Welcome to the Group Kingdom.

Group-based quantum code
Encodes a logical Hilbert space, finite- or infinite-dimensional, into a physical Hilbert space of \(\ell^2\)-normalizable functions on a second-countable unimodular group. For \(K\)-dimensional logical subspace and for groups \(G^{n}\), can be denoted as \(((n,K))_G\). When the logical subspace is the Hilbert space of \(\ell^2\)-normalizable functions on \(G^{ k}\), can be denoted as \([[n,k]]_G\). Ideal codewords may not be normalizable, depending on whether \(G\) is continuous and/or noncompact, so approximate versions have to be constructed in practice.
Parents:
Quantum error-correcting code (QECC).
Parent of:
Group GKP code, Rotor code.
Cousins:
Qubit code, Modular-qudit code, Bosonic code.
Cousin of:
Group-based code.

Group GKP code[1]
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.
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 \(H\).
Parents:
Group-based quantum code.
Parent of:
Molecular code, Quantum-double code.
Cousins:
Bosonic stabilizer code, Calderbank-Shor-Steane (CSS) stabilizer code, Modular-qudit CSS code, Rotor GKP code.
Cousin of:
Covariant code, Linear code over \(G\).

Rotor code
Encodes a logical Hilbert space, finite- or infinite-dimensional, into a physical Hilbert space of \(\ell^2\)-normalizable functions on either the integers \(\mathbb Z\) or the circle group \(U(1)\).
Parents:
Group-based quantum code.
Parent of:
Rotor GKP code, \([[3,1,2]]_{\mathbb Z}\) rotor code, \([[5,1,3]]_{\mathbb Z}\) rotor code.

\([[3,1,2]]_{\mathbb Z}\) rotor code[2]
Extension of the \([[3,1,2]]_3\) qutrit CSS code to the integer alphabet, i.e., the angular momentum states of a planar rotor. The code is \(U(1)\)-covariant and ideal codewords are not normalizable.
Protection: Normalized codewords approximately protect against erasure while maintaining covariance [3].
Parents:
Rotor code, Covariant code.
Cousins:
Three qutrit code.

\([[5,1,3]]_{\mathbb Z}\) rotor code[3]
Extension of the five-qubit stabilizer code to the integer alphabet, i.e., the angular momentum states of a planar rotor. The code is \(U(1)\)-covariant and ideal codewords are not normalizable.
Protection: Normalized codewords approximately protect against erasure while maintaining covariance [3].
Parents:
Rotor code, Covariant code.
Cousins:
Five-qubit perfect code.

Molecular code[1]
Encodes finite-dimensional Hilbert space into the Hilbert space of \(\ell^2\)-normalizable functions on the group \(SO_3\). Construction is based on nested subgroups \(H\subset K \subset SO_3\), where \(H,K\) are finite. The \(|K|/|H|\)-dimensional logical subspace is spanned by basis states that are equal superpositions of elements of cosets of \(H\) in \(K\).
Protection: Protects against generalized bit-flip errors \(g\in SO_3\) 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 \(H\).
Parents:
Group GKP code.

Quantum-double code[4]
A family of topological codes, defined by a finite group \( G \), whose generators are few-body operators associated to the stars and plaquettes, respectively, of a tessellation of a two-dimensional surface (with a qudit of dimension \( |G| \) located at each edge of the tesselation).
Protection: Error-correcting properties established in Ref. [5]. The code distance is the number of edges in the shortest non contractible cycle in the tesselation or dual tesselation [6].
Parents:
Group GKP code, Topological code.
Cousins:
Modular-qudit surface code, String-net code.
Cousin of:
Kitaev surface code.

Rotor GKP code[7][1]
GKP code protecting against small angular position and momentum shifts of a planar rotor.
Parents:
Rotor code.
Cousins:
Gottesman-Kitaev-Preskill (GKP) code.
Cousin of:
Group GKP code, Number-phase code.

## References

- [1]
- V. V. Albert, J. P. Covey, and J. Preskill, “Robust Encoding of a Qubit in a Molecule”, Physical Review X 10, (2020). DOI; 1911.00099
- [2]
- P. Hayden et al., “Error Correction of Quantum Reference Frame Information”, PRX Quantum 2, (2021). DOI; 1709.04471
- [3]
- P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020). DOI; 1902.07714
- [4]
- A. Y. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003). DOI; quant-ph/9707021
- [5]
- S. X. Cui et al., “Kitaev's quantum double model as an error correcting code”, Quantum 4, 331 (2020). DOI; 1908.02829
- [6]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002). DOI; quant-ph/0110143
- [7]
- D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator”, Physical Review A 64, (2001). DOI; quant-ph/0008040