Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1704.07653 · Physical Review A

Robust optimal control of two-level quantum systems

Léo Van Damme, Quentin Ansel, Steffen J. Glaser, Dominique Sugny

Published Jun 2, 2017• 89 citations
arXivHTMLPDF

Abstract

We investigate the time- and energy-minimum optimal solutions for the robust control of two-level quantum systems against offset or control-field uncertainties. Using the Pontryagin maximum principle, we derive the global optimal pulses for the first robustness orders. We show that the dimension of the control landscape is lower than or equal to $2N$ for a field robust to the $N\mathrm{th}$ order, which leads to an estimate of its complexity.

DOI
10.1103/physreva.95.063403

Related papers

Application of the Pontryagin maximum principle to the robust time-optimal control of two-level quantum systems

O. Fresse-Colson, S. Guérin, Xi Chen +1 more

We study the time-optimal robust control of a two-level quantum system subjected to field inhomogeneities. We apply the Pontryagin maximum principle and we introduce a reduced space onto which the optimal dynamics is projected down. This reduction leads to a complete analytical derivation of the optimal solution in terms of elliptic functions and elliptic integrals. Necessary optimality conditions are then obtained for the original system. These conditions are verified numerically and lead to the optimal control protocol. Various examples, ranging from state-to-state transfer to the generation of a NOT gate, illustrate this study. The connection with other geometric optimization approaches that have been used to solve this problem is also discussed. Finally, we show how this method can be combined with numerical optimization algorithms to design robust control for higher-dimensional quantum systems, as illustrated by a Bose-Einstein condensate in an optical lattice.

arXiv:2503.11830•Physical Review A•Aug 6, 2025•7 citations

Time-optimal control of two-level quantum systems by piecewise constant pulses

E Dionis, Dominique Sugny

We apply an extension of the Pontryagin maximum principle to derive time-optimal controls of two-level quantum systems by means of piecewise constant pulses. Global optimal solutions are obtained for state-to-state transfer in the cases with one and two controls. Exact quantum speed limits are established as a function of the sampling period. We observe numerically an exponential convergence towards the minimum time in the continuous limit when this period goes to zero. We show that this convergence is only polynomial for a linearized quantum system. We discuss the experimental impact of this result.

arXiv:2211.09167•Physical Review A•Mar 17, 2023•18 citations

Optimal control and ultimate bounds of 1:2 nonlinear quantum systems

J. J. Zhu, Kaipeng Liu, Xi Chen +1 more

Using optimal control, we establish and link the ultimate bounds in time (referred to as the quantum speed limit) and energy of two- and three-level quantum nonlinear systems which feature 1:2 resonance. Despite the unreachable complete inversion, by using the Pontryagin maximum principle, we determine the optimal time, pulse area, or energy for a given arbitrary accuracy. We show that the third-order Kerr terms can be absorbed in the detuning in order to lock the dynamics to the resonance. In the two-level problem, we determine the nonlinear counterpart of the optimal $\ensuremath{\pi}$-pulse inversion for a given accuracy. In the three-level problem, we obtain an intuitive pulse sequence similar to the linear counterpart but with different shapes. We prove the (slow) logarithmic increasing of the optimal time as a function of the accuracy.

arXiv:2303.15359•Physical Review A•Oct 16, 2023•9 citations

Constructing Noise-Robust Quantum Gates via Pontryagin's Maximum Principle

Joshua Hanson, Dennis G. Lucarelli

Reliable quantum information technologies depend on precise actuation and techniques to mitigate the effects of un-desired disturbances such as environmental noise and imperfect calibration. In this work, we present a general framework based in geometric optimal control theory to synthesize smooth control pulses for implementing arbitrary noise-robust quantum gates. The methodology applies to generic unitary quantum dynamics with any number of qubits or energy levels, any number of control fields, and any number of disturbances, extending existing dynamical decoupling approaches that are only applicable for limited gate sets or small systems affected by one or two disturbances. The noise-suppressing controls are computed via indirect trajectory optimization based on Pontryagin's maximum principle, eliminating the need to make heuristic structural assumptions on parameterized pulse envelopes.

