\(G\)-covariant erasure code[1]

Description

A \(G\)-covariant block code that serves as a proof-of-principle construction to demonstrate the existence of \(G\)-covariant codes where \(G\) is a finite group, and the physical space is finite-dimensional. This construction can be done for any erasure-correcting code.

Consider a finite group \(G\) acting on a finite set \(A\) as a subgroup of the symmetric group on \(|A|\) elements, \(G \subset S_{|A|}\). Let \(U_0: \mathsf{H}_{\text{logical}} \rightarrow \mathsf{H}_{\text{physical}} = \mathsf{H}^{\otimes n}\) be any QECC, possibly non-covariant. Define the covariant encoder \(U \equiv U_0^{\otimes |A|}: \mathsf{H}_{\text{logical}}^{\otimes |A|} \rightarrow \mathsf{H}_{\text{physical}}^{\otimes |A|}\) on \(|A|\). Then, the group acts on codewords by index permutation: \begin{align} V(g) | \phi_{a_1} \rangle | \phi_{a_2} \rangle \cdots | \phi_{a_{|A|}} \rangle = | \phi_{g^{-1} a_1} \rangle | \phi_{g^{-1} a_2} \rangle \cdots | \phi_{g^{-1} a_{|A|}} \rangle~, \tag*{(1)}\end{align} where \(V(g)\) is the unitary representation of \(g \in G\) acting on the physical space. The action of \(V(g)\) is transversal with respect to the partition \(\mathsf{H}_{\text{physical}}^{\otimes |A|}\).