Estimation and control of open quantum systems – Q-COAST

We use non-linear control methods, in particular stochastic Lyapunov methods, stochastic tools, quantum filtering theory for classical and non-classical inputs, quantum stochastic calculus, geometric control, stabilization of quantum systems with quantum feedback; methods for estimating parameters and identifications, perturbation theory, adiabatic elimination, methods of differential geometry and information geometry, quantum statistical mechanics, quantum information theory, etc.

1. Regarding stabilization of N-level quantum systems, we obtained results of robustness of feedback with respect to unknown initial states and also physical parameters. The preliminary result for spin-1/2 was presented at the Conference on Decision and Control 2020 conference. This presents the first result in robustness.
2. We have considered stochastic master equations describing the evolution of a multi-qubit system interacting with electromagnetic fields undergoing continuous time measurements. We give sufficient conditions on the controller and the control Hamiltonian ensuring an almost sure exponential convergence toward maximum entangled of multi-qubit systems (GHZ states). This result is the first result of stabilization of these states exponentially.

3. We have tackled the question of the asymptotic stability of quantum trajectories, also called quantum filters. We determine the limit of quantum fidelity between the true quantum trajectory and the estimated one. Under assumptions of purification and absolute continuity of the initial conditions, we show that this limit is equal to one, which means that the quantum filters are stable. In the general case, under an identifiability assumption, a spectral assumption and an assumption of absolute continuity of the initial conditions, we show that the limit of the Cesaro mean of the estimated trajectory is the same as the true one. Compared to the previous results, this result is stronger because it proves the case where the fidelity is one and also it specifies the limit of fidelity. This result has an impact on the feedback based on the measurement, because the robustness with respect to the initial state is important. For the spin-1/2 system, in the presence of feedback, we have shown asymptotic stability but with another approach which is based on the explicit calculation of the fidelity which has an explicit form in this case.

4. We have developed an identification algorithm to identify a model for unknown linear quantum systems driven by coherent time varying states, based on empirical continuous Homodyne measurement data. The proposed algorithm identifies a model which satisfies the constraints imposed by quantum physics for linear quantum systems, with the constraints not encountered in the identification of classical linear systems. Numerical examples of a multiple input, multiple output optical cavity model are presented to illustrate an application of the identification algorithm.

1. Reduced quantum filters and applications.
2. Estimation of parameters of open quantum systems.
3. Estimation and control of open quantum systems for non-classical input states / Generation of non-classical states.
4. Interaction between quantum filtering theory, quantum feedback, statistical physics and information geometry.
5. Implementation of our feedback results in real experiences.
6. Identification of open quantum systems.

[1] N. H. Amini, M. Bompais, and C. Pellegrini, On asymptotic stability of quantum trajectories and their Cesaro mean, Journal of Physics A: Mathematical and Theoretical, 2021. Link: hal.archives-ouvertes.fr/hal-03308505
[2] W. Liang, N. H. Amini, and P. Mason, Feedback exponential stabilization of GHZ states of multi-qubit systems, IEEE Transactions on Automatic Control, 2021. Link: hal.archives-ouvertes.fr/hal-03015479
[3] W. Liang, N. H. Amini, and P. Mason, Robust feedback stabilization of N-level quantum spin systems, Siam Journal on Control and Optimization, 59(1), 669–692, 2021. Link: hal.inria.fr/hal-02943455
[4] W. Liang, N. H. Amini, and P. Mason, On estimation and feedback control of spin-1/2 systems with unknown initial states, World Congress of the International Federation of Automatic Control (IFAC), 2020. Link: hal.archives-ouvertes.fr/hal-03015497
[5] W. Liang, N. H. Amini, and P. Mason, On the robustness of stabilizing feedbacks for quantum spin- 1/2 systems, 59th IEEE Conference on Decision and Control (CDC), 2020. Link: hal.archives-ouvertes.fr/hal-03015492
[6] H. I. Nurdin, N. Amini, J. Chen, Data-driven system identification of linear quantum systems coupled to time-varying coherent inputs, 59th IEEE Conference on Decision and Control (CDC), 2020. Link: hal.archives-ouvertes.fr/hal-03059873

[7] J. E. Gough, T. S. Ratiu, and O. G. Smolyanov, Wigner measures and coherent quantum control, Proceedings of the Steklov Institute of Mathematics, 313(1), pages 52--59, Springer, 2021.
[8] H. Nurdin and J. Gough, From the Heisenberg to the Schrödinger Picture: Quantum Stochastic Processes and Process Tensors, To appear in 60th IEEE Conference on Decision and Control (CDC), 2021.

Quantum Control attempts to apply and extend the principles already used for classical control systems to the quantum domain. In this way we hope to establish a control theory specifically dedicated to regulating quantum systems.

This proposal addresses some key problems related to the control of open quantum systems by applying quantum feedback control. Open quantum systems are quantum systems in interaction with an environment. This interaction perturbs the system states and causes loss of information from the system to the environment. However by applying quantum feedback control, the system can “fight” against this loss of information. The main obstacle is that standard strategies from classical control are not immediately applicable to quantum systems. While there has been much development on the theoretical side, there remain key open questions concerning optimality, robustness, and best design methods for dealing with generic quantum models which can be implemented in concrete experiments with less difficulties. The first objective of Q-COAST is to develop more efficient and robust strategies for quantum feedback design applied to open quantum systems. As a second objective, we investigate the situation where the inputs are in non-classical states, the case where the generalization from the classical to the quantum case becomes more difficult. Such states are critically important for scalable quantum information processing. Our third objective is to go beyond the existing tools to design estimators and controllers. This will be achieved by introducing new pathways through the interaction between fields of quantum statistical mechanics, quantum information geometry, quantum filtering, and quantum feedback control. The final goal is to develop further numerical simulations of quantum components as well as implementing our proposed strategies in real experiments. The experimental implementations can be realized as the project will involve collaboration with leading experimental groups who have been successfully applying feedback control theoretic principles to actual quantum systems.