# Summary of the project

In recent years, the field of quantum technologies has experienced significant growth. During this expansion, the field of quantum simulation has emerged as a particularly promising one. Quantum simulators are experimental devices that typically take the form of arrays of atoms or other quantum particles, with the important feature that their configuration and the interactions between them can in principle be tuned with a high degree of controllability. This means that they have the potential to reproduce highly complex quantum systems in completely new ways, allowing us to probe previously uncharted realms of physics.

One can think of the quantitative information we get from these experiments in terms of computations: the quantities we can measure in those systems can be understood as the solutions to computational problems, in the same way as we routinely simulate physical systems in standard computers. However, with these new machines, what is the ultimate complexity of the problems that they can solve?

Along those lines, there is a growing consensus among physicists that quantum simulators have already achieved a so-called quantum advantage, in which computations of physical quantities seemingly beyond the scope of our usual computers have already been possible. However, we still have an incomplete theoretical understanding of whether that “quantum advantage” has already been reached, and if so, when and how has it happened. Studies of such an advantage have received considerably less attention and scrutiny than similar recent claims made for digital quantum computers, such as those based on random circuit sampling.

TouQan aims to bridge the gaps in our theoretical understanding of potential quantum advantages in quantum simulators. This is done through an array of overlapping objectives. The first main one is to advance our knowledge of the computational power of simulators from a mathematically rigorous perspective and find ways of characterizing it. This includes considerations such as the impact of hardware noise on the complexity of the experiments at hand. Secondly, we will examine under which conditions classical computers can effectively simulate quantum simulators to narrow down when they do not offer significant advantages. By doing so, a clearer picture of these devices' computational power will emerge. The interdisciplinary nature of this project will require the use and development of advanced tools at the intersection of computer science, physics, and mathematical physics.