arXiv:2409.12287•IEEE International Conference on Quantum Computing and Engineering (QCE)•Sep 15, 2024•2 citations

Time-optimal robust quantum control for partially known errors

Llorenç Balada Gaggioli

Robust quantum control usually assumes one of two extremes: the error is of a known form and only its strength is unknown, or nothing at all is known about it. In practice one often sits in between. The size of a static error is unknown, but its class is known, for example, it only shifts energy levels, or it acts on each subsystem separately. We ask how fast a control can cancel, to first order, every error in such a class. A time-independent Hamiltonian can do this if and only if all errors in the class have zero diagonal in a common basis. In that case an equally spaced energy ladder in this basis does it after one period. A lower bound that holds for any time-dependent control with bounded drive power shows that this simple constant control is optimal for several natural classes, including unknown level shifts and unknown local errors on two subsystems. Completely unknown errors need a time-dependent control, and there, knowing the error class shortens the minimum time by a factor $\sqrt{d^2+1}$. So partial knowledge of the error changes not only how long an optimal robust control takes, but also its form.

quant-pharXiv:2610.21596•Oct 16, 2026•0 citations

Quantum robust control theory for Hamiltonian and control field uncertainty*

Andy Koswara, Vaibhav Bhutoria, Raj Chakrabarti

Abstract Quantum robust control—which can employ fast leading order approximations, slower but more accurate asymptotic methods, or a combination thereof for quantification of robustness—enables control of moments of quantum observables and gates in the presence of Hamiltonian uncertainty or field noise. In this paper, we present a generalized quantum robust control theory that extends the previously described theory of quantum robust control in several important ways. We present robust control theory for control of any moment of arbitrary quantum control objectives, introducing moment-generating functions and transfer functions for quantum robust control that generalize the tools of frequency domain response theory to quantum systems, and extend the Pontryagin maximum principle for quantum control to control optimization in the presence of noise in the manipulated amplitudes or phases used to shape the control field. To provide guidelines as to the types of quantum control systems and control objectives for which asymptotic robustness analysis is important for accuracy, we introduce methods for assessing the Lie algebraic depth of quantum control systems, and illustrate through examples drawn from quantum information processing how such accurate methods for quantification of robustness to noise and uncertainty are more important for control strategies that exploit higher order quantum pathways. In addition, we define the relationship between leading order Taylor expansions and asymptotic estimates for quantum control moments in the presence of Hamiltonian uncertainty and field noise, and apply such leading order approximations to significant pathways analysis and dimensionality reduction of asymptotic quantum robust control calculations, describing numerical methods for implementation of these calculations.

