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

3D surface code | A variant of the Kitaev surface code on a 3D lattice. The closely related solid code [1] consists of several 3D surface codes stitched together in a way that the distance scales faster than the linear size of the system. |

Abelian topological code | Code whose codewords realize topological order associated with an abelian group. Stub. |

Clifford-deformed surface code (CDSC) | A generally non-CSS derivative of the surface code defined by applying a constant-depth Clifford circuit to the original (CSS) surface code. Unlike the surface code, CDSCs include codes whose thresholds and subthreshold performance are enhanced under noise biased towards dephasing. Examples of CDSCs include the XY code, XZZX code, and random CDSCs. |

Color code | A family of abelian topological CSS stabilizer codes defined on a \(D\)-dimensional lattice which satisfies two properties: The lattice is (1) a homogeneous simplicial \(D\)-complex obtained as a triangulation of the interior of a \(D\)-simplex and (2) is \(D+1\)-colorable. Qubits are placed on the \(D\)-simplices and generators are supported on suitable simplices [2]. For 2-dimensional color code, the lattice must be such that it is 3-valent and has 3-colorable faces, such as a honeycomb lattice. The qubits are placed on the vertices and two stabilizer generators are placed on each face [3]. |

Double-semion code | Stub. |

Fibonacci string-net code | Quantum error correcting code associated with the Levin-Wen string-net model with the Fibonacci input category, admitting two types of encodings. |

Five-qubit perfect code | Five-qubit stabilizer code with generators that are symmetric under cyclic permutation of qubits, \begin{align} \begin{split} S_1 &= IXZZX \\ S_2 &= XZZXI \\ S_3 &= ZZXIX \\ S_4 &= ZXIXZ~. \end{split} \end{align} |

Fractal surface code | Kitaev surface code on a fractal geometry, which is obtained by removing qubits from the surface code on a cubic lattice. Stub. |

Freedman-Meyer-Luo code | Hyperbolic surface code constructed using cellulation of a Riemannian Manifold \(M\) exhibiting systolic freedom [4]. Codes derived from such manifolds can achieve distances scaling better than \(\sqrt{n}\), something that is impossible using closed 2D surfaces or 2D surfaces with boundaries [5]. Improved codes are obtained by studying a weak family of Riemann metrics on closed 4-dimensional manifolds \(S^2\otimes S^2\) with the \(Z_2\)-homology. |

Galois-qudit topological code | Abelian topological code, such as a surface [6][7] or color [8] code, constructed on lattices of Galois qudits. |

Generalized color code | Member of a family of non-Abelian topological codes, defined by a finite group \( G \), that serves as a generalization of the color code (for which \(G=\mathbb{Z}_2\)). |

Golden code | Variant of the Guth-Lubotzky hyperbolic surface code that uses regular tessellations for 4-dimensional hyperbolic space. |

Guth-Lubotzky code | Hyperbolic surface code based on cellulations of certain four-dimensional manifolds. The manifolds are shown to have good homology and systolic properties for the purposes of code construction, with corresponding codes exhibiting linear rate. |

Hemicubic code | Stub. |

Higher-dimensional surface code | A family of Kitaev surface codes on planar or toric surfaces of dimension greater than two. Stub. |

Hyperbolic surface code | An extension of the Kitaev surface code construction to hyperbolic manifolds in dimension two or greater. Given a cellulation of a manifold, qubits are put on \(i\)-dimensional faces, \(X\)-type stabilizers are associated with \((i-1)\)-faces, while \(Z\)-type stabilizers are associated with \(i+1\)-faces. |

Hypersphere product code | Stub. |

Kitaev honeycomb code | Code whose logical subspace is labeled by different fusion outcomes of Ising anyons present in the non-Abelian topological phase of the Kitaev honeycomb model [9]. Each logical qubit is constructed out of four Majorana operators, which admit braiding-based gates due to their non-Abelian statistics and which can be used for topological quantum computation. |

