Hamiltonian-learning quantum magnets with non-local impurity tomography

Impurities in quantum materials have provided successful strategies for learning properties of complex states, ranging from unconventional superconductors to topological insulators. In quantum magnetism, inferring the Hamiltonian of an engineered system becomes a challenging open problem in the presence of complex interactions. Here we show how a supervised machine-learning technique can be used to infer Hamiltonian parameters from atomically engineered quantum magnets by inferring fluctuations of the ground states due to the presence of impurities. We demonstrate our methodology both with a fermionic model with spin-orbit coupling, as well as with many-body spin models with long-range exchange and anisotropic exchange interactions. We show that our approach enables performing Hamiltonian extraction in the presence of significant noise, providing a strategy to perform Hamiltonian learning with experimental observables in atomic-scale quantum magnets. Our results establish a strategy to perform Hamiltonian learning by exploiting the impact of impurities in complex quantum many-body states.