Efficient Quantum Algorithm for Nonlinear Reaction–Diffusion Equations and Energy Estimation
Efficient Quantum Algorithm for Nonlinear Reaction–Diffusion Equations and Energy Estimation
复制标题
非线性反应扩散方程和能量估计的高效量子算法
DOI：
--
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发表时间：
2022
影响因子：
2.4
通讯作者：
S. Jordan
中科院分区：
文献类型：
--
作者：
Jin;Dong An;Di Fang;Jiasu Wang;G. Low;S. Jordan
Nonlinear differential equations exhibit rich phenomena in many fields but are notoriously challenging to solve. Recently, Liu et al. (in: Proceedings of the National Academy of Sciences 118(35), 2021) demonstrated the first efficient quantum algorithm for dissipative quadratic differential equations under the condition R<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R < 1$$\end{document}, where R measures the ratio of nonlinearity to dissipation using the ℓ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _2$$\end{document} norm. Here we develop an efficient quantum algorithm based on Liu et al. (2021) for reaction–diffusion equations, a class of nonlinear partial differential equations (PDEs). To achieve this, we improve upon the Carleman linearization approach introduced in Liu et al. (2021) to obtain a faster convergence rate under the condition RD<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_D < 1$$\end{document}, where RD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_D$$\end{document} measures the ratio of nonlinearity to dissipation using the ℓ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{\infty }$$\end{document} norm. Since RD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_D$$\end{document} is independent of the number of spatial grid points n while R increases with n, the criterion RD<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_D<1$$\end{document} is significantly milder than R<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R<1$$\end{document} for high-dimensional systems and can stay convergent under grid refinement for approximating PDEs. As applications of our quantum algorithm we consider the Fisher-KPP and Allen-Cahn equations, which have interpretations in classical physics. In particular, we show how to estimate the mean square kinetic energy in the solution by postprocessing the quantum state that encodes it to extract derivative information.
Nonlinear differential equations exhibit rich phenomena in many fields but are notoriously challenging to solve. Recently, Liu et al. (in: Proceedings of the National Academy of Sciences 118(35), 2021) demonstrated the first efficient quantum algorithm for dissipative quadratic differential equations under the condition R<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R < 1$$\end{document}, where R measures the ratio of nonlinearity to dissipation using the ℓ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _2$$\end{document} norm. Here we develop an efficient quantum algorithm based on Liu et al. (2021) for reaction–diffusion equations, a class of nonlinear partial differential equations (PDEs). To achieve this, we improve upon the Carleman linearization approach introduced in Liu et al. (2021) to obtain a faster convergence rate under the condition RD<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_D < 1$$\end{document}, where RD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_D$$\end{document} measures the ratio of nonlinearity to dissipation using the ℓ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{\infty }$$\end{document} norm. Since RD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_D$$\end{document} is independent of the number of spatial grid points n while R increases with n, the criterion RD<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_D<1$$\end{document} is significantly milder than R<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R<1$$\end{document} for high-dimensional systems and can stay convergent under grid refinement for approximating PDEs. As applications of our quantum algorithm we consider the Fisher-KPP and Allen-Cahn equations, which have interpretations in classical physics. In particular, we show how to estimate the mean square kinetic energy in the solution by postprocessing the quantum state that encodes it to extract derivative information.
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影响因子：
6.4
作者：
Childs, Andrew M.;Liu, Jin-Peng;Ostrander, Aaron
通讯作者：
Ostrander, Aaron
DOI：
10.1103/physreva.105.022415
发表时间：
2020-10
期刊：
ArXiv
影响因子：
--
作者：
B. Kiani;G. Palma;D. Englund;W. Kaminsky;M. Marvian;S. Lloyd
通讯作者：
B. Kiani;G. Palma;D. Englund;W. Kaminsky;M. Marvian;S. Lloyd
基于时间最优绝热量子计算和量子近似优化算法的量子线性系统求解器
DOI：
10.1145/3498331
发表时间：
2022
期刊：
ACM Transactions on Quantum Computing
影响因子：
--
作者：
An, Dong;Lin, Lin
通讯作者：
Lin, Lin
基于时间推进的时间相关线性微分方程的量子求解器
影响因子：
6.4
作者：
Fang, Di;Lin, Lin;Tong, Yu
通讯作者：
Tong, Yu
影响因子：
2.4
作者：
Childs, Andrew M.;Liu, Jin-Peng
通讯作者：
Liu, Jin-Peng