Quantum computers may achieve speedups over their classical counterparts for solving linear algebra problems. However, in some cases -- such as for low-rank matrices -- dequantized algorithms demonstrate that there cannot be an exponential quantum speedup. In this work, we show that quantum computers have provable polynomial and exponential speedups in terms of communication complexity for some fundamental linear algebra problems \update{if there is no restriction on the rank}. We mainly focus on solving linear regression and Hamiltonian simulation. In the quantum case, the task is to prepare the quantum state of the result. To allow for a fair comparison, in the classical case, the task is to sample from the result. We investigate these two problems in two-party and multiparty models, propose near-optimal quantum protocols and prove quantum/classical lower bounds. In this process, we propose an efficient quantum protocol for quantum singular value transformation, which is a powerful technique for designing quantum algorithms. This will be helpful in developing efficient quantum protocols for many other problems.

翻译：量子计算机可能在解决线性代数问题时相较经典计算机实现加速。然而，在某些情况下（例如低秩矩阵），去量子化算法表明不存在指数级量子加速。本研究证明，若对矩阵秩不作限制，量子计算机在解答若干基础线性代数问题（如线性回归和哈密顿量模拟）的通信复杂度上具有确定的多项式与指数级加速优势。我们重点研究线性回归与哈密顿量模拟这两个问题：在量子场景中，任务为制备结果对应的量子态；为公平比较，经典场景中的任务则设定为从结果中采样。我们分别在双方及多方模型下探究这两类问题，提出近最优量子协议，并证明量子/经典下界。在此过程中，我们提出一种高效的量子奇异值变换协议——该技术是设计量子算法的有力工具，将有助于开发众多其他问题的高效量子协议。