Fixing Divergence in Carleman Linearization via Analytical Continuation

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 propose a method based on analytical continuation, where an appropriate conformal mapping ensures convergence along the positive real axis. We validate this divergence-correction method on both the logistic equation and some other nonlinear partial differential equations like KPP–Fisher equations and Phase-Field models under periodic conditions, and further outline the constraints required for successful analytical continuation. Finally, we demonstrate how the proposed method can be implemented on a quantum computer using the Quantum Singular Value Transformation (QSVT) algorithm.