New approaches to control open quantum systems

New approaches to control open quantum systems

This proposal will focus on problems in control and observation for quantum systems. The

project is within the project ANRQ-COAST (estimation and control of open quantum systems).

In the last three decades, the growing interest in quantum technology initiated the development

of a decent quantum control theory. Measuring and controlling quantum systems are experimen-

tally feasible as it has been proven lately in the pioneering work [15]. Automatic control designs

turn out to be promising for quantum computation. The main focus is to apply quantum feed-

back control in order to manipulate the performance of open quantum systems interacting with

an environment with subsequent loss of information. Classical control theory is not immediately

applicable to quantum systems. Several open problems regarding optimality, robustness, and

best design methods for dealing with generic quantum models can be addressed combined with a

concrete implementation.

The main axes of future research are summarized as follows.

• PDE approach: A useful approach to describe concrete physical evolution is provided

by the theory of partial differential equations (PDEs), for instance stochastic Schrödinger

equation [2]. For such systems, continuous observed outputs might be feasible leading to

the question of output feedback control and stabilization. Such designs have appeared

for instance in [3] but for the finite-dimensional realization of such systems. However,

taking the continuum limit of the model, we obtain PDEs not depending on the number

of states leading to scalability and a uniform approach. A PDE approach would lead to

more scalable controllers as the high dimensionality of such systems complicates the design.

Single and interconnected PDE models of quantum systems require the construction of

a physically meaningful theory for efficient and robust control and estimation strategies.

Generic strategies based on Lyapunov theory for interconnected PDEs have appeared for

instance in [7, 8, 9]. It is required to adapt similar approaches in the context of stochastic

PDEs describing the evolution of quantum systems taking into account the specificities

and limitations occurring in such systems. Such methodologies have not been sufficiently

explored yet.

• Mean-field games in quantum control: Mean-field games [11] have physical realizations

in quantum systems, see e.g. [10, 4] for the propagation of chaos. Classical mean-field games

are characterized by the Hamilton-Jacobi-Bellman equation (for the value function u of each

player) combined with the Fokker-Planck equation (for the density function p of the players).

Nevertheless, it becomes nontrivial to describe the distribution of the quantum particles as

it is supported on the set of density matrices Sd which is Lebesgue negligible. Therefore, we

need to identify its support and find a suitable reference measure on its support to establish

the corresponding Fokker-Planck equation. Inspired by [18], we start by considering the

mean-field qubit system, where we may write explicitly the dynamics of each component.

For this system, we aim to show the well-posedness of the Hamilton-Jacobi-Bellman and

Fokker-Planck equations and to derive the propagation of chaos. Furthermore, it is mean-

ingful to investigate state reconstruction problems in the context of inverse problems for

PDEs, we may reconstruct the unknown dynamics of the aggregate distribution of the par-

ticles (Fokker-Planck equation) by observing the optimal control problem of an individual

(Hamilton-Jacobi-Bellman equation).

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• Nonlinear systems: Quantum feedback control when the controller is a quantum system is

treated for linear systems, see [14]. There still remains an optimal design. There are various

problems that should be addressed for quantum feedback control including robust stability

and feedback control design for nonlinear quantum systems (see e.g., [5, 12, 13]), which

capture better the dynamics of such systems. Most of the previous works study the case

when the controller is a classical controller and the controller depends on the measurements

which perturb the quantum system dynamics. For the latter case, we will study the model

reduction strategies to make a classical controller realistic in practice. For the quantum

controller, theories as well as constructive algorithmic realizations for both control and

observer problems of high-dimensional nonlinear systems will be established preserving also

the so-called physical relizability conditions. A classical Luenberger approach was extended

for instance to linear quantum systems. It would be of interest to investigate such extension

to nonlinear systems.

Contact: Nina Amini, Gaoyue Guo, Paolo Mason.

References

[1] D. M. Ambrose, Existence theory for non-separable mean field games in Sobolev spaces,

arXiv-1807.02223, 2020.

[2] A. Barchielli, C. Pellegrini, and F. Petruccione, Quantum trajectories: Memory and contin-

uous observation, Phys. Rev. A 86, 006384, 2012.

[3] G. Cardona, A. Sarlette and P. Rouchon, Exponential Stochastic Stabilization of a Two-Level

Quantum System via Strict Lyapunov Control, IEEE Conference on Decision and Control

(CDC), Miami, FL, USA, pp. 6591-6596,2018.

[4] S. Chalal, N. Amini, and G. Guo, An invitation to qyuantum mean-field filtering and control,

Submitted, 2023.

[5] D. Dong and I. R. Petersen, Quantum estimation, control and learning: Opportunities and

challenges, Annual Reviews in Control, vol. 54, pp. 243–251, 2022.

[6] F. Golse, C. Mouhot, and T. Paul, On the mean field and classical limits of quantum me-

chanics, hal-01119132v3f, 2015.

[7] C. Kitsos, and E. Fridman, Stabilization of underactuated linear coupled reaction-diffusion

PDEs via distributed or boundary actuation, Submitted to IEEE Transactions on Automatic

Control, 2022.

[8] C. Kitsos, E. Cerpa, G. Besancon, and C. Prieur, Output feedback control of a cascade system

of linear Korteweg-de Vries equations, SIAM J. on Control and Optimization (SICON), vol.

59(4), pp. 2955–2976, 2021.

[9] C. Kitsos, G. Besancon, and C. Prieur, Contributions to the problem of high-gain observer

design for hyperbolic systems, Dans: Jiang Z-P., Prieur C., Astolfi A. (eds) Trends in Non-

linear and Adaptive Control. Lecture Notes in Control and Information Sciences, Springer

Nature, vol. 488, pp. 109–134, 2022.

[10] V. N. Kolokoltsov, Quantum mean-field games, Ann. Appl. Probab., vol. 32(3), pp. 2254–

2288, 2022.

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[11] J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire, C. R. Math. Acad.

Sci. Paris, vol. 343 pp. 619—625, 2006.

[12] W. Liang, N. Amini, P. Mason, On exponential stabilization of N-level quantum angular

momentum systems, SIAM j. control optim., 57 (6), p. 3939–3960, 2019.

[13] M. Mirrahimi and R. Van Handel, Stabilizing feedback controls for quantum systems, SIAM

Journal on Control and Optimization, 46, pp. 445–467, 2007.

[14] H. I. Nurdin and N. Yamamoto, Linear Dynamical Quantum Systems, Springer International

Publishing, 2017.

[15] C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, T. Rybarczyk, S. Gleyzes, P. Rouchon,

M. Mirrahimi, H. Amini, M. Brune, J.-M. Raimond, and S. Haroche, Real-time quantum

feedback prepares and stabilizes photon number states, Nature, vol. 477, p. 73, 2011.

[16] I. G. Vladimirov and I. R. Petersen, Coherent quantum LQG controllers with Luenberger

dynamics, arXiv-2211.07097, 2022.

[17] Y. Zhou, J. Hu, and H. Yu, Entanglement dynamics for two-level quantum systems coupled

with massive scalar fields, Physics Letters A, vol. 406, 2021.

[18] A. Sarlette and P. Rouchon, Deterministic submanifolds and analytic solution of the quantum

stochastic differential master equation describing a monitored qubit, Journal of Mathematical

Physics, 58(6), 2017.