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.
December 16, 2019
Laboratoire Jacques Louis Lions (LJLL), salle 15-16-309
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
List of abstracts
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.
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).
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.
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.
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.
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).