This talk will discuss recent advances in quantum algorithms for nonlinear differential equations, with a particular focus on extending quantum differential equation solvers beyond polynomial nonlinearities through Koopman linearization. I will begin with a brief survey of existing approaches to solving nonlinear ordinary differential equations (ODEs) on quantum computers, highlighting the role of linearization techniques such as Carleman linearization. These techniques typically transform a nonlinear ODE into a higher-dimensional linear ODE and then apply a quantum linear differential equation solver. However, these methods are largely restricted to systems whose nonlinearities are polynomial functions of the dependent variables, substantially limiting their applicability.

I will then present a new quantum algorithm for solving nonlinear ODEs with Fourier-type nonlinearities. The algorithm is based on Koopman linearization, a generalization of Carleman linearization that enables the treatment of a broader class of nonlinear systems. I will discuss how Koopman-based representations can be combined with quantum differential equation solvers, present complexity results for this class of ODEs, and describe several methodological advances, including techniques that relax the dissipativity assumptions required for efficient solution extraction and integrated procedures for estimating classical observables from the quantum solution state. The talk will conclude with a discussion of the implications of these results for developing quantum algorithms for a wider range of high-dimensional nonlinear dynamical systems.