Group GKP code[1]

Description

Group code whose construction is based on nested subgroups \(H\subset K \subset G\). Logical subspace is spanned by basis states that are equal superpositions of elements of cosets of \(H\) in \(K\), and can be finite- or infinite-dimensional. Extension of the GKP code construction.

Protection

Protects against generalized bit-flip errors \(g\in G\) that are inside the fundamental domain of \(G/K\). Protection against phase-flip errors determined by branching rules of irreps of \(G\) into those of \(K\), and further into those of \(H\).

Parent

Children

Cousins

  • Bosonic stabilizer code — The group-GKP construction encompasses all bosonic CSS codes. For example, a single-mode qubit GKP code corresponds to the \(2\mathbb{Z}\subset\mathbb{Z}\subset\mathbb{R}\) group construction. For another example, an \([[n,k,d]]_{\mathbb{R}}\) oscillator-into-oscillator stabilizer code corresponds to the \(\mathbb{R}^{\times k_1} \subseteq \mathbb{R}^{\times k_2} \subset \mathbb{R}^{\times n}\) group construction, where \(k=k_2/k_1\).
  • Calderbank-Shor-Steane (CSS) stabilizer code — An \(n\)-qubit CSS code corresponds to the \(C_1^\perp \subseteq C_2 \subset \mathbb{Z}_2^{\times n}\) group construction.

Zoo code information

Internal code ID: group_gkp

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: group_gkp

Cite as:
“Group GKP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/group_gkp
BibTeX:
@incollection{eczoo_group_gkp, title={Group GKP code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/group_gkp} }
Permanent link:
https://errorcorrectionzoo.org/c/group_gkp

References

[1]
V. V. Albert, J. P. Covey, and J. Preskill, “Robust Encoding of a Qubit in a Molecule”, Physical Review X 10, (2020). DOI; 1911.00099
[2]
Victor V. Albert et al., “Spin chains, defects, and quantum wires for the quantum-double edge”. 2111.12096

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/groups/group_gkp.yml.