Séminaires

On s’intéresse au temps nécessaire pour transférer un système quantique fermé de dimension finie d’un état initial vers une cible donnée. On fera le lien avec le contrôle en norme L^1 minimale pour de tels systèmes et on en déduira des stratégies efficaces pour les cas où les approximations riemanniennes usuelles sont inefficaces.
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Dans le cadre des journées thématiques « Quantum Lo  : mécanique quantique en Lorraine »

Venue:
« Salle de conférence »  of the « Institut Élie Cartan de Lorraine: IECL », Nancy.

Objective:
The workshop is a moment of exchange between the Mathematics and Physics communities, specifically focusing on quantum control problems.

Organizers:
Alessandro DUCA, Killian LUTZ, Yannick PRIVAT

Senior speakers:
Gaspard BEUGNOT, Thomas CHAMBRION, Viviana GRASSELLI, Eugenio POZZOLI, Rémi ROBIN, Mario SIGALOTTI, and Dominique SUGNY.

Junior speakers:
Vincent HARDEL, Jean-Gabriel HARTMANN, Denis JANKOVIC and Killian LUTZ.

Planning Monday 17 March:

• 09H45 – 10H30 : Gaspard BEUGNOT
• 10H30 – 11H15 : Rémi ROBIN
• 11H15 – 11H35 : PAUSE
• 11H35 – 12H20 : Dominique SUGNY
• 13H00 – 14H30 : Lunch INRIA
• 14H30 – 15H15 : Viviana GRASSELLI
• 15H15 – 15H45 : Denis JANKOVIC
• 15H45 – 16H30 : COFFEE BREAK + DISCUSSION
• 16H30 – 17H00 : Jean-Gabriel HARTMANN
• 19H30 : Dinner at Café FOY

Planning Tuesday 18 March:

• 09H30 – 10H15 : Thomas CHAMBRION
• 10H15 – 10H45 : COFFEE BREAK
• 10H45 – 11H30 : Mario SIGALOTTI
• 11H30 – 12H00 : PAUSE
• 12H00 – 12H30 : Vincent HARDEL
• 13H00 – 14H30 : Lunch INRIA
• 14H30 – 15H15 : Eugenio POZZOLI
• 15H15 – 15H45 : Killian LUTZ
• 15H45 – : COFFEE BREAK

Abstracts :

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T. Chambrion
Title: Contrôle en temps petit des systèmes bilinéaires conservatifs en
dimension finieAbstract : on s’intéresse au temps nécessaire pour transférer un système
quantique fermé de dimension finie d’un état initial vers une cible
donnée. On fera le lien avec le contrôle en norme L^1 minimale pour de
tels systèmes et on en déduira des stratégies efficaces pour les cas où
les approximations riemanniennes usuelles sont inefficaces.

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V. Grasselli
Title: Semirelativistic Hartree equation and speed of propagationAbstract: We will consider the semirelativistic Hartree equation, which is the effective equation describing a boson star, a large group of gravitating bosons.
First, we will briefly describe under which conditions this non linear equation admits a global solution. Then we will study some dispersive properties of the solution, in particular how fast it propagates and its maximal and minimal velocities of propagation. These results are a joint work with Sébastien Breteaux and Jérémy Faupin.

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Jean-Gabriel Hartmann
Title: Optimal control techniques for quantum computing with qudits.
Abstract: In this study, we investigate pulse-level optimal control techniques for quantum gates in qudit systems using the Givens Rotation Decomposition (GRD) and the GRAPE algorithm. These methods are evaluated using benchmark gates including the Quantum Fourier Transform (QFT), Grover’s Search Algorithm and Haar-random gates.  In this talk we present some initial results of our numerical simulations, demonstrating the differences in performance between these approaches, as well as the robustness of each method to environmental noise and control errors. These results highlight the potential of qudits in offering competitive gate efficiencies, suggesting pathways for optimality in gate design as well as extensions to multi-qudit systems.

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Denis Jankovic
Titre: Gradient-based optimization of the PMP-informed generator of optimal pulses for qudit control.Abstract: This presentation explores a hybrid approach to designing control pulses for multi-level quantum systems (qudits). By combining Pontryagin’s Maximum Principle (PMP) with gradient-based optimization, we aim to guide the pulse-generation process more effectively. We discuss the theoretical foundations, outline the algorithmic framework, and present initial numerical results that suggest potential improvements in both accuracy and computational efficiency. The goal is to provide insight into how a PMP-informed strategy may offer a promising direction for quantum control of fast pulses.

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K. Lutz
Title: Estimations a priori pour la synthèse de porte logique optimal en temps et précision.Abstract: Quel temps incompressible est nécessaire à l’exécution d’un calcul quantique et quelle précision peut-on attendre de son résultat ? Ce travail porte sur le contrôle en dimension finie de l’équation de GKS-Lindblad modélisant la décohérence des états quantiques, une source d’erreurs de calculs.
En se restreignant à des contrôles bornés ponctuellement et étant donné une porte logique idéale, nous mettons en évidence l’existence d’un temps minimal de contrôle permettant son implémentation avec une précision maximale. Pour le contrôle correspondant, nous établissons des estimations a priori qui minorent et majorent à la fois le temps d’exécution et les erreurs de calculs. Ces estimations, explicites et facilement calculables à partir des données, permettent d’évaluer a posteriori la pertinence de contrôles obtenus par optimisation numérique.

