Group GKP code

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 CSS
\(n\) 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

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

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.
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 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].

Member of code lists

Primary Hierarchy

Quantum Domain

Group Kingdom

Group-based quantum codeQECC Quantum

Parents

Group-based quantum code

Group GKP code

Children

Group-based cluster-state code

Group-based cluster states are stabilized by group-based error operators [5,6].

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 codeCDSC Kitaev surface

Quantum-double Hamiltonians can be expressed in terms of \(X\)- and \(Z\)-type operators of group-GKP codes; see [4; Sec. 3.3].

Bosonic CSS code

The group-GKP construction encompasses all bosonic CSS codes. 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 \(m<n\). 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 \(k=k_2-k_1\).

Modular-qudit CSS codeColor BB Homological Kitaev surface Quantum Reed-Muller

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 codeQuantum Reed-Muller Color Homological Quantum QR code BB Kitaev surface

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.

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
[4]: V. V. Albert, D. Aasen, W. Xu, W. Ji, J. Alicea, and J. Preskill, “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
[6]: C. Fechisin, N. Tantivasadakarn, and V. V. Albert, “Non-invertible symmetry-protected topological order in a group-based cluster state”, (2024) arXiv:2312.09272

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

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