Further improving quantum algorithms for nonlinear differential equations via higher-order methods and rescaling

The solution of large systems of nonlinear differential equations is needed for many applications in science and engineering. In this study, we present three main improvements to existing quantum algorithms based on the Carleman linearisation technique. First, by using a high-precision technique for the solution of the linearised differential equations, we achieve logarithmic dependence of the complexity on the error and near-linear dependence on time. Second, we demonstrate that a rescaling technique can considerably reduce the cost, which would otherwise be exponential in the Carleman order for a system of ODEs, preventing a quantum speedup for PDEs. Third, we provide improved, tighter bounds on the error of Carleman linearisation. We apply our results to a class of discretised reaction-diffusion equations using higher-order finite differences for spatial resolution. We show that providing a stability criterion independent of the discretisation can conflict with the use of the rescaling due to the difference between the max-norm and 2-norm. An efficient solution may still be provided if the number of discretisation points is limited, as is possible when using higher-order discretisations.