Several quantum algorithms for linear algebra problems, and in particular quantum machine learning problems, have been "dequantized" in the past few years. These dequantization results typically hold when classical algorithms can access the data via length-squared sampling. In this work we investigate how robust these dequantization results are. We introduce the notion of approximate length-squared sampling, where classical algorithms are only able to sample from a distribution close to the ideal distribution in total variation distance. While quantum algorithms are natively robust against small perturbations, current techniques in dequantization are not. Our main technical contribution is showing how many techniques from randomized linear algebra can be adapted to work under this weaker assumption as well. We then use these techniques to show that the recent low-rank dequantization framework by Chia, Gily\'en, Li, Lin, Tang and Wang (JACM 2022) and the dequantization framework for sparse matrices by Gharibian and Le Gall (STOC 2022), which are both based on the Quantum Singular Value Transformation, can be generalized to the case of approximate length-squared sampling access to the input. We also apply these results to obtain a robust dequantization of many quantum machine learning algorithms, including quantum algorithms for recommendation systems, supervised clustering and low-rank matrix inversion.

翻译：近年来，针对线性代数问题（尤其是量子机器学习问题）的若干量子算法已被“去量子化”。这些去量子化结果通常成立的前提是经典算法能够通过长度平方采样访问数据。本文研究了这些去量子化结果的鲁棒性。我们引入了近似长度平方采样的概念，即经典算法仅能从一个与理想分布总变差距离较小的分布中进行采样。尽管量子算法天然对小扰动具有鲁棒性，但当前的去量子化技术却缺乏这种性质。我们的主要技术贡献是展示了如何将随机线性代数中的多种技术适配至这一较弱假设下。进而利用这些技术证明：Chia、Gilyén、Li、Lin、Tang与Wang（JACM 2022）近期提出的低秩去量子化框架，以及Gharibian与Le Gall（STOC 2022）提出的稀疏矩阵去量子化框架（两者均基于量子奇异值变换），可推广至输入数据仅能通过近似长度平方采样访问的情形。我们还应用这些结果，实现了对包括推荐系统量子算法、监督聚类量子算法及低秩矩阵求逆量子算法在内的多种量子机器学习算法的鲁棒性去量子化。