Fast-forwarding quantum algorithms for weakly nonlinear dissipative differential equations and beyond

Yixiang Li; Dong An

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arXiv · 2608.25822

Fast-forwarding quantum algorithms for weakly nonlinear dissipative differential equations and beyond

Yixiang Li·Dong An

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Abstract

We study a fast-forwarded quantum algorithm for solving weakly nonlinear dissipative ordinary differential equations. Our approach is a combination of the Carleman embedding technique and the linear combination of Hamiltonian simulation algorithm for linearized systems with fast-forwarded scaling. The complexity of our algorithm does not explicitly depend on the evolution time $T$, thus greatly improving the previous state-of-the-art $\widetilde{\mathcal{O}}(\sqrt{T})$ to $\mathcal{O}(1)$, and any remaining time dependence enters through the output norm and forcing parameters. We rigorously analyze the performance of this approach by convergence guarantees of the Carleman embedding for time-dependent coefficient matrices and detailed complexity estimates, and improve the realization of the Carleman-embedding-based algorithms by simplifying the post-selection step. In addition, we perform a numerical study on differential equations beyond the weakly nonlinear case, and identify possibility of achieving fast-forwarding scaling for systems with stronger nonlinearity or linear non-resonant effect.

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Yixiang Li, Dong An. 2026-08-26. Fast-forwarding quantum algorithms for weakly nonlinear dissipative differential equations and beyond. https://arxiv.org/abs/2608.25822

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