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:2303.15359 · Physical Review A

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

J. J. Zhu, Kaipeng Liu, Xi Chen, S. Guérin

Published Oct 16, 2023• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.108.042610

Related papers

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

Robust optimal control of two-level quantum systems

Léo Van Damme, Quentin Ansel, Steffen J. Glaser +1 more

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.

arXiv:1704.07653•Physical Review A•Jun 2, 2017•89 citations

Estimation of optimal control for two-level and three-level quantum systems with bounded amplitude

Xikun Li

Abstract A systematic scheme is proposed to numerically estimate the quantum speed limit and temporal shape of optimal control in two-level and three-level quantum systems with bounded amplitude. For the two-level system, two quantum state transitions are studied as illustration. Comparisons between numerical and analytical results are made, and deviations are significantly small. For the three-level system, two critical time points are determined with high accuracy, and optimal controls are obtained for different durations. The shape of optimized control field is simple and does not switch frequently, thus are easy to implement in experiment. In addition, we compare our method with the chopped random basis (CRAB), and the performance of our method is much better than that of CRAB. Our scheme is of importance in estimating the quantum speed limit and optimal control for cases in which analytical solution is absent.

arXiv:2208.13377•Physica Scripta•Sep 8, 2023•1 citation

Control of nonlinear quantum systems: From nonlinear Rabi oscillations to robust two-stage optimal inverse engineering

J. J. Zhu, S. Guérin

Nonlinear quantum systems behave radically different from their linear counterparts and obstructions, identified as separatrix crossing and bifurcations in the phase space, can prevent their efficient control by standard approaches using adiabatic passage. We provide the exact resonant solution by solving the time-dependent nonlinear Schr\"odinger equation in terms of a Jacobi elliptic integral, interpreting the dynamics with the motion of a planar pendulum. This is referred to as nonlinear Rabi oscillation, which shows an obstruction to complete population transfer above a certain critical value of the nonlinearity strength, corresponding to an oscillating regime of the pendulum. Below this threshold, the so-called $2K$ pulse induces a complete population transfer, associated with a rotating regime of the pendulum, and representing the nonlinear counterpart of the $\ensuremath{\pi}$ pulse. We next propose a fast, robust, and high-fidelity control technique by a two-stage optimal inverse engineering strategy. It consists in dynamically compensating (diagonal) third-order nonlinearities, referred to as inverse linearization, combined with robust inverse optimization (RIO) developed for linear systems. We show that a moderate nonlinearity strength dramatically improves robustness.

Physical Review A•Dec 2, 2024•3 citations

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

Optimal Control of Quantum Systems by Pontryagin Maximum Principle

Nahid Binandeh Dehaghani, Фернандо Лобо Перейра

Optimal control provides powerful tools and concepts that can be applied to control quantum systems. It has been used extensively to improve the performance of quantum processes in a variety of active areas in quantum technologies. This paper reviews the necessary conditions of optimality in the form of a Pontryagin Maximum Principle (PMP), and introduces its formulations for quantum systems. After formulating an optimal control problem for a two-level quantum system, we then study the application of PMP to obtain an optimal control strategy.

U Porto Journal of Engineering•Apr 28, 2022•9 citations
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 \ensuremathπ\ensuremath{\pi}\ensuremathπ-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.
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.
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.
Abstract A systematic scheme is proposed to numerically estimate the quantum speed limit and temporal shape of optimal control in two-level and three-level quantum systems with bounded amplitude. For the two-level system, two quantum state transitions are studied as illustration. Comparisons between numerical and analytical results are made, and deviations are significantly small. For the three-level system, two critical time points are determined with high accuracy, and optimal controls are obtained for different durations. The shape of optimized control field is simple and does not switch frequently, thus are easy to implement in experiment. In addition, we compare our method with the chopped random basis (CRAB), and the performance of our method is much better than that of CRAB. Our scheme is of importance in estimating the quantum speed limit and optimal control for cases in which analytical solution is absent.
Nonlinear quantum systems behave radically different from their linear counterparts and obstructions, identified as separatrix crossing and bifurcations in the phase space, can prevent their efficient control by standard approaches using adiabatic passage. We provide the exact resonant solution by solving the time-dependent nonlinear Schr\"odinger equation in terms of a Jacobi elliptic integral, interpreting the dynamics with the motion of a planar pendulum. This is referred to as nonlinear Rabi oscillation, which shows an obstruction to complete population transfer above a certain critical value of the nonlinearity strength, corresponding to an oscillating regime of the pendulum. Below this threshold, the so-called 2K2K2K pulse induces a complete population transfer, associated with a rotating regime of the pendulum, and representing the nonlinear counterpart of the $\ensuremath{\pi}$ pulse. We next propose a fast, robust, and high-fidelity control technique by a two-stage optimal inverse engineering strategy. It consists in dynamically compensating (diagonal) third-order nonlinearities, referred to as inverse linearization, combined with robust inverse optimization (RIO) developed for linear systems. We show that a moderate nonlinearity strength dramatically improves robustness.
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.
Optimal control provides powerful tools and concepts that can be applied to control quantum systems. It has been used extensively to improve the performance of quantum processes in a variety of active areas in quantum technologies. This paper reviews the necessary conditions of optimality in the form of a Pontryagin Maximum Principle (PMP), and introduces its formulations for quantum systems. After formulating an optimal control problem for a two-level quantum system, we then study the application of PMP to obtain an optimal control strategy.