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 $$G^{\times n}$$, can be denoted as $$((n,K))_G$$. When the logical subspace is the Hilbert space of $$\ell^2$$-normalizable functions on $$G^{\times k}$$, 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. Cousins: Qubit code, Modular-qudit code, Bosonic code. Cousin of: Group-based code.
Group 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. Extension of the GKP code construction. 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, Rotor GKP code.
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.
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. [3]. The code distance is the number of edges in the shortest non contractible cycle in the tesselation or dual tesselation [4]. Parents: Group GKP code, Topological code.
GKP code protecting against small angular position and momentum shifts of a planar rotor. Parents: Group GKP code. Cousins: Gottesman-Kitaev-Preskill (GKP) code. Cousin of: 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]
A. Y. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003). DOI; quant-ph/9707021
[3]
S. X. Cui et al., “Kitaev's quantum double model as an error correcting code”, Quantum 4, 331 (2020). DOI; 1908.02829
[4]
E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002). DOI; quant-ph/0110143
[5]
D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator”, Physical Review A 64, (2001). DOI; quant-ph/0008040