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

Description

A \(G\)-covariant code constructed in a physical space consisting of a tensor product of identical subsystems (e.g., qubits, modular qudits, or Galois qudits). The code is 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.

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~, \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 \(\mathsf{H}_{\text{physical}}^{\otimes |A|}\).

Protection

Depends on the base encoding \(U_0\).

Parent

  • Covariant code — In a proof of principle demonstration, error-correcting codes that are finite-\(G\) covariant can be constructed from a base encoding \(U_0\).

References

[1]
P. Hayden et al., “Error Correction of Quantum Reference Frame Information”, PRX Quantum 2, (2021). DOI; 1709.04471

Zoo code information

Internal code ID: g_covariant_erasure

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Zoo Code ID: g_covariant_erasure

Cite as:
“\(G\)-covariant erasure code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/g_covariant_erasure
BibTeX:
@incollection{eczoo_g_covariant_erasure, title={\(G\)-covariant erasure code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/g_covariant_erasure} }
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Cite as:

“\(G\)-covariant erasure code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/g_covariant_erasure

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/properties/covariant/g_covariant-erasure.yml.