Q-Math Seminar

Logo Q-Math

Adrián Pérez-Salinas (Universiteit Leiden)

Quantum variational re-uploading for machine learning

Friday the 12th of April, 2024, 11:00, Room 2.2.D08

Variational quantum algorithms have emerged as a promising framework for Noisy Intermediate-Scale Quantum computing. While originally conceived to find ground states, variational quantum algorithms were modified to include data through embedding maps, opening the path to variational quantum machine learning. The re-uploading scheme provided a recipe to systematically introduce data, allowing for optimization of the embeddings. In this talk we will review the re-uploading framework, with in-depth focus on the expressivity and trainability of the functions output by the machine learning models. We also explore the connection to dataless variational models. Finally, we will discuss about educated guesses for future research.

Link for online session (Active on request):

Kia Romero Hojjati (National Autonomous University of Mexico)

Fisher information and Physics

Friday the 26th of January, 2024, 11:00, Room 2.2.D08

The talk will be structured into two parts, the first will quickly review some known topics on the relation between physics and information theory. Secondly, we will be in a position to discuss ideas that will be published in an upcoming paper.

First, we will discuss the work of Roy Frieden, an informational-theoretic variational principle that allows for a rough derivation of equations in physics; the so-called Extreme Physical Information (EPI) principle. We will swiftly discuss the derivation of the Klein-Gordon equation using this principle. Following up, a fragment of the stochastic derivation of quantum mechanics will be presented. After a short break, we will begin by considering a thought experiment that will argue that quantum fluctuations of the vacuum can help determine timelike directions in a background spacetime. Then ideas from the first part of the talk will be used to help realize the former ideas discussed.

Link for online session (Active on request):

Kia Romero Hojjati (National Autonomous University of Mexico)

Scaling symmetries and the contact reduction of a symplectic system

Thursday the 25th of January, 2024, 11:00, Room 2.2.D08

The talk is aimed to present a new kind of symmetry; namely scaling symmetries in the context of symplectic-Hamiltonian systems. These symmetries allow a geometrical reduction of the system into a scale-invariant one. The mathematical framework takes place in the realm of symplectic, and contact geometry- the odd-dimensional cousin of symplectic geometry. The talk will briefly present the former geometry and define the Hamiltonian dynamics in such a context.

The main result of the talk will show exactly how the dynamics of a scale-dependent system can be projected onto the scale-independent one. The generality of this formalism can be widely applied, however, its use becomes conceptually obvious when applied to Big Bang cosmological models, since they encounter singularities when the scale of the system approaches zero or infinity. This last application will be briefly discussed if time allows it.

Link for online session (Active on request):

Javier Lafuente López (Universidad Complutense de Madrid )

Geometría para Un Universo de Juguete

Tuesday the 12th of December, 2023, 13:00, Room 2.2.D08

En 1983 Hartley y Hawking dan a conocer en el contexto de la teoría de la gravedad cuántica su propuesta de ausencia de borde. El interés por el estudio de la geometría de las métricas con cambio de signatura tiene su origen en el intento de modelizar esta propuesta, usando espaciotiempos con métrica que tiene una parte Lorentziana y otra Riemanniana, donde el tiempo se ha vuelto imaginario. Los trabajos de Kossowski, Kriele, yo mismo y otros, sobre las métricas con cambio transversal de signatura, constituyen nuestro punto de partida. Aunque es posible que la propuesta de Hawking, no pueda modelizarse mediante las métricas con cambio de signatura ¿Qué nos impide jugar con este tipo de métricas para construir una universo de juguete? El modelo de universo que se propone aquí, no es más que un juego geométrico -sin más pretensiones- que describe una cosmología en la que la velocidad $c$ de la luz no es constante, pero que sin embargo sí mantiene el principio de que cada observador sea instantáneamente el centro del frente de ondas de la luz que emite. Las hipersuperficies $c=cte$, serían hipersuperficies de Cauchy y a nosotros nos habría tocado vivir en la hipersuperficie $c=3\times10^{8}m/seg,$ y nos parecería que es constante pues en un amplio margen en torno a esta hipersuperficie la variación de $c$ sería indetectable a escala humana. La función $T=cte/c$ hace el papel de tiempo universal y en el Bigbang, $T=0$, es en donde se produce el cambio de signatura y la velocidad de la luz se hace infinita.

Link for online session (Active on request):

T. Chambrion (IMB, U. Bourgogne)

Impulsive control of single-input bilinear quantum systems

Tuesday the 14th of November, 2023, 12:00, Room 2.2.D08

Control systems are said to be bilinear when the dynamics takes the form x’=Ax+uBx, with x the state (in some Hilbert space), A and B two linear (possibly unbounded) operators and u a real valued function of the time (the control law). When the operator B is bounded and the operator A generates a C^0 semi-group, a standard fixed-point argument shows that solutions are well-defined for locally integrable control laws.
In this talk, we investigate the construction of propagators for conservative quantum systems (A and B are skew-symmetric) when the control u is a Radon measure whose atoms can be understood as shocks for the system (hence the name: ”impulsive control”). We present a construction valid for unbounded operators B, under regularity conditions met by most of the quantum systems encountered in the literature.
Under the same regularity conditions, we also present an extension of a celebrated result by Ball, Marsden and Slemrod in 1982 (“when B is bounded, the attainable set with locally integrable controls is meager, hence exact controllability does not hold”), for the case where control laws are Radon measures.

This is a joint work with Nabile Boussaïd from LMB (Besançon) and Marco Caponigro from Università degli Studi di Roma 'Tor Vergata'.

Link for online session (Active on request):

N. Boussaïd (LMB, U. Franche Comté)

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

Tuesday the 14th of November, 2023, 12:45, Room 2.2.D08

We give sufficient conditions for the exact controllability in projection of bilinear Schrödinger equations with minimal regularity of the control (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 Università degli Studi di Roma 'Tor Vergata' and Thomas Chambrion from the IMB (Dijon).

Link for online session (Active on request):