# Metrological code[1]

## Description

Linear 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} \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)~. \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 code defined in a physical space consisting of a tensor product of \(n\) subsystems (e.g., qubits, modular qudits, or Galois qudits) 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.

## Child

- Quantum error-correcting code (QECC) — Metrological codes satisfy the Knill-Laflamme conditions conditions only partially, and codes that satisfy them fully are QECCs.

## 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].

## References

- [1]
- Philippe Faist et al., “Time-energy uncertainty relation for noisy quantum metrology”. 2207.13707

## Zoo code information

## 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/tree/main/codes/quantum/metrological.yml.