Dimension Liftings for Quantum Computation of Partial Differential Equations and Related Problems
This lecture presents a systematic framework for developing quantum simulation algorithms for general differential equations, using dimension lifting to transform nonlinear and time-dependent problems into forms solvable by simulating the Schrödinger equation.
Overview
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 will be interesting to explore what other problems in scientific computing, such as ODEs, PDEs, and linear algebra that arise in both classical and quantum systems, can be handled by quantum simulation.
We will present a systematic way to develop quantum simulation algorithms for general differential equations. Our basic framework is dimension lifting, that transfers nonlinear PDEs to linear ones, and linear ones to Schrodinger type PDEs. For non-autonomous PDEs and ODEs, or Hamiltonian systems with time-dependent Hamiltonians, we also add an extra dimension to transform them into autonomous PDEs that have only time-independent coefficients, thus quantum simulations can be done without using the cumbersome Dyson’s series and time-ordering operators. Our formulation allows both qubit and qumode (continuous-variable) formulations, and their hybridizations, and provides the foundation for analog quantum computing.
Presenters
Shi Jin, Director of Institute of Natural Sciences, Chair Professor of Mathematics, Shanghai Jiao Tong University
Brief Biography
Shi Jin is the Director of Institute of Natural Sciences, and Chair Professor of Mathematics, at Shanghai Jiao Tong University. He also serves as a co-director of the Shanghai National Center for Applied Mathematics, director of Ministry of Education Key Lab on Scientific and Engineering Computing, and director of Shanghai Jiao Tong University Chongqing Artificial Intelligence Institute.
He received a Feng Kang Prize of Scientific Computing in 2001., a Morningside Silver Medal in 2007. He is an inaugural Fellow of the American Mathematical Society (AMS) (2012), a Fellow of Society of Industrial and Applied Mathematics (SIAM) (2013), an inaugural Fellow of the Chinese Society of Industrial and Applied Mathematics (CSIAM) (2020), and an Invited Speaker at the International Congress of Mathematicians in 2018. In 2021 he was elected a Foreign Member of Academia Europaea and a Fellow of European Academy of Sciences. In 2024 he was awarded a Shanghai Natural Science Prize (first class).
His research interests include kinetic theory, hyperbolic conservation laws, quantum dynamics, uncertainty quantification, interacting particle systems， computational fluid dynamics, machine learning and quantum computing, etc.
