Abstract: It is well-known that plug-in statistical estimation of optimal transport suffers from the curse of dimensionality. Despite recent efforts to improve the rate of estimation with the smoothness of the problem, the computational complexity of these recently proposed methods still degrade exponentially with the dimension. In this paper, thanks to an infinite-dimensional sum-of-squares representation, we derive a statistical estimator of smooth optimal transport which achieves a precision $\varepsilon$ from $\tilde{O}(\varepsilon^{-2})$ independent and identically distributed samples from the distributions, for a computational cost of $\tilde{O}(\varepsilon^{-4})$ when the smoothness increases, hence yielding dimension-free statistical \emph{and} computational rates, with potentially exponentially dimension-dependent constants.