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理解量子计算机在处理实际问题（尤其是在机器学习领域）中的计算能力提供了新视角。过去几年中，针对各类任务的高效算法不断涌现，但其下界分析仍属空白。本文借助通信复杂度理论，首次提出了研究此类任务下界的方法。我们主要关注线性回归、监督聚类、主成分分析、推荐系统及哈密顿量模拟等问题的下界证明。针对这些问题，我们基于相关矩阵的弗罗贝尼乌斯范数证明了二次下界。由于量子算法在这些问题上具有弗罗贝尼乌斯范数的线性复杂度，我们的结果表明量子-经典计算分离度至少达到二次量级。作为方法拓展，我们进一步将量子通信复杂度应用于矩阵相关问题的量子查询算法下界分析，并给出了若干应用案例。