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arXiv:2201.05266 · Quantum

Model predictive control for robust quantum state preparation

Andy J. Goldschmidt, Jonathan L. DuBois, Steven L. Brunton, J. Nathan Kutz

Published Oct 13, 2022• 17 citations
arXivHTMLPDF

Abstract

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.

DOI
10.22331/q-2022-10-13-837

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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.
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.
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.
This paper presents a novel approach to dynamic quantum control utilizing feedback learning algorithms. The core idea is to develop a system capable of real-time adjustment of control parameters based on feedback information obtained from the quantum system itself. This approach addresses the inherent challenges in precisely controlling quantum systems, which are often highly sensitive to environmental noise and difficult to model accurately. The proposed framework leverages the principles of quantum control and feedback learning to achieve optimal control performance. The system architecture incorporates a closed-loop control mechanism where the system's output is continuously monitored, and the control parameters are adjusted accordingly. Mathematical formulations detailing the control process, feedback mechanisms, and optimization strategies are provided. This work contributes to advancements in quantum control by offering a robust and adaptable method for manipulating quantum states, with potential applications in quantum computing, quantum sensing, and quantum communication. The algorithm is designed to minimize the influence of external disturbances and maximize the accuracy of state preparation and manipulation. The presented approach provides a pathway towards more reliable and efficient quantum control strategies.
This article presents a robust control strategy using Time-Optimal Model Predictive Control (TOMPC) for a two-level quantum system subject to bounded uncertainties. In this method, the control field is optimized over a finite horizon using a nominal quantum system as the reference and then the optimal control for the first time interval is applied and a projective measurement is implemented on the uncertain system. The new control field for the next time interval will be iteratively optimized based on the measurement result. We present theoretical results to guarantee the stability of the TOMPC algorithm. We also characterize the robustness and the convergence rate of the TOMPC strategy for the control of two-level systems. Numerical simulations further demonstrate that, in the presence of uncertainties, our quantum TOMPC algorithm enhances robustness and steers the state to the desired state with high fidelity. This work contributes to the progress of Model Predictive Control in quantum control and explores its potential in practical applications of quantum technology.
The precise control of quantum systems enables the development of a wide range of promising quantum technologies. These technologies allow for beating the limits of their classical pendants and outperform them in various tasks. This thesis employs and expands a variety of methods and concepts of optimal control theory in order to address different problems in quantum technologies and work towards their realization. Schrödinger cat states present a particularly versatile set of states, that have a broad range of applications for all types of quantum technologies. We employ optimal control theory to investigate how cat states can be prepared in different physical systems. This allows us to investigate the role of higher order nonlinearities in anharmonic resonators. Further, this enables the identification of strategies for cat state preparation in the presence of higher order anharmonicities and an analysis if they can be exploited as resource. Additionally, we develop a set of functionals, which allow for optimizing towards arbitrary cat states. This facilitates the preparation of cat states even in systems, where the reachable set of states is not immediately apparent. Quantum computing is a particularly prominent quantum technology and promises a completely new paradigm for computing. Crosstalk, the undesired interaction between different parts of a quantum processor, poses a major challenge across various architectures. Therefore, we utilize concepts from optimal control to introduce the perfect entangler spectrum as a tool to detect operational crosstalk, caused by spectator qubits. This allows us to identify parameter regimes of superconducting qubits, which exhibit crosstalk and determine the underlying mechanisms. In addition, we utilize optimal control and parameter optimizations to explore mitigation strategies for the different crosstalk mechanisms. Our work paves the way for systematically eliminating crosstalk, when scaling up quantum computing architectures.