Model predictive control for robust quantum state preparation

Andy J. Goldschmidt1, Jonathan L. DuBois2, Steven L. Brunton3, and J. Nathan Kutz4

1Department of Physics, University of Washington, Seattle, WA 98195
2Lawrence Livermore National Laboratory, Livermore, CA 94550
3Department of Mechanical Engineering, University of Washington, Seattle, WA 98195
4Department of Applied Mathematics, University of Washington, Seattle, WA 98195

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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 $\textit{model predictive control}$ (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.

Model predictive control (MPC) is a popular approach for control design in many areas of engineering. This is due to its capability to include constraints, and its robustness against noise and coherent modelling errors. In this work, we adapt classic model predictive control for the design of quantum control sequences to address the problem of planning over mischaracterized models of quantum devices. The MPC perspective is to solve the quantum control problem as if it is an $\textit{online}$ optimization, where the control sequence is optimized sequentially over a receding horizon. We demonstrate the practical benefits of this MPC perspective by studying three representative examples in quantum control engineering.

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