Continuous Variable Hamiltonian Learning at Heisenberg Limit via Displacement-Random Unitary Transformation

Characterizing the Hamiltonians of continuous-variable (CV) quantum systems remains a fundamental challenge due to the infinite-dimensional Hilbert space and the presence of unbounded operators. Existing learning protocols are often restricted to low-order Hamiltonian structures and can be sensitive to experimental noise, leaving generic multi-mode settings largely unresolved. In this work, we introduce the Displacement-Random Unitary Transformation (D-RUT), an experimentally accessible protocol for learning the coefficients of generic multi-mode bosonic Hamiltonians of arbitrary finite order. We prove that D-RUT achieves Heisenberg-limited scaling while remaining robust to state preparation and measurement (SPAM) errors. To extend the method efficiently to multi-mode systems, we develop a hierarchical coefficient recovery strategy that yields superior statistical efficiency compared to existing simultaneous estimation schemes. Importantly, we further show that our framework applies naturally to Hamiltonian coefficient learning in the first-quantized formulations, substantially broadening its scope beyond prior CV approaches. Numerical experiments validate the predicted Heisenberg scaling our approach in both single- and multi-mode nonlinear systems.