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arXiv 2011.03185quant-phcs.NAmath.NAphysics.plasm-ph
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用于耗散非线性微分方程的高效量子算法

Efficient quantum algorithm for dissipative nonlinear differential equations

Institute for Advanced Computer Studies, University of Maryland（马里兰大学高级计算机研究所）；Department of Mathematics, University of Maryland（马里兰大学数学系）；Norwegian University of Science and Technology（挪威科技大学）；Plasma Science and Fusion Center, Massachusetts Institute of Technology（麻省理工学院等离子体科学与聚变中心）；Raytheon BBN Technologies（雷神BBN技术公司）；Institute for Physical Science and Technology, University of Maryland（马里兰大学物理科学与技术研究所）；Department of Computer Science, University of Maryland（马里兰大学计算机科学系）

Jin-Peng Liu, Herman Øie Kolden, Hari K. Krovi, Nuno F. Loureiro, Konstantina Trivisa, Andrew M. Childs

备注：
36 pages, 1 figure. Updated to reflect a correction to the version published in PNAS (https://doi.org/10.1073/pnas.2615307123)
发表信息：
Proceedings of the National Academy of Sciences 118, e2026805118 (2021)
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AI总结：

研究针对耗散非线性微分方程提出量子算法，采用卡尔曼线性化方法，在\(R < 1\)时复杂度有指数级提升，相比此前算法有显著改进，还给出复杂度下界并探讨了潜在应用。

摘要：

非线性微分方程能描述多种现象，但求解困难。此前量子算法在线性微分方程上成果丰硕，而量子力学的线性限制了非线性情况的进展。本文针对耗散二次\(n\)维常微分方程开发了量子算法，在\(R < 1\)时，其复杂度为\(T^2 q~\mathrm{poly}(\log T, \log n, \log 1/\epsilon)/\epsilon\)，相比此前算法有指数级提升。算法采用卡尔曼线性化方法并给出新的收敛定理，还给出了一般二次微分方程量子算法最坏情况复杂度的下界。最后讨论了潜在应用。

英文摘要：

Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic $n$-dimensional ordinary differential equations. Assuming $R < 1$, where $R$ is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity $T^2 q~\mathrm{poly}(\log T, \log n, \log 1/ε)/ε$, where $T$ is the evolution time, $ε$ is the allowed error, and $q$ measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in $T$. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a novel convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for $R \ge \sqrt{2}$. Finally, we discuss potential applications, showing that the $R < 1$ condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of $R$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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