Here is a list of codes related to and generalizing the color code.

Code | Description |
---|---|

2D color code | Color code defined on a two-dimensional trivalent planar graph with three-colorable faces. Each face hosts two stabilizer generators, a Pauli-\(X\) and a Pauli-\(Z\) string acting on all the qubits of the face. |

3D color code | Color code defined on a four-valent four-colorable tiling of 3D space. Logical dimension is determined by the genus of the underlying surface (for closed surfaces) and types of boundaries (for open surfaces). |

Ball color code | A color code defined on a \(D\)-dimensional colex. This family includes hypercube color codes (color codes defined on balls constructed from hyperoctahedra) and 3D ball color codes (color codes defined on duals of certain Archimedean solids). |

Color code | Member of a family of qubit CSS codes defined on a \(D\)-dimensional graph which satisfies two properties: (1) the graph is a homogeneous simplicial \(D\)-complex obtained as a triangulation of the interior of a \(D\)-simplex, and (2) the graph is \(D+1\)-colorable. Qubits are placed on the \(D\)-simplices and generators are supported on suitable simplices [1–3]. Admissible graphs can be obtained via a fattening procedure [4]; see also a construction based on the more general quantum pin codes [5]. |

Cubic honeycomb color code | 3D color code defined on a four-colorable bitruncated cubic honeycomb uniform tiling. |

Honeycomb (6.6.6) color code | Triangular color code defined on a patch of the 6.6.6 (honeycomb) tiling. |

Hyperbolic color code | An extension of the color code construction to hyperbolic manifolds. As opposed to there being only three types of uniform three-valent and three-colorable lattice tilings in the 2D Euclidean plane, there is an infinite number of admissible hyperbolic tilings in the 2D hyperbolic plane [6]. Certain double covers of hyperbolic tilings also yield admissible tilings [7]. Other admissible hyperbolic tilings can be obtained via a fattening procedure [4]; see also a construction based on the more general quantum pin codes [5]. |

Square-octagon (4.8.8) color code | Triangular color code defined on a patch of the 4.8.8 (square-octagon) tiling, which itself is obtained by applying a fattening procedure to the square lattice [4]. |

Stellated color code | A non-CSS color code on a lattice patch with a single twist defect at the center of the patch. |

Tetrahedral color code | 3D color code defined on select tetrahedra of a 3D tiling. Qubits are placed on the vertices, edges, triangles, and in the center of each tetrahedron. The code has both string-like and sheet-like logical operators [8]. |

Truncated trihexagonal (4.6.12) color code | Triangular color code defined on a patch of the 4.6.12 (truncated trihexagonal or square-hexagon-dodecagon) tiling. |

Twist-defect color code | A non-CSS extension of the 2D color code whose non-CSS stabilizer generators are associated with twist defects of the associated lattice. |

XYZ color code | Non-CSS variant of the 6.6.6 color code whose generators are \(XZXZXZ\) and \(ZYZYZY\) Pauli strings associated to each hexagonal in the hexagonal (6.6.6) tiling. |

\([[15,1,3]]\) quantum Reed-Muller code | \([[15,1,3]]\) CSS code that is most easily thought of as a tetrahedral 3D color code. This code contains 15 qubits, represented by four vertices, four face centers, six edge centers, and one body center. The tetrahedron is cellulated into four identical polyhedron cells by connecting the body center to all four face centers, where each face center is then connected by three adjacent edge centers. Each colored cell corresponds to a weight-8 \(X\)-check, and each face corresponds to a weight-4 \(Z\)-check. A logical \(Z\) is any weight-3 \(Z\)-string along an edge of the entire tetrahedron. The logical \(X\) is any weight-7 \(X\)-face of the entire tetrahedron. |

\([[2^D,D,2]]\) hypercube code | Member of a family of codes defined by placing qubits on a \(D\)-dimensional hypercube, \(Z\)-stabilizers on all two-dimensional faces, and an \(X\)-stabilizer on all vertices. These codes realize gates at the \((D-1)\)-st level of the Clifford hierarchy. It can be generalized to a \([[4^D,D,4]]\) error-correcting family [9]. Various other concatenations give families with increasing distance (see cousins). |

\([[2^r-1,1,3]]\) simplex code | Member of color-code code family constructed with a punctured first-order RM\((1,m=r)\) \([2^r-1,r+1,2^{r-1}-1]\) code and its even subcode for \(r \geq 3\). Each code transversally implements a diagonal gate at the \((r-1)\)st level of the Clifford hierarchy [10,11]. Each code is a color code defined on a simplex in \(r-1\) dimensions [12,13], where qubits are placed on the vertices, edges, and faces as well as on the simplex itself. |

\([[2m,2m-2,2]]\) error-detecting code | CSS stabilizer code for \(m\geq 2\) with generators \(\{XX\cdots X, ZZ\cdots Z\} \) acting on all \(2m\) physical qubits. The code is constructed via the CSS construction from an SPC code and a repetition code [14; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [15]. |

\([[4,2,2]]\) CSS code | Four-qubit CSS stabilizer code is the smallest qubit stabilizer code to detect a single-qubit error. |

\([[6,4,2]]\) error-detecting code | Error-detecting six-qubit code with rate \(2/3\) whose codewords are cat/GHz states. A set of stabilizer generators is \(XXXXXX\) and \(ZZZZZZ\). It is the unique code for its parameters, up to local equivalence [15; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [16]. |

\([[7,1,3]]\) Steane code | A \([[7,1,3]]\) CSS code that is the smallest qubit CSS code to correct a single-qubit error. The code is constructed using the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors. |

\([[8,2,2]]\) hyperbolic color code | An \([[8,2,2]]\) hyperbolic color code defined on the projective plane. |

\([[8,3,2]]\) CSS code | Smallest 3D color code whose physical qubits lie on vertices of a cube and which admits a (weakly) transversal CCZ gate. |

## References

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- A. R. Calderbank et al., “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
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- H. Goto, “Many-hypercube codes: High-rate quantum error-correcting codes for high-performance fault-tolerant quantum computation”, (2024) arXiv:2403.16054