Quantum Riemannian Cubics with Obstacle Avoidance for Quantum Geometric Model Predictive Control

We propose a geometric model predictive control framework for quantum systems subject to smoothness and state constraints. By formulating quantum state evolution intrinsically on the projective Hilbert space, we penalize covariant accelerations to generate smooth trajectories in the form of Riemannian cubics, while incorporating state-dependent constraints through potential functions. A structure-preserving variational discretization enables receding-horizon implementation, and a Lyapunov-type stability result is established for the closed-loop system. The approach is illustrated on the Bloch sphere for a two-level quantum system, providing a viable pathway toward predictive feedback control of constrained quantum dynamics.

💡 Research Summary

The paper introduces a geometric model predictive control (MPC) framework tailored for quantum systems that must satisfy smoothness and state‑dependent constraints. By representing pure quantum states on the projective Hilbert space (P(\mathcal H)) equipped with the Fubini–Study metric, the authors eliminate the global phase and obtain a coordinate‑free Riemannian manifold ((Q,g)). On this manifold they formulate a second‑order variational problem that penalizes the squared norm of the covariant acceleration (D_t\dot\gamma). The resulting optimal trajectories are Riemannian cubics, i.e., critical points of the functional
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