Efficient quantum algorithm for dissipative nonlinear differential equations

Differential equations are ubiquitous throughout mathematics, natural and social science, and engineering. There has been extensive previous work on efficient quantum algorithms for linear differential equations. However, analogous progress for nonlinear differential equations has been severely limited due to the linearity of quantum mechanics. We give the first quantum algorithm for dissipative nonlinear differential equations that is efficient provided the dissipation is sufficiently strong relative to the nonlinearity and the inhomogeneity. We also establish a lower bound showing that 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. Finally, we discuss potential applications of this approach to problems arising in biology as well as in fluid and plasma dynamics.
ArXiv link: https://arxiv.org/abs/2011.03185