Courses

Course I -  Boson sampling and signatures of multiparticle interference in bosonic systems

Leonardo Novo - INL

While the static properties of systems of non-interacting bosonic systems are easy to compute (e.g. properties of ground states or thermal states) the simulation of their dynamics is computationally expensive. The reason is that the symmetrisation of the bosonic multiparticle wavefunction implies that the probabilities of observing particular configuration of the bosons are related to matrix permanents. The fermionic counterpart of this problem involves the computation of determinants, which is an easy computational problem.  In contrast, the computation of the permanent, despite being a matrix function as is the determinant, takes exponential time in the size of the matrix using the best-known algorithms.
This fact inspired a proposal to demonstrate quantum advantage over classical computers usually referred to as the boson sampling problem. In this tutorial, I will define this problem and mention recent experimental efforts to construct boson samplers in linear-optical platforms or ultracold atoms aiming at demonstrating such a quantum computational advantage. I will also address the following question: how can we verify that a boson sampler is working correctly if we cannot simulate it? Namely, I will focus on explaining typical signatures of interference of multiple bosons, such as suppressions of certain outcomes due to destructive interference or boson bunching phenomena due to constructive interference.  

Course II - Bootstrapping Quantum Mechanics

João Vilas Boas - School of Mathematical Sciences/Queen Mary University of London (QMUL). 

The bootstrap approach is a constraint-oriented view of physics. The philosophy aims to constrain the observables we are interested in by requiring them to fulfill certain general criteria.

Initially proposed to study strong interactions involving mesons and baryons, the bootstrap philosophy was particularly successful when applied to two-dimensional conformal field theories 50 years ago and, more notably, to higher dimensions in the recent revival of the conformal bootstrap. The imposition of constraints in the form of a semidefinite optimization problem contributed greatly to this success. Today, we know that this formulation can be applied to many other contexts and the bootstrap approach is now paving the way into new territories, such as the revival of the S-matrix bootstrap, matrix models or quantum mechanics.

In this course, we will review some of the successes of the bootstrap program and introduce the basic concepts of semidefinite programming. In addition to its traditional applications, we'll see how to constrain the behavior of a quantum mechanical system by solving a semidefinite problem and we will comment on some potential issues that we need to handle carefully.

Course III - A Brief Introduction to Density Functional Theory

Fernando Nogueira, Pedro Borlido e Jaime Silva - U Coimbra/CFisUC

Density Functional Theory (DFT) is a widely used and efficient approach to solving the many-fermion problem. This reformulation of the many-body Schrödinger equation, which is a priori exact, replaces the wavefunction as the central quantity with the electronic density, expressing all observables as functionals of the ground-state density. In this course, we will review the theoretical foundations of DFT, introduce its practical formulation through the Kohn-Sham method, and explore some of its unique features and limitations. Additionally, we will provide a brief introduction to using several DFT-based electronic structure codes.