Title: Efficient quantum algorithms for nonlinear ODEs and PDEs
Abstract: Nonlinear dynamics play a prominent role in many domains and are notoriously difficult to solve. Whereas previous quantum algorithms for general nonlinear equations have been severely limited due to the linearity of quantum mechanics, we gave the first efficient quantum algorithm for nonlinear differential equations with sufficiently strong dissipation. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in the evolution time. We also established a lower bound showing that nonlinear differential equations with sufficiently weak dissipation have worst-case complexity exponential in time, giving an almost tight classification of the quantum complexity of simulating nonlinear dynamics. Furthermore, numerical results suggest that our algorithm may potentially address complex nonlinear phenomena even in regimes with weaker dissipation. We applied the quantum algorithm for nonlinear reaction-diffusion equations with an improved convergence rate. We post-processed the quantum encoding of the solutions of the Fisher-KPP equation and the Allen-Cahn equation for estimating thermodynamic free energies, laying the groundwork for practical applications of quantum computers for classical nonlinear systems. Recently, we showed that generic quantum algorithms for solving differential equations suffer from computational overheads due to their “non-quantumness”, and determined how to fast-forward quantum algorithms for solving special classes of non-quantum dynamics with improved efficiency.
 Efficient quantum algorithm for dissipative nonlinear differential equations, Proceedings of the National Academy of Science 118, 35 (2021), arXiv:2011.03185.
 Efficient quantum algorithm for nonlinear reaction-diffusion equations and free energy estimation, arXiv:2205.01141.
 A theory of quantum differential equation solvers: limitations and fast-forwarding, arXiv:2211.05246.
Biography: Jin-Peng is a Simons Quantum Postdoctoral Fellow at Simons Institute, UC Berkeley in 2022-2023, hosted by Umesh Vazirani and Lin Lin. He will be a Postdoctoral Associate at the Center for Theoretical Physics, MIT in 2023 fall. He received a PhD in applied mathematics at University of Maryland in 2022 spring, advised by Andrew Childs. He focuses mainly on the design and analysis of quantum algorithms for scientific computational problems, including topics such as linear and nonlinear differential equations, quantum dynamics, and stochastic processes, with applications in areas such as biology and epidemiology, fluid dynamics, quantum chemistry, finance, and machine learning.
His focus is mainly on the design and analysis of quantum algorithms for scientific computational problems, including topics such as linear and nonlinear differential equations, quantum dynamics, and stochastic processes, with applications in areas such as biology and epidemiology, fluid dynamics, quantum chemistry, finance, and machine learning.