Precision Limits of Hamiltonian Learning for Quantum Processors under Calibration Uncertainty

Learning the Hamiltonian of a quantum processor is important for controlling and certifying quantum devices, but the state preparation and measurement used to probe it are themselves imperfect. Even when exact data identify an interaction uniquely, identifiability alone does not determine finite-data precision. Here we show, for n-qubit Markovian dynamics under geometric regularity conditions, that the local uncertainty of a Hamiltonian coefficient scales either linearly or as the square root of the entrywise error in the generator matrix in the readout gauge. Square-root scaling occurs exactly when a gauge transformation moves the coefficient at first order while noise positivity can be restored at second order, and real noise correlations drive this first-order change. As a consequence, under suitable dynamical and readout conditions, square-root scaling forces uniformly valid confidence intervals from N shots to have expected length Ω(N⁻¹⁄⁴), even for adaptive local experiments. Under a common generator norm bound, estimating this matrix attains this rate up to logarithmic factors. Experiments on three superconducting processors show that additional measurement bases increase the visibility of injected rotations at a matched number of shots. Whether a coupling admits the usual N⁻¹⁄² scaling is therefore set by how noise correlations and positivity constrain its gauge directions.