{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"1861ec14602907ec7ba3fa9e136dee12153259380e657fd98ab35796c4bef1cf","created":"2026-10-03T07:17:54Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"ae44219c5182ba9dc88d73048c85d25b15ebec44945a8601017f2dbc5ba596b1","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"amo.noisy-quantum-metrology.qec-heisenberg-practical","kind":"well-posed","literature_status":"partially-resolved","n":"1","parents":[],"plain":"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.","posed_since":"","precise":"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.","problem_ref":null,"references":"","settled_by":"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_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.","title":"Heisenberg scaling with error correction under imperfect 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