Fixing Divergence in Carleman Linearization via Analytical Continuation

Abstract

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.