AI 中文总结
本研究提出一种结合Carleman嵌入技术与线性化系统哈密顿量模拟算法线性组合的量子算法，可快进求解弱非线性耗散微分方程，将算法复杂度从与演化时间T相关的O~(√T)提升至显性不依赖T的O(1)，还拓展至更强非线性等场景的数值研究。
AI 中文摘要
我们研究一种用于求解弱非线性耗散常微分方程的快进量子算法。该方法结合了Carleman嵌入技术与线性化系统的哈密顿量模拟算法的线性组合，且具备快进缩放特性。本算法的复杂度不显性依赖演化时间T，将此前最优的\\(\widetilde{\mathcal{O}}(\sqrt{T})\\)提升至\\(\mathcal{O}(1)\\)，剩余时间依赖性通过输出范数与强迫参数体现。我们针对时变系数矩阵的Carleman嵌入收敛性保障开展严格分析，并通过简化后选择步骤改进基于Carleman嵌入的算法的实现。此外，我们对超出弱非线性场景的微分方程进行数值研究，发现针对更强非线性或线性非共振效应的系统，具备实现快进缩放的可能性。
英文摘要
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.
