  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.
