Heisenberg scaling with error correction under imperfect control
In plain words
Quantum error correction can restore the ideal 1/N gain when the signal and the noise act differently on the sensor. It is open in general whether this still works when the correcting operations are themselves noisy.
Precise statement
For a sensor Hamiltonian $H$ and Markovian noise with Lindblad operators $L_k$ obeying the HNLS condition ($H \text{ not in } \operatorname{span}\{1, L_k, L_k^{\mathrm{dag}}, L_k^{\mathrm{dag}} L_j\}, 1 = \text{identity}$), a fault-tolerant threshold for Heisenberg scaling is known for a Pauli-Z signal under bit-flip noise with a repetition code and noisy state preparation and measurement (Sahu, Xu, Zhou, arXiv 2601.05457, 2026). Determine whether a threshold exists for every HNLS-satisfying noise model with noisy ancillas, syndrome extraction and recovery, and how the qubit and time overhead scales with $N$ and the target precision.
What would settle it
A proof of a threshold theorem for metrology with fault-tolerant recovery covering general HNLS noise, or a counterexample noise model for which precision saturates at any nonzero control-error rate.
Status in the literature
Unverified note
A January 2026 preprint (arXiv 2601.05457) proves Heisenberg scaling below a threshold for repetition-code protocols with noisy operations; general noise models, overheads and experiments beyond a few qubits remain open.