Unconditionally stable time discretization of Lindblad master equations in infinite dimension using quantum channels

We examine the time discretization of Lindblad master equations in infinite-dimensional Hilbert spaces. Our study is motivated by the fact that, with unbounded Lindbladian, projecting the evolution onto a finite-dimensional subspace using a Galerkin approximation inherently introduces stiffness, leading to a Courant--Friedrichs--Lewy type condition for explicit integration schemes. We propose and establish the convergence of a family of explicit numerical schemes for time discretization adapted to infinite dimension. These schemes correspond to quantum channels and thus preserve the physical properties of quantum evolutions on the set of density operators: linearity, complete positivity and trace. Numerical experiments inspired by bosonic quantum codes illustrate the practical interest of this approach when approximating the solution of infinite dimensional problems by that of finite dimensional problems of increasing dimension.