Abstract: Quantum computers have the potential to gain algebraic and even up to exponential speed up compared with its classical counterparts, and can lead to technology revolution in the 21st century. Since quantum computers are designed based on quantum mechanics principle, they are most suitable to solve the Schrodinger equation, and linear PDEs (and ODEs) evolved by unitary operators.  The most efficient quantum PDE solver is quantum simulation based on solving the Schrodinger equation. It became challenging for general PDEs, more so for nonlinear ones. Our talk will cover three topics:

1) We introduce the “warped phase transform” to map general linear PDEs and ODEs to Schrodinger equation or with unitary evolution operators in higher dimension so they are suitable for quantum simulation;

2) For (nonlinear) Hamilton-Jacobi equation and scalar nonlinear hyperbolic equations we use the level set method to map them—exactly—to phase space linear PDEs so they can be implemented with quantum algorithms and we gain quantum advantages for various physical and numerical parameters. 

3) For PDEs with uncertain coefficients, we introduce a transformation so the uncertainty only appears in the initial data, allowing us to compute ensemble averages with multiple initial data with just one run, instead of multiple runs as in Monte-Carlo or stochastic collocation type sampling algorithms

报告人：金石，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。