Leverage score sampling is crucial to the design of randomized algorithms for large-scale matrix problems, while the computation of leverage scores is a bottleneck of many applications. In this paper, we propose a quantum algorithm to accelerate this useful method. The speedup is at least quadratic and could be exponential for well-conditioned matrices. We also prove some quantum lower bounds, which suggest that our quantum algorithm is close to optimal. As an application, we propose a new quantum algorithm for rigid regression problems with vector solution outputs. It achieves polynomial speedups over the best classical algorithm known. In this process, we give an improved randomized algorithm for rigid regression.

翻译：杠杆分数抽样对大规模矩阵问题的随机算法设计至关重要，而杠杆分数的计算是许多应用的瓶颈。本文提出一种量子算法来加速这一实用方法。加速效果至少是二次的，对于良态矩阵甚至可能达到指数级。我们同时证明了若干量子下限，表明所提量子算法接近最优。作为应用，我们提出了一种面向向量解输出的刚性回归问题量子新算法，其相对于已知最佳经典算法实现了多项式加速。在此过程中，我们还给出了改进的刚性回归随机算法。