In the framework of the ANR project QUACO and the ITN project QUSCO we organize a thematic day about analysis and control of quantum dynamics.

Date

December 16, 2019

Venue

Laboratoire Jacques Louis Lions (LJLL), salle 15-25-309

Program

9:45-10:45 H. Teismann - Some mathematical questions (and fewer answers) in quantum control

10:45-11:00 Coffee break

11:00-12:00 N. Boussaïd - Exact controllability in projections of bilinear Schrödinger
equations: the mixed spectrum case

14:00-14:30 J. Guillaud - Fault-tolerant quantum computation with repetition cat-qubits

14:30-15:30 M. Zworski - Viscosity Limits for 0th order operators

15:30-16:00 Coffee break

16:00-16:30 I. Beschastnyi - Semi-classical obstructions to small time controllability of the
Schroedinger equation

16h30-17h30 C. Laurent - Bilinear control of Schrödinger equations

Organizers

Ugo Boscain

Jean-Michel Coron

Mario Sigalotti

List of abstracts

**Beschastnyi Ivan**

Title: *Semi-classical obstructions to small time controllability of the
Schroedinger equation*

Abstract: In this talk we will discuss relations between the
controllability properties of a controlled classical Hamiltonian system
and its quantization. In particular, we will see some concrete examples,
where one can deduce that a quantum system is not small time
controllable from the behavior of the corresponding classical system.

**Boussaïd Nabile**

Title: *Exact controllability in projections of bilinear Schrödinger
equations: the mixed spectrum case*

Abstract: We give sufficient conditions for the exact controllability
in projection of bilinear Schrödinger equations with minimal regularity
(switching control) in the case where the spectrum of the free
Hamiltonian is mixed (a discrete and an essential part).

The idea behind the proof is to use a Galerkin approximation to reduce
the problem to the finite dimensional case. The natural Galerkin basis
is the one provided by a orthonormal family of eigenvectors. The latter
is never complete if the essential spectrum is continuous. When such a
situation happens, we use averaging methods and a generalization of the
RAGE theorem to decouple the dynamics with respect to the sum of
eigenspaces and the one with respect to the continuous spectrum.

This is a joint work with Marco Caponigro from the CNAM (Paris) and
Thomas Chambrion from the IMB (Dijon).

**Guillaud Jérémie**

Title: *Fault-tolerant quantum computation with repetition cat-qubits*

Abstract: Quantum error correcting codes provide, when operated below the threshold, an arbitrary good protection against noise, thus solving the decoherence problem for quantum information processing. However, the actual implementation of the most promising ones, such as the surface code, comes at the price of tremendous physical resources to reach a sufficient level of protection. We present a 1D repetition code based on the so-called cat-qubits as a viable candidate for a massive reduction in the hardware requirements for universal and fault-tolerant quantum computation. The cat-qubits that are stabilized by a two-photon driven dissipative process, exhibit a tunable noise bias where the effective bit-flips are exponentially suppressed with the average number of photons. Exploiting this noise bias, we build, at the level of the repetition cat-qubit, a universal set of fully protected logical gates. Remarkably, this construction avoids the costly magic states preparation, distillation and injection, even for non-Clifford gates.

**Laurent Camille**

Title: *Bilinear control of Schrödinger equations*

Abstract: In this talk, we will discuss about the controllability of the Schrödinger equation when the control act as a potential term in the equation. We will discuss two cases:

-the 1D case where we only control the amplitude of a potential with a fixed profile. The proof will make use of a "regularizing effect".

-the 2D case where the potential satisfies a Poisson equation and the control is the boundary value of the potential. We will make the link with the more standard boundary control of the Schrödinger equation.

This is joint work with Karine Beauchard.

**Teismann Holger**

Title: *Some mathematical questions (and fewer answers) in quantum control*

Abstract: In this talk several problems arising in quantum control theory will be described, which are in various stages of analysis, ranging from (sort of) solved to completely open. The talk will touch on topics such as quantum speed limit and minimal control time, Hamiltonian amplification, and dispersion.

**Zworski Maciej**

Title: *Viscosity limits for 0th order operators*

Abstract: For self-adjoint pseudodifferential operators of order 0,
Colin de Verdiere and Saint-Raymond introduced natural dynamical
conditions (motivated by the study of internal waves in fluids)
guaranteeing absolute continuity of the spectrum. I will present an
alternative approach to obtaining such results based on Melrose's
radial propagation estimates from scattering theory (joint work with S
Dyatlov). I will then explain how an adaptation of the
Helffer-Sjoestrand theory of scattering resonances shows that in a
complex neighbourhood of the continuous spectrum viscosity eigenvalues
have limits as viscosity goes to 0. Here the viscosity eigenvalues are
the eigen-values of the original operator to which an
anti-self-adjoint elliptic 2nd order operator is added (joint work
with J Galkowski).

ANR QUACO

ITN project QuSCo