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.
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.
This paper explores the application of Model Predictive Control (MPC) strategies to quantum systems undergoing continuous-time measurement. The MPC control law is formulated based on the estimated state obtained from quantum filtering. As quantum filtering is affected by uncertainties, applying MPC for real-time monitoring enhances the stability of the filtering process. Our analysis demonstrates that the MPC framework ensures exponential stability in mean square. Additionally, we establish that a nonlinear MPC model guarantees system stability under specified conditions. Furthermore, the inherent robustness of MPC allows the system to remain exponentially bounded in mean square, even in the presence of environmental disturbances. Numerical simulations are provided to validate these control properties of 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.
This review develops a control-theoretic perspective on measurement-based feedback for continuously monitored open quantum systems, with the main analysis focused on finite-dimensional systems governed by diffusive stochastic master equations. We introduce the relevant state-space, invariant-subspace, and quantum non-demolition structures, interpret quantum filtering as nonlinear observer dynamics, and review open-loop asymptotics, filter stability, state-feedback stabilization, robustness, and reduced-order observer-based control. Particular emphasis is placed on a recurrence--contraction framework, in which Hamiltonian feedback removes non-target invariant obstructions while measurement-induced dynamics provide local exponential contraction. Although the detailed analysis is developed for finite-dimensional diffusive models, the underlying measurement--estimation--feedback architecture is relevant across a broad range of quantum platforms. We further discuss implementation challenges and open problems involving scalable estimation, sampling and delay, adaptation, hybrid and non-Markovian dynamics, practical stability, optimal control, and learning-based design. By organizing these developments around invariance, estimation, recurrence, and contraction, the review provides a tutorial bridge between measurement-based quantum feedback and nonlinear stochastic control.
A critical engineering challenge in quantum technology is the accurate control of quantum dynamics. Model-based methods for optimal control have been shown to be highly effective when theory and experiment closely match. Consequently, realizing high-fidelity quantum processes with model-based control requires careful device characterization. In quantum processors based on cold atoms, the Hamiltonian can be well-characterized. For superconducting qubits operating at milli-Kelvin temperatures, the Hamiltonian is not as well-characterized. Unaccounted for physics (i.e., mode discrepancy), coherent disturbances, and increased noise compromise traditional model-based control. This work introduces model predictive control (MPC) for quantum control applications. MPC is a closed-loop optimization framework that (i) inherits a natural degree of disturbance rejection by incorporating measurement feedback, (ii) utilizes finite-horizon model-based optimizations to control complex multi-input, multi-output dynamical systems under state and input constraints, and (iii) is flexible enough to develop synergistically alongside other modern control strategies. We show how MPC can be used to generate practical optimized control sequences in representative examples of quantum state preparation. Specifically, we demonstrate for a qubit, a weakly-anharmonic qubit, and a system undergoing crosstalk, that MPC can realize successful model-based control even when the model is inadequate. These examples showcase why MPC is an important addition to the quantum engineering control suite.
Quantum engineering has seen rapid growth in the past two decades. Physicists, mathematicians and engineers have been working in unison to control a number of diverse systems in the quantum regime. Quantum control involving feedback has become particularly topical, as using information gained from a system can lead to more stable operation of a control protocol. Quantum feedback can be split into two paradigms: measurement based feedback and coherent feedback. Although it is increasingly evident that retaining the coherence of the feedback signal provides an intrinsic advantage over measurement-based feedback, coherent feedback is still a new paradigm. In particular, there are a limited number of options for coherently estimating a state within a feedback loop. As a step towards better understanding and implementation of quantum feedback, this thesis reports on modelling, estimation and control of quantum systems. The first topic is physical realisability of a variety of quantum systems, especially finite level systems and their outgrowths. Bilinear quantum stochastic differential equations (QSDEs) have to be employed to characterise these systems, which is a significant complement to the previous work on linear QSDEs. In light of the fact that direct and indirect couplings play a vital role in quantum network and control, we also provide state-space models for different classes of coupled open quantum systems incorporating both bidirectional and directional interactions. The second topic is coherent observers and optimal filtering. It is well established that classically estimation using the Kalman filter can provide improved performance over direct feedback schemes, and similar demonstrations have been performed for measurement-based quantum feedback. In this thesis I present coherent observers, including least mean squares estimators, which are driven by the coherent output of a specified quantum plant and designed such that some subset of the observer and plant's expectation values converge in the asymptotic limit. Not only can estimators "observe" mean quantities of quantum plant, but also they can "estimate" quantum correlations such as entanglement. This is of much importance to coherent feedback design in the absence of measurement steps without loss of fidelity. Last but not least, several topics regarding stabilisation and control design of quantum systems are discussed in my thesis. To be specific, tools for quantum stability analysis (e.g. Lyapunov conditions) are provided. Both measurement-based feedback and coherent feedback control design are taken into account. A measurement-based optimal controller for opto-mechanical systems aimed at synchronising different mechanical modes is presented, which is application-oriented. Moreover, A general observer-based coherent feedback control framework is studied, and I demonstrate a pole-placement approach via coherent observers that can be applied to various scenarios.
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.