  Neufeld and Wu (arXiv:2310.12545) developed a multilevel Picard (MLP)
algorithm which can approximately solve general semilinear parabolic PDEs with
gradient-dependent nonlinearities, allowing also for coefficient functions of
the corresponding PDE to be non-constant. By introducing a particular
stochastic fixed-point equation (SFPE) motivated by the Feynman-Kac
representation and the Bismut-Elworthy-Li formula and identifying the first and
second component of the unique fixed-point of the SFPE with the unique
viscosity solution of the PDE and its gradient, they proved convergence of
their algorithm. However, it remained an open question whether the proposed MLP
schema in arXiv:2310.12545 does not suffer from the curse of dimensionality. In
this paper, we prove that the MLP algorithm in arXiv:2310.12545 indeed can
overcome the curse of dimensionality, i.e. that its computational complexity
only grows polynomially in the dimension $d\in \mathbb{N}$ and the reciprocal
of the accuracy $\varepsilon$, under some suitable assumptions on the nonlinear
part of the corresponding PDE.
