Quantum communication complexity of linear regression

IF 0.8 Q3 COMPUTER SCIENCE, THEORY & METHODS ACM Transactions on Computation Theory Pub Date : 2023-09-22 DOI:10.1145/3625225
Ashley Montanaro, Changpeng Shao
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引用次数: 8

Abstract

Quantum computers may achieve speedups over their classical counterparts for solving linear algebra problems. However, in some cases – such as for low-rank matrices – dequantised 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 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. We feel this will be helpful in developing efficient quantum protocols for many other problems.
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量子通信复杂度线性回归
在解决线性代数问题时,量子计算机可能比经典计算机实现更快的速度。然而,在某些情况下——比如对于低秩矩阵——去量子化算法证明不可能有指数级的量子加速。在这项工作中,我们表明,如果没有秩限制,量子计算机在一些基本线性代数问题的通信复杂性方面具有可证明的多项式和指数加速。我们主要集中在求解线性回归和哈密顿模拟。在量子情况下,任务是准备结果的量子态。为了进行公平的比较,在经典情况下,任务是从结果中抽样。我们在两方和多方模型中研究了这两个问题,提出了近最优量子协议,并证明了量子/经典下界。在此过程中,我们提出了一种高效的量子奇异值变换的量子协议,这是设计量子算法的有力技术。我们认为这将有助于为许多其他问题开发有效的量子协议。
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来源期刊
ACM Transactions on Computation Theory
ACM Transactions on Computation Theory COMPUTER SCIENCE, THEORY & METHODS-
CiteScore
2.30
自引率
0.00%
发文量
10
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