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Group GKP code[1]

Description

Group-based quantum code whose construction is based on nested subgroups \(H\subset K \subset G\). 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.

Group GKP codes are stabilized by \(X\)-type group-based right-multiplication error operators representing \(H\), all \(Z\)-type operators that are constant on \(K\), and any left-multiplication error operators that are constant on the codespace. The \(Z\)-type operators are non-unitary matrix product operators for non-Abelian groups. The codes' 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 depending on the chosen groups.

The construction encompasses all Pauli-type CSS codes since those are Abelian group GKP codes; relevant \(G\) and \(H\) are tabulated in Table I. 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. An \([[n,k,d]]_{\mathbb{R}}\) analog CSS code corresponds to the \(\mathbb{R}^{ k_1} \subseteq \mathbb{R}^{ k_2} \subset \mathbb{R}^{n}\) group construction, where \(k=k_2-k_1\). A single-mode qubit GKP CSS code corresponds to the \(2\mathbb{Z}\subset\mathbb{Z}\subset\mathbb{R}\) group construction, and multimode GKP CSS codes can be similarly described. Oscillator-into-oscillator GKP CSS codes for \(n\) modes correspond to subgroups \(\mathbb{Z}^m\) for \(m<n\). Rotor GKP codes correspond to the \(\mathbb{Z}_{k_1} \subseteq \mathbb{Z}_{k_2} \subset U(1)\) group construction, where \(k=k_2/k_1\).

Space

\(G\)

\(H\)

Related code

\(n\) qubits

\(\mathbb{Z}_2^n\)

\(\mathbb{Z}_2^m\)

qubit CSS

\(n\) modular qudits

\(\mathbb{Z}_q^n\)

\(H\)

modular-qudit CSS

\(n\) Galois qudits

\(GF(q)^n\)

\(GF(q)^m\)

Galois-qudit CSS

\(n\) modes

\( \mathbb{R}^n \)

\( \mathbb{R}^m \)

analog CSS

\(n\) modes

\( \mathbb{R}^n \)

\( \mathbb{Z}^n \)

multimode GKP

\(n\) modes

\( \mathbb{R}^n \)

\( \mathbb{Z}^{m<n} \)

oscillator-into-oscillator GKP

\(n\) rotors

\( \mathbb{Z}^n \)

\( \mathbb{Z}^m \)

homological rotor

rotor

\(U(1)\)

\(\mathbb{Z}_n\)

rotor GKP

rigid body

\(SO(3)\)

\(H\)

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].

Cousins

Primary Hierarchy

Parents
Group GKP code
Children
CSS codes are Abelian group GKP codes, i.e., group GKP codes constructed out of Pauli-type operators.

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, S. Nezami, V. V. Albert, G. Salton, F. Pastawski, P. Hayden, and J. Preskill, “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
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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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Cite as:

“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/group_gkp.yml.