Efficient quantum algorithm for dissipative nonlinear differential equations

Nonlinear differential equations appear in many domains and are notoriously difficult to solve. Whereas previous quantum algorithms for general nonlinear differential equations have complexity exponential in the evolution time, we give the first quantum algorithm for dissipative nonlinear differential equations that is efficient provided the dissipation is sufficiently strong relative to nonlinear and forcing terms and the solution does not decay too rapidly. 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. Furthermore, numerical results for the Burgers equation suggest that our algorithm may potentially address complex nonlinear phenomena even in regimes with weaker dissipation.