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

Table I: Special cases of group GKP codes

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 \(H\).

Transversal Gates

Group-GKP codes corresponding to the \(G^{k_1} \subseteq G^{ k_2} \subset G^{n}\) group construction admit \(X\)-type transversal Pauli gates representing \(G\) [2].

Realizations

Cryptographic applications stemming from the monogamy of entanglement of group GKP code and error words [3].

Parent

Children

  • 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 [4; 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 \(k=k_2/k_1\).

Cousins

  • Calderbank-Shor-Steane (CSS) stabilizer code — Group GKP codes are stabilized by \(X\)-type Pauli matrices 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 \(m<n\).
  • Cluster-state code — Cluster states can be generalized to finite groups [5].
  • 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 [4; Sec. 3.3].
  • 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
[5]
C. G. Brell, “Generalized cluster states based on finite groups”, New Journal of Physics 17, 023029 (2015) arXiv:1408.6237 DOI
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Zoo Code ID: group_gkp

Cite as:
“Group GKP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/group_gkp
BibTeX:
@incollection{eczoo_group_gkp, title={Group GKP code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/group_gkp} }
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“Group GKP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/group_gkp

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