Structure-Preserving Optimal Control of Open Quantum Systems via a Discrete Contact PMP

We develop a discrete Pontryagin Maximum Principle (PMP) for controlled open quantum systems governed by Lindblad dynamics, and introduce a second--order contact Lie--group variational integrator (contact LGVI) that preserves both the CPTP (completely positive and trace--preserving) structure of the Lindblad flow and the contact geometry underlying the discrete PMP. A type--II discrete contact generating function produces a strict discrete contactomorphism under which the state, costate, and cost propagate in exact agreement with the variational structure of the discrete contact PMP. We apply this framework to the optimal control of a dissipative qubit and compare it with a non--geometric explicit RK2 discretization of the Lindblad equation. Although both schemes have the same formal order, the RK2 method accumulates geometric drift (loss of trace, positivity violations, and breakdown of the discrete contact form) that destabilizes PMP shooting iterations, especially under strong dissipation or long horizons. In contrast, the contact LGVI maintains exact CPTP structure and discrete contact geometry step by step, yielding stable, physically consistent, and geometrically faithful optimal control trajectories.