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 error operators representing \(H\) and all \(Z\)-type operators that are constant on \(K\). 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 a wide variety of codes, whose \(G\) 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 |
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
- 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
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, “Noninvertible Symmetry-Protected Topological Order in a Group-Based Cluster State”, Physical Review X 15, (2025) arXiv:2312.09272 DOI
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