AI 中文总结
研究针对卡尔曼线性化在求解非线性微分方程时出现的指数发散问题，通过解析延拓插入正则化函数进行校正，在逻辑斯谛方程等上验证了方法，并使用量子算法实现逻辑斯谛方程求解，给出复杂度和误差分析。
AI 中文摘要
非线性微分方程在众多现象建模中起关键作用，但其解难以获得。随着量子计算发展，人们探索有效求解此类方程的量子算法，基于卡尔曼线性化的方法颇具潜力，它能将非线性微分方程转化为线性系统，但该方法在一定时间尺度后会出现指数发散。通过用特征值和特征向量重新表述解，发现发散源于收敛邻域外的洛朗展开。为解决此问题，通过解析延拓对发散解插入正则化函数。在逻辑斯谛方程、KPP - Fisher方程和相场模型等偏微分方程上验证了该发散校正方法，并使用量子算法实现逻辑斯谛方程的方法，提供了详细的复杂度和误差分析。
英文摘要
Nonlinear differential equations play a crucial role in modeling a wide range of phenomena, yet their solutions remain notoriously difficult to obtain. With the rapid development of quantum computing, quantum algorithms for efficiently solving such equations are actively being explored. One promising approach is based on Carleman linearization, which transforms nonlinear differential equations into linear systems. However, this method suffers from exponential divergence beyond a certain time scale. By reformulating the solutions in terms of eigenvalues and eigenvectors, we identify that this divergence originates from the Laurent expansion outside its neighborhood of convergence. To address this issue, we insert a regularized function to the divergent solution hinted by analytical continuation. We validate this divergence-correction method on both the logistic equation and some other partial differential equations like KPP-Fisher equations and Phase-Field models under periodic conditions. We implement our method for the logistic equation using the Linear Combination of Unitaries (LCU) quantum algorithm, providing a detailed complexity and error analysis.
Comments35 pages. 18 figures, this work is an extended version of a presentation delivered at the 16th Quantum Software Research Presentation Meeting
