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arXiv:2602.08881

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

Leonardo Colombo

math-pheess.SYmath.OCquant-phSubmitted Feb 9, 2026• Announced Feb 10, 2026• 0 citations
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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.

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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.
Quantum splines are curves in a Hilbert space or, equivalently, in the corresponding Hilbert projective space, which generalize the notion of Riemannian cubic splines to the quantum domain. In this paper, we present a generalization of this concept to general density matrices with a Hamiltonian approach and using a geometrical formulation of quantum mechanics. Our main goal is to formulate an optimal control problem for a nonlinear system on [Formula: see text] which corresponds to the variational problem of quantum splines. The corresponding Hamiltonian equations and interpolation conditions are derived. The results are illustrated with some examples and the corresponding quantum splines are computed with the implementation of a suitable iterative algorithm.
Quantum splines are curves in a Hilbert space or, equivalently, in the corresponding Hilbert projective space, which generalize the notion of Riemannian cubic splines to the quantum domain. In this paper, we present a generalization of this concept to general density matrices with a Hamiltonian approach and using a geometrical formulation of quantum mechanics. Our main goal is to formulate an optimal control problem for a nonlinear system on the dual of the Lie algebra of the unitary group which corresponds to the variational problem of quantum splines. The corresponding Hamiltonian equations and interpolation conditions are derived. The results are illustrated with some examples and the corresponding quantum splines are computed with the implementation of a suitable iterative algorithm.
We introduce a new hybrid quantum-classical algorithm inspired by an advanced control strategy known as model predictive control (MPC). This algorithm unifies the optimization-based design of variational quantum algorithms (VQAs) with the feedback-based design of feedback-based quantum algorithms (FQAs). Variational circuit parameters are optimized using a layer-wise receding horizon strategy, where observable measurements after every layer initialize a classically simulated dynamic model used to predict quantum state evolution and optimize over future parameterized gates. This hybrid algorithm can be used for applications such as ground state preparation and approximate combinatorial optimization, and presents an ideal use case for the integration of quantum computers with high-performance computing, where the latter resource can be used to increase the scale and efficiency of the predictions critical to MPC. We show through mathematical proof and numerical evidence that the MPC-based algorithm can be guaranteed to at least match the performance of FQAs. Through simulations on Max-Cut problems and a two-dimensional transverse-field Ising model, we demonstrate that relaxed implementations of the MPC-based algorithm can also provide improved performance in practice compared to an FQA.
The application of Model Predictive Control (MPC) to quantum systems faces fundamental challenges arising from measurement-induced disturbances and the probabilistic nature of quantum dynamics. Unlike classical nonlinear systems, quantum measurements inherently disturb the system state, making feedback control intrinsically model-dependent and stochastic. Within the framework of Positive Operator-Valued Measures (POVMs), this thesis focuses on strong quantum measurements, where each observation induces a complete state collapse on the system. When uncertainty is introduced, the measurement process itself becomes a dominant source of disturbance. To address this, a lower bound on the measurement uncertainty is imposed, quantified as the predicted deviation between the measured quantum state and the nominal MPC trajectory, ensuring that the quantum state can be steered along the designed MPC trajectory while maintaining physical realizability and stability. The thesis further develops a stochastic MPC (SMPC) formulation based on quantum filtering equations, describing the conditional evolution of quantum states under continuous observation. A theoretical equivalence between the proposed SMPC scheme and an optimal control problem is established, providing a tractable computational approach for feedback design in quantum stochastic systems. Finally, to address the limitations imposed by explicit identification of quantum systems in complex quantum systems, a data-driven MPC framework is introduced. Using locally observable dynamics and trajectory data, a data-driven MPC formulation is introduced to realize predictive quantum control without explicit system identification. This framework lays the foundation for measurement-aware and physically consistent quantum MPC.
Model predictive control has shown potential to enhance the robustness of quantum control systems. In this work, we propose a tractable Stochastic Model Predictive Control (SMPC) framework for finite-dimensional quantum systems under continuous-time measurement and quantum filtering. Using the almost-sure eigenstate reduction of quantum trajectories, we prove that the infinite-horizon stochastic objective collapses to a fidelity term that is computable in closed form from the one-step averaged state. Consequently, the online SMPC step requires only deterministic propagation of the filter and a terminal fidelity evaluation. An advantage of this method is that it eliminates per-horizon Monte Carlo scenario sampling and significantly reduces computational load while retaining the essential stochastic dynamics. We establish equivalence and mean-square stability guarantees, and validate the approach on multi-level and Ising-type systems, demonstrating favorable scalability compared to sampling-based SMPC.
We extend the geometric tracking infrastructures established in preceding works to formalize the strict coordinate and dynamical limitations that govern parameterized quantum steering protocols. By constructing a differential-geometric framework mapping smooth control parameters directly to the projective Hilbert space, we isolate the architectural conditions under which optimization protocols undergo structural failure. First, we state and prove the Control Immersion Criterion, demonstrating that any trajectory intersecting a localized coordinate caustic interface triggers a sharp rank deficiency in the pulled-back information metric tensor. This metric non-invertibility forces a catastrophic, asymptotic divergence in the local Levi Civita Christoffel connection symbols (Γmij → ∞), completely destabilizing gradient-driven updates. Second, we evaluate the global phase-space volume element under cyclic parameters, proving that the anti-Hermiticity of the connection generator ensures a strictly real-valued Berry connection and subsequent structural closure of the modified symplectic form (dω=0), maintaining topological conservation despite severe localized metric warping. Finally, we derive a sharp kinematic speed limit bounding non-adiabatic escape rates out of targeted operating subspaces as a direct penalty scaled by the metric tensor. Crucially, this bound exposes an operational duality linking statistical estimation constraints to state-leakage rates through the Quantum Fisher Information Metric structure (gQF IM = 4gij ). Dense grid calculations and dynamic fourth-order Runge-Kutta numerical simulations confirm these analytical constraints down to the machine precision floor, validating this framework for structural pre-mapping routines inside advanced, singularity-aware quantum compilation engines.