Error-corrected sensing code[1,2] 

Code that can be obtained via an optimization procedure that ensures correction against a set \(\cal{E}\) of errors as well as guaranteeting optimal precision in locally estimating a parameter using a noiseless ancilla. For tensor-product spaces consisting of \(n\) subsystems (e.g., qubits, modular qudits, or Galois qudits), the procedure can yield a code whose parameter estimation precision satisfies Heisenberg scaling, i.e., scales quadratically with the number \(n\) of subsystems.

The conditions required for a code are that it corrects errors in the set \(\cal{E}\) and admits a continuous-parameter \(U(1)\) group of logical gates generated by some signal Hamiltonian \(H\) (with the time of evolution by \(H\) the parameter that is to be estimated). This means that \(H\) cannot itself be a detectable error, i.e., \(H\) cannot be expressed as a linear combination of the errors, a condition known as the Hamiltonian-not-in-Kraus-span condition [3] (alternatively, Hamiltonian-not-in-Lindblad-span for Markovian noise [2]). If these conditions are satisfied, a semidefinite-program based optimization procedure yields a metrologically optimal code. The procedure has been generalized to more general groups, corresponding to multiparameter estimation [4]. If these conditions are not satisfied, Heisenberg scaling is not achievable, but metrologically optimal codes can still be obtained via another semidefinite-program based optimization procedure [3,5].

Metrologically optimal QECCs require error-free ancillas for optimal local parameter estimation using an entangling gate. In this sense, such codes can be thought of as being entanglement-assisted. Ancilla-free versions exist in the case when the noise commutes with the signal Hamiltonian [6,7].