Covariant code[1] 

A block code on \(n\) subsystems that admits a group \(G\) of transversal gates. The group has to be finite for finite-dimensional codes due to the Eastin-Knill theorem. Continuous-\(G\) covariant codes, necessarily infinite-dimensional, are relevant to error correction of quantum reference frames [1] and error-corrected parameter estimation.

Denoting the code's encoding map as \(U\), covariance is equivalent to \begin{align} \left(\bigotimes_{j=1}^{n}V_{j}\left(g\right)\right)U=UV_{L}\left(g\right)\quad\quad\forall g\in G\,, \tag*{(1)}\end{align} where \(V_j(g)\) is a unitary representation of \(g\) acting on the \(j\) subsystem, and \(V_L\) is a unitary representation acting on the unencoded logical information. In this way, covariant encoding maps are equivariant (i.e., commute) with group actions on the logical and physical spaces.

Almost always, the physical representation is defined to be the transversal one (with respect to some tensor-product decomposition), but can reduce to any representation when the code is a subspace of a larger space that is not expressed as a tensor product (\(n=1\)). More generally, a code is sometimes said to be time-covariant if it admits a continuous-parameter \(U(1)\) family of gates, not necessarily transversal [2].