发表机构
Freie Universität Berlin; Center for Theoretical Physics – a Leinweber Institute, MIT; Google Quantum AI（柏林自由大学; 麻省理工学院莱因韦伯理论物理中心; 谷歌量子AI）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究将量子线性系统问题的查询复杂度精确确定为Θ(κ√d log(1/ε))，并解决了Berry和Childs的开放问题，证明任意N×N酉矩阵可用O(√N)次查询实现。
AI 中文摘要
在量子线性系统问题（QLSP）中，我们被给予对一个条件数为κ的d-稀疏N×N矩阵A的查询访问，以及制备一个与向量b成正比的量子态的能力。目标是输出与方程A x = b的解x成正比的量子态的ε近似。经过一系列长期的研究工作，此前已知的最佳量子算法在QLSP上的查询复杂度分别为O(κ d log(1/ε))和κ√d(κd/ε)^{o(1)}，而已知的最佳下界为Ω(κ log(1/ε))和Ω(κ√d)。我们改进了这些界限，并证明QLSP的复杂度为Θ(κ√d log(1/ε))。我们还解决了Berry和Childs的一个开放问题，表明任何N×N酉矩阵都可以通过对其矩阵元素的O(√N)次查询以有界误差实现。
英文摘要
In the quantum linear systems problem (QLSP), we are given query access to a $d$-sparse $N\times N$ matrix $A$ with condition number $κ$, and the ability to prepare a quantum state proportional to a vector $\vec b$. The goal is to output an $ε$-approximation to the quantum state proportional to the solution $\vec x$ of $A\vec{x}=\vec{b}$. Following a long line of work, the best previously known quantum algorithms for the QLSP had query complexities $O(κd\log(1/ε))$ and $κ\sqrt d(κd/ε)^{o(1)}$, while the best known lower bounds were $Ω(κ\log(1/ε))$ and $Ω(κ\sqrt d)$. We improve these bounds and show that the complexity of the QLSP is $Θ(κ\sqrt d\log(1/ε))$. We also resolve an open problem of Berry and Childs by showing that any $N\times N$ unitary can be implemented with bounded error using $O(\sqrt N)$ queries to its matrix entries.
Comments20 pages
