# Metrological code[1]

## Description

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.

## Protection

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.

## Parent

## Cousins

- Error-corrected sensing code — Error-corrected sensing codes are required to satisfy the Knill-Laflamme conditions, while metrological codes need only satisfy the conditions partially.
- Covariant code — Any time-covariant QECC, i.e., a code admitting a continuous-parameter \(U(1)\) family of gates, is automatically a metrological code.
- Qubit stabilizer code — A joint \(+1\) and \(-1\) eigenstate of a set of stabilizer can form a metrological stabilizer code [1].
- Quantum error-correcting code (QECC) — Metrological codes are logical-qubit codes that satisfy the Knill-Laflamme conditions conditions only partially, and codes that satisfy them fully are QECCs.
- Error-transmuting code (QETC) — Metrological codes are also codes which satisfy a generalization of the Knill-Laflamme conditions, albeit a different one.

## References

- [1]
- P. Faist et al., “Time-Energy Uncertainty Relation for Noisy Quantum Metrology”, PRX Quantum 4, (2023) arXiv:2207.13707 DOI

## Page edit log

- Victor V. Albert (2022-07-30) — most recent

## Cite as:

“Metrological code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/metrological

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/metrological.yml.