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

TL;DR

The paper develops a fully geometric framework for optimal control of open quantum systems governed by Lindblad dynamics by merging a discrete contact Pontryagin Maximum Principle with Lie-group variational integrators. It introduces a second-order contact LGVI that preserves CPTP structure and discrete contact geometry, and demonstrates its effectiveness on a dissipative qubit compared with a non-geometric RK2 scheme. Key contributions include a discrete contact PMP on manifolds, exact/discrete contact Lagrangians on Lie groups, and a Strang-split Lindblad LGVI that yields stable, physically consistent optimal trajectories. Numerical results show that geometric preservation prevents trace/positivity drift and maintains the discrete contact identity, especially on long horizons. The framework promises robust, geometry-faithful control of noisy quantum devices and motivates extensions to multi-qubit systems and higher-order integrators.

Abstract

We develop a discrete Pontryagin Maximum Principle (PMP) for controlled open quantum systems governed by Lindblad dynamics, and introduce a second--order \emph{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.

Figures (8)

• Figure 1: Bloch trajectories for $T=10$, $\Delta t=0.01$. Both schemes start from $\rho_0=\ket1\bra1$ and are driven by the same sinusoidal control. The trajectories remain close and confined to the Bloch ball; the dynamics show the expected contraction towards the ground state.
• Figure 2: Left: Trace drift $|\tr(\rho_k)-1|$ for $T=10$, $\Delta t=0.01$. The contact LGVI keeps the trace at roundoff level, while RK2 exhibits a small but systematic drift, reflecting its lack of exact CPTP structure. Right: Positivity violation $\delta_{\mathrm{pos}}(k)$ for $T=10$, $\Delta t=0.01$. The contact LGVI remains positive semidefinite to machine precision, whereas RK2 displays slight negativity in the eigenvalues of $\rho_k$.
• Figure 3: Global error $\varepsilon_{\mathrm{glob}}(k) =\|\rho_k-\rho_{\mathrm{ref}}(t_k)\|_F$ for $T=10$, $\Delta t=0.01$, using a fine–step contact LGVI trajectory as reference. Both schemes exhibit second–order accuracy; the contact LGVI has slightly smaller error while maintaining exact CPTP structure.
• Figure 4: Left: Trace drift for $T=100$, $\Delta t=0.01$, $\gamma=10$. The contact LGVI remains at machine precision, whereas the RK2 method loses trace preservation catastrophically. Right: Positivity drift for $T=100$, $\Delta t=0.01$, $\gamma=10$. The contact LGVI stays positive semidefinite; RK2 produces massive violations of positivity
• Figure 5: Contact–form defect $\theta_k$ for $T=100$, $\Delta t=0.01$, $\gamma=10$. LGVI remains geometrically consistent; RK2 diverges dramatically.

Theorems & Definitions (31)

• Remark 1: Relation to Ohsawa's contact PMP
• Remark 2: Relation to the presymplectic/contact approach of de León--Lainz--Muñoz-Lecanda
• Theorem 1
• proof
• Remark 3
• Remark 3
• Remark 4
• Remark 5
• Theorem 2
• proof