New Journal of Physics•May 24, 2021•24 citations
We investigate the time- and energy-minimum optimal solutions for the robust control of two-level quantum systems against offset or control-field uncertainties. Using the Pontryagin maximum principle, we derive the global optimal pulses for the first robustness orders. We show that the dimension of the control landscape is lower than or equal to 2N2N2N for a field robust to the NthN\mathrm{th}Nth order, which leads to an estimate of its complexity.
We study the time-optimal robust control of a two-level quantum system subjected to field inhomogeneities. We apply the Pontryagin maximum principle and we introduce a reduced space onto which the optimal dynamics is projected down. This reduction leads to a complete analytical derivation of the optimal solution in terms of elliptic functions and elliptic integrals. Necessary optimality conditions are then obtained for the original system. These conditions are verified numerically and lead to the optimal control protocol. Various examples, ranging from state-to-state transfer to the generation of a NOT gate, illustrate this study. The connection with other geometric optimization approaches that have been used to solve this problem is also discussed. Finally, we show how this method can be combined with numerical optimization algorithms to design robust control for higher-dimensional quantum systems, as illustrated by a Bose-Einstein condensate in an optical lattice.
We apply an extension of the Pontryagin maximum principle to derive time-optimal controls of two-level quantum systems by means of piecewise constant pulses. Global optimal solutions are obtained for state-to-state transfer in the cases with one and two controls. Exact quantum speed limits are established as a function of the sampling period. We observe numerically an exponential convergence towards the minimum time in the continuous limit when this period goes to zero. We show that this convergence is only polynomial for a linearized quantum system. We discuss the experimental impact of this result.
Using optimal control, we establish and link the ultimate bounds in time (referred to as the quantum speed limit) and energy of two- and three-level quantum nonlinear systems which feature 1:2 resonance. Despite the unreachable complete inversion, by using the Pontryagin maximum principle, we determine the optimal time, pulse area, or energy for a given arbitrary accuracy. We show that the third-order Kerr terms can be absorbed in the detuning in order to lock the dynamics to the resonance. In the two-level problem, we determine the nonlinear counterpart of the optimal $\ensuremath{\pi}$-pulse inversion for a given accuracy. In the three-level problem, we obtain an intuitive pulse sequence similar to the linear counterpart but with different shapes. We prove the (slow) logarithmic increasing of the optimal time as a function of the accuracy.
Reliable quantum information technologies depend on precise actuation and techniques to mitigate the effects of un-desired disturbances such as environmental noise and imperfect calibration. In this work, we present a general framework based in geometric optimal control theory to synthesize smooth control pulses for implementing arbitrary noise-robust quantum gates. The methodology applies to generic unitary quantum dynamics with any number of qubits or energy levels, any number of control fields, and any number of disturbances, extending existing dynamical decoupling approaches that are only applicable for limited gate sets or small systems affected by one or two disturbances. The noise-suppressing controls are computed via indirect trajectory optimization based on Pontryagin's maximum principle, eliminating the need to make heuristic structural assumptions on parameterized pulse envelopes.
Robust quantum control usually assumes one of two extremes: the error is of a known form and only its strength is unknown, or nothing at all is known about it. In practice one often sits in between. The size of a static error is unknown, but its class is known, for example, it only shifts energy levels, or it acts on each subsystem separately. We ask how fast a control can cancel, to first order, every error in such a class. A time-independent Hamiltonian can do this if and only if all errors in the class have zero diagonal in a common basis. In that case an equally spaced energy ladder in this basis does it after one period. A lower bound that holds for any time-dependent control with bounded drive power shows that this simple constant control is optimal for several natural classes, including unknown level shifts and unknown local errors on two subsystems. Completely unknown errors need a time-dependent control, and there, knowing the error class shortens the minimum time by a factor d2+1\sqrt{d^2+1}d2+1​. So partial knowledge of the error changes not only how long an optimal robust control takes, but also its form.
Abstract Quantum robust control—which can employ fast leading order approximations, slower but more accurate asymptotic methods, or a combination thereof for quantification of robustness—enables control of moments of quantum observables and gates in the presence of Hamiltonian uncertainty or field noise. In this paper, we present a generalized quantum robust control theory that extends the previously described theory of quantum robust control in several important ways. We present robust control theory for control of any moment of arbitrary quantum control objectives, introducing moment-generating functions and transfer functions for quantum robust control that generalize the tools of frequency domain response theory to quantum systems, and extend the Pontryagin maximum principle for quantum control to control optimization in the presence of noise in the manipulated amplitudes or phases used to shape the control field. To provide guidelines as to the types of quantum control systems and control objectives for which asymptotic robustness analysis is important for accuracy, we introduce methods for assessing the Lie algebraic depth of quantum control systems, and illustrate through examples drawn from quantum information processing how such accurate methods for quantification of robustness to noise and uncertainty are more important for control strategies that exploit higher order quantum pathways. In addition, we define the relationship between leading order Taylor expansions and asymptotic estimates for quantum control moments in the presence of Hamiltonian uncertainty and field noise, and apply such leading order approximations to significant pathways analysis and dimensionality reduction of asymptotic quantum robust control calculations, describing numerical methods for implementation of these calculations.