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E. Pozzoli
Title: Small-time approximate controllability of bilinear Schrödinger equations and diffeomorphismsAbstract: We propose a new method to prove the global approximate controllability of bilinear Schrödinger PDEs posed on a manifold M. Contrarily to previous ones, it works in arbitrarily small time and does not require a discrete spectrum.
This approach consists in controlling separately the radial and the angular part of the wavefunction thanks to the control of the group of diffeomorphisms and the control of phases. The control of the radial part uses the transitivity of the group action of diffeomorphisms on positive densities proved by Moser.
We develop this approach on two examples of Schrödinger equations, posed tori and euclidean spaces, for which the small-time control of phases was recently proved by Duca and Nersesyan. We prove that it implies the small-time control of flows of vector fields thanks to Lie bracket techniques. Combining this property with the simplicity of the diffeomorphism group proved by Thurston, we obtain the result.

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R. Robin
Title: Numerical analysis of the Lindblad equationAbstract : In this talk, we will present recent advances in the numerical approximation of the Lindblad master equation, focusing on infinite-dimensional Hilbert spaces. After reviewing the key properties of the evolution of the Lindblad equation, we will discuss both spatial discretization using Galerkin approximations and temporal discretization methods. The first highlighted contribution is the introduction of an a posteriori error estimate. The second is the analysis of new structure-preserving time discretization schemes. These contributions are the result of joint works with Paul-Louis Etienney, Pierre Rouchon and Lev-Arcady Sellem.

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M. Sigalotti
Title: Eigenvalue intersections and controllabilityAbstract: Studying the spectrum of the Hamiltonian as a function of the control parameters plays a fundamental role in
establishing which states can be joined one to another by adiabatic control. More, generally,
conditions based on the behavior of the eigenvalue intersections can be used to establish operator controllability (not necessarily using adiabatic steering). In this talk we will present some results in this direction and also discuss how the spectrum can be used to deduce controllability when some control parameters are imposed to be constant.

In this talk I present a joint paper with Umberto De Maio (Università degli Studi di Napoli « Federico II », Italy) and Catalin Lefter (Al.I.Cuza University and Octav Mayer Institute of Mathematics, Iasi, Romania).

This paper is devoted to studying the null internal controllability of a Kirchhoff-Love thin plate with a middle surface having a comb-like shaped structure with a large number of thin fingers described by a small positive parameter $\varepsilon$. It is often impossible to directly approach such a problem numerically, due to the large number of thin fingers. So an asymptotic analysis is needed. In this paper, we first prove that the problem is null controllable at each level $\varepsilon$. We then prove that the sequence of the respective controls with minimal $L^2$ norm converges, as $\varepsilon$ vanishes, to a limit control function ensuring the optimal null controllability of a degenerate limit problem set in a domain without fingers.

We consider a rigid body moving in an inviscid compressible fluid within a bounded domain. The fluid is thereby described by the compressible Euler equations, while the rigid body obeys the conservation of linear and angular momentum. This gives us a coupled system comprising an ODE and the initial boundary value problem (IBVP) of a hyperbolic system with characteristic boundary, where the fluid velocity matches the solid velocity along the normal direction of the solid boundary.
We establish the existence of a unique classical solution to this coupled system. Our approach involves constructing an approximate system with a non-characteristic boundary, which enables the decoupling of the fluid and solid equations. To obtain uniform norm control, we employ the conormal vector fields to derive the conormal and vorticity estimates, by using the structure of Euler equations. Finally, we are able to obtain the solution by compactness principle.

La journée en l’honneur de Georges Rhin aura lieu le vendredi 7 mars. Plus de détails ici.

Dans cet exposé, nous montrerons un résultat de contrôlabilité à zéro
d’une équation parabolique d’ordre quatre sous des conditions aux
limites générales satisfaisant la condition de Lopatinskii-Sapiro. Pour
ce faire, nous établirons une inégalité spectrale pour la solution du
système parabolique, ce qui nous conduira au résultat de contrôlabilité
à zéro. Cette inégalité spectrale découle d’une inégalité
d’interpolation obtenue grâce à une inégalité de Carleman pour
l’opérateur d’ordre quatre sous les conditions aux limites satisfaisant
la condition de Lopatinskii-Sapiro.

C’est un travail en collaboration avec Luc Robbiano.

Our work focuses on the derivation of effective models for a quantum system interacting with a large reservoir of particles through a mean-field scaling. Since this scaling is semiclassical in nature, we also employ techniques from the semiclassical analysis of bosonic fields. Furthermore, we examine key properties of open quantum systems, such as decoherence and non-Markovianity. While a complete analysis of the system-environment ensemble is feasible for simple models, an effective description is a crucial step in the analysis of systems interacting with large environments.

During this presentation, I will try to draw up a panel of existing results until now
concerning the mathematical justification of multiphase systems for compressible viscous
flows. We will end up the presentation with a recent work with Cosmin Burtea, Frédéric
Lagoutière and Pierre Gonin–Joubert.

Attention : horaires inhabituels, le groupe de travail aura lieu de 10h45 à 12h15 (une séance d’une heure et demie) et sera précédé d’une pause café-gâteau de 10h15 à 10h45

Références et résumé peuvent se trouver ici : https://arxiv.org/abs/2410.21068