Quantum algorithms for nonlinear partial differential equations

主讲人：金石 上海交通大学教授

时间：2022年6月2日15：30

地点：腾讯会议 208 500 303

举办单位：数理学院

主讲人介绍：金石，欧洲人文和自然科学院外籍院士和欧洲科学院院士，国际数学家大会45分钟报告者、上海交通大学讲席教授、博士生导师。现任上海交通大学自然科学院院长,CSIAM会士、AMS会士、SIAM会士。曾获冯康科学计算奖、第四届世界华人数学家大会晨兴数学奖银奖；在计算流体力学，动力学方程，双曲型守恒律方程高频波计算，计算物理和多尺度问题的计算方法等领域发表了200余篇SCI论文。

内容介绍：Nonlinear partial differential equations (PDEs) are crucial to modelling important problems in science but they are computationally expensive and suffer from the curse of dimensionality. Since quantum algorithms have the potential to resolve the curse of dimensionality in certain instances, some quantum algorithms for nonlinear PDEs have been developed. However, they are fundamentally bound either to weak nonlinearities,valid to only short times,or display no quantum advantage. We construct new quantum algorithms--based on level sets --for nonlinear Hamilton-Jacobi and scalar hyperbolic PDEs that can be performed with quantum advantages on various critical numerical parameters, even for computing the physical observables, for arbitrary nonlinearity and are valid globally in time. These PDEs are important for many appli- cations like optimal control,machine learning, semi-classical limit of Schrodinger equations, mean-field games and many more. Depending on the details of the initial data, it can display up to exponential advantage in both the dimension of the PDE and the error in computing its observables. For general nonlinear PDEs, quantum advantage with respect to M,for computing the ensemble averages of solutions corresponding to M different initial data,is possible in the large $M$ limit. This is a joint work with Nana Liu.