Quantum-inspired classical algorithms provide us with a new way to understand the computational power of quantum computers for practically-relevant problems, especially in machine learning. In the past several years, numerous efficient algorithms for various tasks have been found, while an analysis of lower bounds is still missing. Using communication complexity, in this work we propose the first method to study lower bounds for these tasks. We mainly focus on lower bounds for solving linear regressions, supervised clustering, principal component analysis, recommendation systems, and Hamiltonian simulations. For those problems, we prove a quadratic lower bound in terms of the Frobenius norm of the underlying matrix. As quantum algorithms are linear in the Frobenius norm for those problems, our results mean that the quantum-classical separation is at least quadratic. As a generalisation, we extend our method to study lower bounds analysis of quantum query algorithms for matrix-related problems using quantum communication complexity. Some applications are given.

翻译：量子启发经典算法为我们提供了理解量子计算机在实用相关问题（特别是机器学习）上计算能力的新途径。过去数年间，针对各类任务已有大量高效算法被提出，但下界分析仍属空白。本研究首次提出利用通信复杂度来研究这些任务下界的方法。我们重点分析了解线性回归、有监督聚类、主成分分析、推荐系统和哈密顿量模拟的下界。针对这些问题，我们证明了基于底层矩阵弗罗贝尼乌斯范数的二次下界。由于量子算法在这些问题上的复杂度与弗罗贝尼乌斯范数呈线性关系，我们的结果表明量子-经典分离至少是二次的。作为推广，我们将该方法拓展至利用量子通信复杂度分析矩阵相关问题量子查询算法的下界，并给出了若干应用实例。