# 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, and has turned out to encompass 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 |

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

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

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

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

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

## Protection

## Transversal Gates

## Realizations

## Parents

- Group-based quantum code
- Stabilizer code — Group GKP codes are stabilized by \(X\)-type Pauli matrices representing \(H\) and all \(Z\)-type operators that are constant on \(K\).

## Children

- Molecular code
- Quantum-double code — Quantum-double models admit stabilizer-like \(X\)- and \(Z\)-type operators [4], and the codes can be formulated as group GKP codes.

## Cousins

- Bosonic stabilizer code — The group-GKP construction encompasses all bosonic CSS codes. A singlemode 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 \(m<n\).
- Calderbank-Shor-Steane (CSS) stabilizer code — An \(n\)-qubit CSS code corresponds to the \(C_1^\perp \subseteq C_2 \subset \mathbb{Z}_2^{n}\) group construction.
- 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 \(k=k_2/k_1\).
- Rotor GKP code — Rotor GKP codes correspond to the \(\mathbb{Z}_{k_1} \subseteq \mathbb{Z}_{k_2} \subset U(1)\) group construction, where \(k=k_2/k_1\).
- Linear code over \(G\)
- Covariant 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]
- 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 (2022-09-28) — most recent
- 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.), 2022. https://errorcorrectionzoo.org/c/group_gkp

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