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. More precisely, we show that for linear regressions, in the row-sparse case, the lower bound is quadratic in the Frobenius norm of the underlying matrix, which is tight. In the dense case, with an extra assumption on the accuracy we obtain that the lower bound is quartic in the Frobenius norm, which matches the upper bound. For supervised clustering, we obtain a tight lower bound that is quartic in the Frobenius norm. For the other three tasks, we obtain a lower bound that is quadratic in the Frobenius norm, and the known upper bound is quartic in the Frobenius norm. Through this research, we find that large quantum speedup can exist for sparse, high-rank, well-conditioned matrix-related problems. Finally, we extend our method to study lower bounds analysis of quantum query algorithms for matrix-related problems. 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实际问题（尤其是机器学习问题）上的计算能力提供了新视角。过去数年间，研究者已发现多种针对不同任务的高效算法，但下界分析仍较为缺失。本文利用通信复杂度，首次提出研究这些任务下界的方法体系。我们重点分析了线性回归、监督聚类、主成分分析、推荐系统及哈密顿模拟的下界。具体而言：对于行稀疏线性回归，我们证明下界为基础矩阵Frobenius范数的二次函数，且该下界是紧的；对于稠密情形，在增加精度假设后，下界达到Frobenius范数的四次方，与上界匹配。监督聚类得到了Frobenius范数四次方的紧下界。其余三项任务的下界为Frobenius范数的二次函数，而已知上界为四次方。通过本研究，我们发现对于稀疏、高秩、良态矩阵相关问题，存在显著的量子加速潜力。最后，我们将方法扩展至矩阵相关问题量子查询算法的下界分析，并给出若干应用实例。