The proposed CECAM advanced School (alternative format) will focus on molecular simulations that are based on the path integral formulation of quantum mechanics. Relying on the isomorphism [1] between the partition function of a quantum system and a fictitious classical system of ring polymers, early pioneering work developed Path Integral Molecular Dynamics (MD) [2-3] and Monte Carlo (MC) [4] algorithms to obtain equilibrium thermal properties of quantum condensed phases. Important progress was made when it was also shown that the classical dynamics of the ring polymers can be used to approximate real-time quantum correlation functions using Centroid [5] or Ring Polymer MD [6-7], providing access also to quantum response properties such as diffusion coefficients, reaction rates and spectra. These methods have been applied more widely in recent years, thanks to the development of highly efficient algorithms to include important nuclear quantum effects, such as delocalization, zero-point energy and tunneling in molecular simulations [8]. Many of them have been implemented in the open-source software i-PI [9] and are used routinely by a global community of computational chemists, physicists and material scientists, which has grown substantially in the past 15 years.

CECAM has previously supported several advanced Schools (2012, 2016, 2018, 2021, 2023) on path integral quantum mechanics, which helped train a young generation of graduate students and ECR in this expanding field. Together with senior faculty, they have recently contributed to several important advances in the field, extending path integral methods to describing phenomena such as nonadiabatic dynamics [15-17], excitons and quasiparticles in condensed phases [13-14], real-time dynamics of molecular aggregates and extended systems [15] and nonlinear spectroscopy [16]. The field has also benefited from important work on combining machine learning algorithms to describe the interparticle interaction with path integral methods that significantly accelerate the simulations. This powerful combination has been used in a range of exciting applications, including an accurate description of the thermodynamic stabilities of molecular crystals [17], predicting a supersolid phase of deuterium at high pressure [18], understanding the supercritical behavior of liquid hydrogen [19] and a superionic phase of water at planetary conditions [20]. Furthermore, in recent years, machine learning is revolutionizing the disciplines in the field that have stood for decades, introducing new algorithms [21-23] potentially reshaping future path-integral simulations. Therefore, different from previous several advanced Schools, this School emphasizes path-integral simulations in the context of machine learning advancements.

This surge of innovative activity highlights the need for a CECAM advanced School to showcase cutting-edge advancements and address current challenges in the field. The event will preserve the successful novel format experimented in 2023, combining the educational rigor of an advanced school with the in-depth discussions and academic exchange of a workshop. The schedule includes: Two sessions dedicated to PhD lightning talks on the first day; Courses on path-integral simulations, paired with a hands-on tutorial using i-PI on the second day; Advanced workshop presentations and discussions on days 3-5.