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

• URL: http://arxiv.org/abs/2608.25822v1
• Date: Wed, 26 Aug 2026 14:03:50 GMT
• Title: Fast-forwarding quantum algorithms for weakly nonlinear dissipative differential equations and beyond
• Abstract summary: We study a fast-forwarded quantum algorithm for solving weakly nonlinear dissipative ordinary differential equations.<n>Our approach is a combination of the Carleman embedding technique and the linear combination of Hamiltonian simulation algorithm for linearized systems.
• Score: 5.509525921849405
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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