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

Characterizing continuous-variable (CV) Hamiltonians can be formulated as Hamiltonian learning under quantum measurement constraints: finite operator coefficients are inferred from noisy measurement outcomes obtained by probing an infinite-dimensional system. Existing Heisenberg-limited CV protocols are often limited to low-order structures, vulnerable to noise, or unresolved for generic multi-mode settings. We introduce Displacement-Random Unitary Transformation (D-RUT), an active data acquisition protocol with pre-specified probes and number-preserving transformations that reduce finite-order bosonic Hamiltonian learning to polynomial recovery. We prove Heisenberg-limited total evolution time with robustness to state preparation and measurement (SPAM) errors, and develop hierarchical multi-mode coefficient recovery with better statistical efficiency than simultaneous estimation. We also extend D-RUT to first-quantized Hamiltonian coefficient learning, and numerical experiments on single- and multi-mode nonlinear systems validate the predicted Heisenberg scaling.

An important goal of machine learning is to infer compact predictive models from limited and noisy observations that generalize to unseen data and dynamics. In quantum systems, this learning problem corresponds to Hamiltonian learning: reconstructing the Hamiltonian, the generator of quantum dynamics, from measurement data. Accurate Hamiltonian models are essential for quantum simulation, control, calibration, and the reliable execution of quantum algorithms. The problem is particularly challenging in continuous-variable quantum systems, such as photonic and superconducting microwave platforms, where the underlying state space is infinite-dimensional and nonlinear interactions can cause small modeling errors to accumulate rapidly over time. These features make Hamiltonian learning a high-dimensional and strongly correlated inference problem that demands scalable, robust, and sample-efficient machine learning methods. We introduce Displacement-Random Unitary Transformation (D-RUT), a method that learns the key coefficients in this model by actively selecting probes that convert hidden quantum behavior into a simple scalar signal, from which the unknown coefficients can be reconstructed even with noisy data. Its measurement time efficiency reaches the best scaling allowed by quantum mechanics: Heisenberg limit, which goes beyond the central quantum limit obtained by naive sample averaging. The same framework works for both single/multi systems under first and second quantization. This gives researchers a practical way to calibrate, verify, and benchmark continuous-variable quantum hardware, helping make quantum simulators, quantum processors, and future quantum machine learning systems more reliable.