Identifying topological phases of strongly correlated electrons by machine learning the geometry of the quantum phase diagram

Discovering novel quantum phases of matter is a central aim in condensed matter physics. The limitations of analytical and classical numerical techniques in simulating quantum matter, especially in the strongly interacting regime, provide a promising opportunity for quantum processors. Although quantum processors can help overcome the exponential scaling of classical resources in simulating these systems, efficiently learning the properties of the simulated systems and their phase diagrams still remains a challenge in cases where we do not know a priori what observables to measure, or the observables have an exponential sample complexity. In this talk, I will present a method for characterizing quantum phase diagrams from ground state classical shadows data, utilizing machine learning and discrete geometry. We find that the phases and phase transitions can be identified from the Riemannian geometric properties of the sampled state manifold. Our approach provides a path to combining the power of quantum processors and classical machine learning techniques to enable the discovery of novel phases of matter.