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

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.

Quantum Physics (quant-ph); Numerical Analysis (math.NA)

arXiv:2608.25822 [quant-ph]

(or arXiv:2608.25822v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.25822 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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