Hamiltonian Learning for Hybrid Quantum Systems at Heisenberg Limit*

Hybrid spin-boson quantum systems are fundamental in quantum materials and quantum information science. In this work, we demonstrate that Hamiltonian learning in such systems can achieve the Heisenberg limit. Specifically, we establish a rigorous theoretical framework proving that, given access to an unknown hybrid Hamiltonian system, our algorithm can estimate the Hamiltonian coupling parameters up to root mean square error (RMSE) ϵ with a total evolution time scaling as T~O(1/ϵ) using only O(polylog(1/ϵ)) measurements. Furthermore, it remains robust against small state preparation and measurement (SPAM) errors. To validate our method, we apply it to the generalized Dicke model and the spin-boson model for spectral learning, demonstrating its efficiency in practical quantum systems. These results provide a scalable and robust framework for precision quantum sensing and Hamiltonian characterization in hybrid quantum platforms.