Full- and low-rank exponential midpoint schemes for forward and adjoint Lindblad equations

The Lindblad equation is a widely used quantum master equation to model the dynamical evolution of open quantum systems whose states are described by density matrices. This equation is also a fundamental building block to design optimal control functions. In this paper we develop full- and low-rank exponential midpoint integrators for solving both the forward and adjoint Lindblad equations. These schemes are applicable to optimize-then-discretize approaches for optimal control of open quantum systems. We show that the proposed schemes preserve positivity and trace unconditionally. Furthermore, convergence of these numerical schemes is proved theoretically and verified numerically.