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

TL;DR

The paper addresses robust, smooth quantum trajectory generation under state constraints by formulating quantum dynamics intrinsically on the projective Hilbert space with a second-order variational principle. It develops obstacle-avoiding Riemannian cubics and structure-preserving Lie-group variational integrators, and embeds them in a geometric model predictive control (QGMPC) framework, with a Lyapunov-type stability guarantee for the closed-loop system. The approach is demonstrated on the Bloch sphere for a qubit, showing how receding-horizon optimization yields constraint-aware, smooth quantum trajectories that respect geometric structure. Together, the work provides a rigorous, geometry-aware pathway to predictive feedback control of constrained quantum dynamics with practical stability guarantees and robust numerical schemes.

Abstract

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.

Figures (6)

• Figure 1: Bloch representation of a qubit state $\rho = \frac{1}{2}\left(\mathbf{1} + \vec{n}\cdot \vec{\sigma}\right)$. Pure states lie on the surface ($\|\vec{n}\|=1$), while mixed states correspond to points in the interior ($\|\vec{n}\|<1$).
• Figure 2: Obstacle potential on the Bloch sphere. Top row: orthographic top views of the lifted obstacle potential $V(\vec{n})$, aligned with the obstacle direction $\vec{n}_o$, for fixed $\tau = 1.0$ and $D = 0.393$, comparing the cases $N=2$ (left) and $N=6$ (right). The black contour indicates the level set $\theta = D$, defining the effective obstacle boundary. Bottom row: meridional sections of the same potential in terms of the polar angle $\theta$ for $\theta \leq 2D$, highlighting the increasing steepness and localisation of the repulsive barrier as $N$ increases.
• Figure 3: Quantum Riemannian cubic connecting the south pole to a terminal state located just outside a forbidden spherical cap around the north pole. The red cap denotes the forbidden region induced by the obstacle potential, while the blue curve shows the resulting cubic-shape trajectory.
• Figure 4: Top: Constraint preservation on $S^2$. The LGVI preserves the constraint intrinsically, whereas extrinsic explicit schemes exhibit constraint drift. Bottom: Exact preservation of the discrete momentum map by the LGVI.
• Figure 5: Closed-loop QGMPC for a single qubit.

Theorems & Definitions (24)

• Definition 3.1
• Theorem 3.2
• Remark 3.3
• Lemma 4.1
• Definition 4.2
• Proposition 4.3
• Remark 4.4
• Remark 4.5
• Remark 4.6
• Remark 4.7