-范数采样能够在与矩阵乘积维度无关的时间内近似这些乘积。我们将所有主要结果归结为这一事实，使阐述简洁、自足且直观。

英文摘要：

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 Gilyén, 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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