Metrological code[1] 


Two-dimensional subspace of a Hilbert space whose basis states satisfy only a part of the Knill-Laflamme conditions. The satisfied part of the conditions ensures that the code can be used for local parameter estimation.

Letting \(\Pi = U U^\dagger\) be the codespace projector for encoding isometry \(U\) and projecting a pair of errors \(E_i,E_j\) from an error set \(\cal E\) into the two-dimensional codespace yields \begin{align} \Pi E_{i}^{^{\dagger}}E_{j}\Pi=c_{ij}\,\Pi+x_{ij}\overline{X}+y_{ij}\overline{Y}+z_{ij}\overline{Z} \tag*{(1)}\end{align} with error-matrix element \(c_{ij}\) and logical-error coefficients \begin{align} \left\{ x,y,z\right\} _{ij}={\textstyle \frac{1}{2}}\text{Tr}\left(\left\{ \overline{X},\overline{Y},\overline{Z}\right\} E_{i}^{^{\dagger}}E_{j}\right)~. \tag*{(2)}\end{align} If all three logical-error coefficients are zero, then the Knill-Laflamme conditions are satisfied, and the code is a QECC. If only one of the three coefficients is zero, then the code is the more general metrological code.


Physical noise can cause logical errors along one of the three axes, i.e., either logical-\(X\), \(Y\), or \(Z\), depending on what basis is used. Codes protect against logical errors along the remaining two axes.

A metrological block quantum code has distance \(d\) if the above conditions are satisfied for an error set \(\cal E\) consisting of errors supported on \(d-1\) subsystems of less.




P. Faist et al., “Time-Energy Uncertainty Relation for Noisy Quantum Metrology”, PRX Quantum 4, (2023) arXiv:2207.13707 DOI
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Zoo Code ID: metrological

Cite as:
“Metrological code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_metrological, title={Metrological code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Metrological code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.