题  目：Quantum algorithms for nonlinear partial differential equations

报告人：金石 教授

单  位：上海交通大学

时  间：2022年10月20日 15:00-4:00

地  点：腾讯593-625-376

摘要：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 applications 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.

报告人简介：现为上海交通大学自然科学研究院院长，数学学院讲席教授。他同时担任上海国家应用数学中心联合主任与上海交通大学重庆人工智能研究院院长。他是美国数学会首批会士， 美国工业与应用数学学会会士和2018年国际数学家大会邀请报告人， 并于2021年当选为欧洲人文与自然科学院(Academia Europaea)外籍院士与欧洲科学院(European Academy of Sciences)院士。他的研究方向包括科学计算，动理学理论，多尺度计算，计算流体力学, 不确定性量化，机器学习与量子计算等。