Hamiltonian Learning at Heisenberg Limit for Hybrid Quantum Systems

Hybrid quantum systems with different particle species are fundamental in quantum materials and quantum information science. In this work, we establish a rigorous theoretical framework proving that, given access to an unknown spin-boson type Hamiltonian, our algorithm achieves Heisenberg-limited estimation for all coupling parameters up to error $\epsilon$ with a total evolution time ${O}(\epsilon^{-1})$ using only ${O}({\rm polylog}(\epsilon^{-1}))$ measurements. It is also robust against small state preparation and measurement errors. In addition, we provide an alternative algorithm based on distributed quantum sensing, which significantly reduces the evolution time per measurement. To validate our method, we demonstrate its efficiency in hybrid Hamiltonian learning and spectrum learning, with broad applications in AMO, condensed matter and high energy physics. Our results provide a scalable and robust framework for precision Hamiltonian characterization in hybrid quantum platforms.