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 planar graph. 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 particular \(D\)-dimensional graphs. |

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 [1]. Certain double covers of hyperbolic tilings also yield admissible tilings [2]. Other admissible hyperbolic tilings can be obtained via a fattening procedure [3]; see also a construction based on the more general quantum pin codes [4]. |

Quasi-hyperbolic color code | An extension of the color code construction to quasi-hyperbolic manifolds, e.g., a product of a 2D hyperbolic surface and a circle. |

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 [3]. |

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 [5]. |

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. |

Union-Jack color code | Triangular color code defined on a patch of the Tetrakis square tiling (a.k.a. the Union Jack 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. |

\([[16,6,4]]\) Tesseract color code | A 4D color code defined on a tesseract, with stabilizer generators of both types supported on each cube. |

\([[2^D,D,2]]\) hypercube quantum 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 [6]. 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 [7,8]. Each code is a color code defined on a simplex in \(r-1\) dimensions [9,10], where qubits are placed on the vertices, edges, and faces as well as on the simplex itself. |

\([[2m,2m-2,2]]\) error-detecting code | Self-complementary CSS 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 [11; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [12]. |

\([[4,2,2]]\) Four-qubit 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 [12; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [13]. |

\([[7,1,3]]\) Steane code | A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [14]. 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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- E. B. da Silva and W. S. Soares Jr, “Hyperbolic quantum color codes”, (2018) arXiv:1804.06382
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- N. Delfosse, “Tradeoffs for reliable quantum information storage in surface codes and color codes”, 2013 IEEE International Symposium on Information Theory (2013) arXiv:1301.6588 DOI
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- H. Bombin and M. A. Martin-Delgado, “Exact topological quantum order inD=3and beyond: Branyons and brane-net condensates”, Physical Review B 75, (2007) arXiv:cond-mat/0607736 DOI
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- C. Vuillot and N. P. Breuckmann, “Quantum Pin Codes”, IEEE Transactions on Information Theory 68, 5955 (2022) arXiv:1906.11394 DOI
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- A. Kubica et al., “Three-Dimensional Color Code Thresholds via Statistical-Mechanical Mapping”, Physical Review Letters 120, (2018) arXiv:1708.07131 DOI
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- D. Hangleiter et al., “Fault-tolerant compiling of classically hard IQP circuits on hypercubes”, (2024) arXiv:2404.19005
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- B. Zeng et al., “Local unitary versus local Clifford equivalence of stabilizer and graph states”, Physical Review A 75, (2007) arXiv:quant-ph/0611214 DOI
- [8]
- S. X. Cui, D. Gottesman, and A. Krishna, “Diagonal gates in the Clifford hierarchy”, Physical Review A 95, (2017) arXiv:1608.06596 DOI
- [9]
- H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”, (2015) arXiv:1311.0879
- [10]
- B. J. Brown, N. H. Nickerson, and D. E. Browne, “Fault-tolerant error correction with the gauge color code”, Nature Communications 7, (2016) arXiv:1503.08217 DOI
- [11]
- N. Rengaswamy et al., “Synthesis of Logical Clifford Operators via Symplectic Geometry”, 2018 IEEE International Symposium on Information Theory (ISIT) (2018) arXiv:1803.06987 DOI
- [12]
- A. R. Calderbank et al., “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
- [13]
- H. Goto, “Many-hypercube codes: High-rate quantum error-correcting codes for high-performance fault-tolerant quantum computing”, (2024) arXiv:2403.16054
- [14]
- B. Shaw et al., “Encoding one logical qubit into six physical qubits”, Physical Review A 78, (2008) arXiv:0803.1495 DOI