  We study the rate of convergence of the empirical spectral distribution of
products of independent non-Hermitian random matrices to the power of the
Circular Law. The distance to the deterministic limit distribution will be
measured in terms of a uniform Kolmogorov-like distance. First, we prove that
for products of Ginibre matrices, the optimal rate is given by $\mathcal O
(1/\sqrt n)$, which is attained with overwhelming probability up to a
logarithmic correction. Avoiding the edge, the rate of convergence of the mean
empirical spectral distribution is even faster. Second, we show that also
products of matrices with independent entries attain this optimal rate in the
bulk up to a logarithmic factor. In the case of Ginibre matrices, we apply a
saddlepoint approximation to a double contour integral representation of the
density and in the case of matrices with independent entries we make use of
techniques from local laws.
