Model predictive control for robust quantum state preparation

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

doi:10.22331/q-2022-10-13-837

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

Published:	2022-10-13, volume 6, page 837
Eprint:	arXiv:2201.05266v2
Doi:	https://doi.org/10.22331/q-2022-10-13-837
Citation:	Quantum 6, 837 (2022).

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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 $\textit{model predictive control}$ (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.

Featured image: $\textbf{(a)}$ In the circuit model, a qubit is represented by a line segment, and a box is draw to indicate a quantum process applied to the qubit. Here, we represent a quantum state preparation abstractly by $\textbf{A}$ and posit uncharacterized modifications that make $\textbf{A}$’s model unreliable for open-loop control. $\textbf{(b)}$ Under the hood of the $\textbf{A}$ operation, we design robust control pulses using model predictive control (MPC). In $\textbf{(b) ii-iii.}$, MPC synthesizes a robust control input by incorporating state feedback into a sequence of receding-horizon control laws (in $\textbf{(b) ii.}$, the first three MPC iterations are labeled 1., 2., and 3. and colored with increasing grayscale value). At each MPC iteration, an open-loop control problem is solved over the current prediction horizon T using $\textbf{A}$’s unreliable system model to yield $\mathbf{x}^\textrm{opt}(t)$ and $\mathbf{u}^\textrm{opt}(t)$. The first entry of the open-loop solution $\mathbf{u}^\textrm{opt}(t)$ is applied as $\mathbf{u}(t)$ in $\textbf{(b) iii.}$. The resulting state $\mathbf{x}(t + 1)$ is recorded, and the next MPC iteration begins from this recorded state.

Popular summary

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.

► BibTeX data

@article{Goldschmidt2022modelpredictive,
  doi = {10.22331/q-2022-10-13-837},
  url = {https://doi.org/10.22331/q-2022-10-13-837},
  title = {Model predictive control for robust quantum state preparation},
  author = {Goldschmidt, Andy J. and DuBois, Jonathan L. and Brunton, Steven L. and Kutz, J. Nathan},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {837},
  month = oct,
  year = {2022}
}

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[1] Lennart Maximilian Seifert, Ziqian Li, Tanay Roy, David I. Schuster, Frederic T. Chong, and Jonathan M. Baker, "Exploring ququart computation on a transmon using optimal control", Physical Review A 108 6, 062609 (2023).

[2] Ettore Canonici, Stefano Martina, Riccardo Mengoni, Daniele Ottaviani, and Filippo Caruso, "Machine Learning based Noise Characterization and Correction on Neutral Atoms NISQ Devices", Advanced Quantum Technologies 7 1, 2300192 (2024).

[3] Yuzuru Kato and Hiroya Nakao, 2023 62nd IEEE Conference on Decision and Control (CDC) 5912 (2023) ISBN:979-8-3503-0124-3.

[4] Jason D. Chadwick and Frederic T. Chong, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 1286 (2023) ISBN:979-8-3503-4323-6.

[5] Maison Clouatre, Mark Balas, Vinod Gehlot, and John Valasek, "Linear Quantum State Observers", IEEE Transactions on Quantum Engineering 3, 1 (2022).

[6] Yuzuru Kato and Hiroya Nakao, 2022 IEEE 61st Conference on Decision and Control (CDC) 5141 (2022) ISBN:978-1-6654-6761-2.

[7] Christiane P. Koch, Ugo Boscain, Tommaso Calarco, Gunther Dirr, Stefan Filipp, Steffen J. Glaser, Ronnie Kosloff, Simone Montangero, Thomas Schulte-Herbrüggen, Dominique Sugny, and Frank K. Wilhelm, "Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe", EPJ Quantum Technology 9 1, 19 (2022).

[8] Jason D. Chadwick and Frederic T. Chong, "Efficient control pulses for continuous quantum gate families through coordinated re-optimization", arXiv:2302.01553, (2023).

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