Motivated by the Derrida-Lebowitz-Speer-Spohn (DLSS) quantum drift equation, which is the Wasserstein gradient flow of the Fisher information, we study in details solutions of the corresponding implicit Euler scheme. We also take advantage of the convex (but non-smooth) nature of the corresponding variational problem to propose a numerical method based on the Chambolle-Pock primal-dual algorithm. Dedicated to Giuseppe Buttazzo, a master of the calculus of variations, on the occasion of his 70th birthday.