Limitations of Quantum Algorithms for Nonlinear Partial Differential Equations

We investigate the limitations of obtaining an exponential quantum advantage for solving nonlinear partial differential equations (PDEs) on a quantum computer. In particular we tighten the worst-case bounds of the algorithm introduced by Liu et al. [Liu et al. PNAS 2020], closing one of their open questions. We also show that no PDE that has positive Lyapunov exponent and solutions that grow sub-exponentially can be solved in time scaling faster than exponentially.