# Group GKP code[1]

## Description

Group-based quantum 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.

The group GKP code was originally formulated as an extension of the GKP code construction to other group-valued spaces. In other words, the only requirement to construct group GKP codes is that the configuration space \(G\) is a group under some operation. The construction encompasses a wide variety of codes, tabulated in Table I.

Space | \(G\) | \(H\) | Related code |
---|---|---|---|

\(n\) qubits | \(\mathbb{Z}_2^n\) | \(\mathbb{Z}_2^m\) | qubit CSS |

\(n\) modular qudits | \(\mathbb{Z}_q^n\) | \(\mathbb{Z}_q^m\) | modular-qudit CSS |

\(n\) Galois qudits | \(GF(q)^n\) | \(GF(q)^m\) | Galois-qudit CSS |

\(n\) modes | \( \mathbb{R}^n \) | \( \mathbb{R}^m \) | analog stabilizer |

\(n\) modes | \( \mathbb{R}^n \) | \( \mathbb{Z}^n \) | multimode GKP |

\(n\) modes | \( \mathbb{R}^n \) | \( \mathbb{Z}^{m<n} \) | oscillator-into-oscillator GKP |

planar rotor | \(U(1)\) | \(\mathbb{Z}_n\) | rotor GKP |

rigid body | \(SO(3)\) | point group | molecular |

## Protection

## Transversal Gates

## Realizations

## Parent

## Children

- Group-based cluster-state code — Group-based cluster states are stabilized by group-based error operators [4,5].
- Group-based QPC
- Molecular code
- Rotor GKP code — Rotor GKP codes correspond to the \(\mathbb{Z}_{k_1} \subseteq \mathbb{Z}_{k_2} \subset U(1)\) group construction, where \(k=k_2/k_1\).
- Quantum-double code — Quantum-double Hamiltonians can be expressed in terms of \(X\)- and \(Z\)-type operators of group-GKP codes; see [6; Sec. 3.3].
- Modular-qudit CSS code — An \(n\) modular-qubit CSS code corresponds to the \(\mathbb{Z}_q^{k_1} \subseteq \mathbb{Z}_q^{k_2} \subset \mathbb{Z}_q^{n}\) group construction, where \(k=k_2/k_1\), and where the group operation is addition. This construction should be extendable to additive \(q\)-ary codes over \(\mathbb{Z}_q\) since those are also groups under addition.
- Galois-qudit CSS code — An \(n\) Galois-qubit CSS code corresponds to the \(GF(q)^{k_1} \subseteq GF(q)^{k_2} \subset GF(q)^{n}\) group construction, where \(k=k_2/k_1\), and where the group operation is addition. This construction should be extendable to additive \(q\)-ary codes since those are also groups under addition.

## Cousins

- Calderbank-Shor-Steane (CSS) stabilizer code — Group GKP codes are stabilized by \(X\)-type group-based error operators representing \(H\) and all \(Z\)-type operators that are constant on \(K\). However, the \(Z\)-type operators are not unitary for non-Abelian groups.
- Bosonic stabilizer code — The group-GKP construction encompasses all bosonic CSS codes. A single-mode qubit GKP code corresponds to the \(2\mathbb{Z}\subset\mathbb{Z}\subset\mathbb{R}\) group construction, and multimode GKP codes can be similarly described. An \([[n,k,d]]_{\mathbb{R}}\) analog stabilizer code corresponds to the \(\mathbb{R}^{ k_1} \subseteq \mathbb{R}^{ k_2} \subset \mathbb{R}^{n}\) group construction, where \(k=k_2/k_1\). GKP stabilizer codes for \(n\) modes correspond to subgroups \(\mathbb{Z}^m\) for \(m<n\).
- Linear code over \(G\)
- Generalized 2D color code — Generalized color-code Hamiltonians should be expressable in terms of \(X\)- and \(Z\)-type operators of group-GKP codes; see [6; Sec. 3.3].
- Covariant block quantum code — Group-GKP codes corresponding to the \(G^{k_1} \subseteq G^{k_2} \subset G^{n}\) group construction admit \(X\)-type transversal Pauli gates that represent the group \(G\), and are thus \(G\)-covariant [2].

## References

- [1]
- V. V. Albert, J. P. Covey, and J. Preskill, “Robust Encoding of a Qubit in a Molecule”, Physical Review X 10, (2020) arXiv:1911.00099 DOI
- [2]
- P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
- [3]
- E. Culf, T. Vidick, and V. V. Albert, “Group coset monogamy games and an application to device-independent continuous-variable QKD”, (2022) arXiv:2212.03935
- [4]
- C. G. Brell, “Generalized cluster states based on finite groups”, New Journal of Physics 17, 023029 (2015) arXiv:1408.6237 DOI
- [5]
- C. Fechisin, N. Tantivasadakarn, and V. V. Albert, “Non-invertible symmetry-protected topological order in a group-based cluster state”, (2024) arXiv:2312.09272
- [6]
- V. V. Albert et al., “Spin chains, defects, and quantum wires for the quantum-double edge”, (2021) arXiv:2111.12096

## Page edit log

- Victor V. Albert (2023-06-03) — most recent
- Alexander Cowtan (2023-06-03)
- Victor V. Albert (2022-09-28)
- Philippe Faist (2022-09-27)
- Victor V. Albert (2021-11-29)

## Cite as:

“Group GKP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/group_gkp