Before revision:  More concretely, we show that the ODE approximation achieves a 2-Wasserstein error of \varepsilon in O\big(d^{2{5}}/\varepsilon^{2{5}}\big) and O\big(d/\varepsilon^{1{3}}\big) steps when Lipschitz continuity is assumed for the Hessian and third derivative of f. By discretizing this ODE using a fourth order splitting method, we  obtain a practical MCMC method that requires just three additional gradient evaluations in each step.
After revision:  More concretely, we show that the ODE approximation achieves a 2-Wasserstein error of \varepsilon in O\big(d^{2{5}}/\varepsilon^{2{5}}\big) and O\big(d/\varepsilon^{1{3}}\big) steps when Lipschitz continuity is assumed for the Hessian and third derivative of f. By discretizing this ODE using a fourth order splitting method, we obtain a practical MCMC method that requires just three additional gradient evaluations per step.