Heisenberg scaling with error correction under imperfect control · revision history
Revision 1
Accepted · first revision · record
- Title
- Heisenberg scaling with error correction under imperfect control
- 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.
- 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.
- 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.
- Kind
- well-posed
- Answer type
- proof
- References
- (empty)
- Literature status
- partially-resolved
- 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.
- Posed since
- (empty)