Quantum Algorithms for Nonlinear Differential Equations via Pivot-Shifted Carleman Linearization

TL;DR

This paper introduces a pivot-shifted Carleman linearization method for quantum algorithms that efficiently solves a broader class of nonlinear differential equations, with proven convergence and improved stability.

Contribution

It develops a novel pivot-shifted Carleman linearization framework that extends quantum simulation capabilities to more nonlinear systems, including unstable ones, with rigorous convergence guarantees.

Findings

Logarithmic scaling of truncation order with simulation time

Exponential error decay with truncation order in numerical experiments

Enhanced stability and accuracy through pivot shifting

Abstract

We develop a pivot-shifted Carleman linearization framework for quantum algorithms solving quadratic nonlinear ordinary differential equations. By shifting the dynamics by a pivot state prior to Carleman lifting, and combining this with a Lyapunov transform and rescaling, we enlarge the class of nonlinear systems that can be efficiently simulated on quantum computers. For systems that exhibit stability in the shifted coordinates, we establish long time convergence of the truncated Carleman embedding. We prove that the truncation order scales only logarithmically with the simulation time and target precision, and we derive end-to-end quantum query complexity bounds for preparing a state proportional to the final solution. By introducing a modified nonlinearity condition, this framework entirely removes the conventional lower bound requirement on the initial condition. For more general systems that remain unstable after shifting, we provide short time convergence guarantees that are similarly free from the initial condition constraints. Numerical experiments on the logistic and the Lotka-Volterra equations demonstrate that an appropriate pivot choice improves stability and accuracy, and yields exponential error decay with truncation order. These results show that pivot shifting provides a practical and theoretically justified route for extending Carleman-based quantum algorithms to a broader class of nonlinear dynamical systems.