Certified predictions for quantum states of many particles

A set of particles that interact with each other tends to minimize its energy. When the whole system actually reaches the minimum energy, its description is given by the so-called ‘ground-state’, or state of minimal energy. Lots of efforts are devoted to discovering and analyzing its properties, which of course include -but are not limited to- the energy. For instance, magnetic and conduction properties are other worthwhile knowing features, to name just a few. However, discovering the exact value of these ground-state properties becomes harder as the number of particles increases, soon reaching a point, say for a few tens of particles, where even a supercomputer would not be able to find the solution. To circumvent this obstacle, approximate approaches have been developed, with variational methods being the main adopted solution. Unfortunately, variational methods only provide estimates to ground-state properties, with no guarantee about how close they are to the actual properties of the unknown true ground state. In our work, we provide a new approach to derive certifiable bounds on any ground-state properties of interest for large many-body systems. The main idea consists of combining standard variational approaches with methods for non-commutative polynomial optimization pioneered in the group and well established in quantum information theory. The developed approach was benchmarked in several paradigmatic many-body spin models, demonstrating its viability and making it a powerful tool to understand the properties of systems made of interacting quantum particles.