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IEEE Conference on Decision and Control (CDC)

Model Predictive Control of Two-level Open Quantum Systems

Yunyan Lee, Ian R. Petersen, Daoyi Dong

Published Dec 16, 2024• 2 citations
Publisher

Abstract

This paper presents a robust control approach for a two-level open quantum system subject to bounded uncertainties through the application of model predictive control. We demonstrate two common cases: depolarizing decoherence and phase-damping decoherence in open quantum systems. We develop a model predictive control method using quantum measurements and provide a lower bound on the probability of obtaining the predicted state. Numerical results illustrate the effectiveness of this model predictive control approach in achieving stability and robustness in open quantum systems.

DOI
10.1109/cdc56724.2024.10886028

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This paper presents a robust control approach for a two-level open quantum system subject to bounded uncertainties through the application of model predictive control. We demonstrate two common cases: depolarizing decoherence and phase-damping decoherence in open quantum systems. We develop a model predictive control method using quantum measurements and provide a lower bound on the probability of obtaining the predicted state. Numerical results illustrate the effectiveness of this model predictive control approach in achieving stability and robustness in open quantum systems.
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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.
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.
We exploit how to overcome phase damping decoherence when the phase damping parameters are uncertain. It is shown that the “relaxed” control goal may be achieved by combing simple open-loop coherent control with periodic projective measurements. Both the sufficient conditions and constructive design procedures are presented. The control design is reduced to selecting the period of projective measurements and fidelity-truncated time based on two lower-bound functions for both probability and fidelity performance. When both probability and fidelity performances are specified, the stabilization time is further maximized by searching the “optimal” period of projective measurements. It is also exemplified that larger uncertainties may lead to shorter stabilization time for the given performance level.
Abstract Model predictive control (MPC) is one of the most successful modern control methods. It relies on repeatedly solving a finite-horizon optimal control problem and applying the beginning piece of the optimal input. In this paper, we develop a modular framework for improving efficiency and robustness of quantum optimal control (QOC) via MPC. We first provide a tutorial introduction to basic concepts of MPC from a QOC perspective. We then present multiple MPC schemes, ranging from simple approaches to more sophisticated schemes which admit stability guarantees. This yields a modular framework which can be used 1) to improve efficiency of open-loop QOC and 2) to improve robustness of closed-loop quantum control by incorporating feedback. We demonstrate these benefits with numerical results, where we benchmark the proposed methods against competing approaches.
To make full use of the amazing characteristics of quantum states, control of open quantum systems is of great significance in practical applications. This paper studies a more practical control problem where the decoherence parameter is not known exactly. The problem is considered in two phases. First, the unknown decoherence strength is estimated based on continuous measurements and an appropriately designed control. Then, Markovian feedback control is designed to deal with the decoherence effect based on the accurate parameter estimation. It is shown that state transfer with a high probability can be achieved by designing the feedback gains and the measurement. Moreover, good state transfer performance can be guaranteed even there exists an estimation error on the decoherence strength. Simulation studies are included to demonstrate the effectiveness of the proposed approach.