Kitaev surface code | A family of abelian topological CSS stabilizer codes whose generators are few-body \(X\)-type and \(Z\)-type Pauli strings associated to the stars and plaquettes, respectively, of a cellulation of a two-dimensional surface (with a qubit located at each edge of the cellulation). Toric code often either refers to the construction on the two-dimensional torus or is an alternative name for the general construction. The construction on surfaces with boundaries is often called the planar code [10]. |

Matching code | Member of a class of qubit stabilizer codes based on the abelian phase of the Kitaev honeycomb model. |

Projective-plane surface code | A family of Kitaev surface codes on the non-orientable 2-dimensional compact manifold \(\mathbb{R}P^2\) (in contrast to a genus-\(g\) surface). Whereas genus-\(g\) surface codes require \(2g\) logical qubits, qubit codes on \(\mathbb{R}P^2\) are made from a single logical qubit. |

Quantum-double 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). |

Rotated surface code | Also called a checkerboard code. CSS variant of the surface code defined on a square lattice that has been rotated 45 degrees such that qubits are on vertices, and both \(X\)- and \(Z\)-type check operators occupy plaquettes in an alternating checkerboard pattern. |

String-net code | Also called a Turaev-Viro or Levin-Wen model code. A family of topological codes, defined by a finite unitary fusion category \( \mathcal{C} \), whose generators are few-body operators acting on a cell decomposition dual to a triangulation of a two-dimensional surface (with a qudit of dimension \( |\mathcal{C}| \) located at each edge of the decomposition). |

Surface-17 code | A \([[9,1,3]]\) rotated surface code named for the sum of its 9 data qubits and 8 syndrome qubits. It uses the smallest number of qubits to perform error correction on a surface code with parallel syndrome extraction. |

Topological code | Stub. |

Two-dimensional hyperbolic surface code | Hyperbolic surface codes based on a tessellation of a closed 2D manifold with a hyperbolic geometry (i.e., non-Euclidean geometry, e.g., saddle surfaces when defined on a 2D plane). |

XY surface code | Also called the tailored surface code (TSC). Non-CSS derivative of the surface code whose generators are \(XXXX\) and \(YYYY\), obtained by mapping \(Z \to Y\) in the surface code. |

XYZ\(^2\) hexagonal stabilizer code | An instance of the matching code based on the Kitaev honeycomb model. It is described on a hexagonal lattice with \(XYZXYZ\) stabilizers on each hexagonal plaquette. Each vertical pair of qubits has an \(XX\), \(YY\), or \(ZZ\) link stabilizer depending on the orientation of the plaquette stabilizers. |

XZZX surface code | Non-CSS variant of the rotated surface code whose generators are \(XZXZ\) Pauli strings associated, clock-wise, to the vertices of each face of a two-dimensional lattice (with a qubit located at each vertex of the tessellation). |

\([[15,1,3]]\) 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. |

\([[4,2,2]]\) CSS code | Also known as the \(C_4\) code. Four-qubit CSS stabilizer code with generators \(\{XXXX, ZZZZ\} \) and codewords \begin{align} \begin{split} |\overline{00}\rangle = (|0000\rangle + |1111\rangle)/\sqrt{2}~{\phantom{.}}\\ |\overline{01}\rangle = (|0011\rangle + |1100\rangle)/\sqrt{2}~{\phantom{.}}\\ |\overline{10}\rangle = (|0101\rangle + |1010\rangle)/\sqrt{2}~{\phantom{.}}\\ |\overline{11}\rangle = (|0110\rangle + |1001\rangle)/\sqrt{2}~. \end{split} \end{align} This code is the smallest single-qubit error-detecting code. It is also the smallest instance of the toric code, and its various single-qubit subcodes are small planar surface codes. |

\([[7,1,3]]\) Steane code | A \([[7,1,3]]\) CSS code that uses the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors. The parity-check matrix for the \([7,4,3]\) Hamming code is \begin{align} H = \left(\begin{matrix} 1&0&0&1&0&1&1\\ 0&1&0&1&1&0&1\\ 0&0&1&0&1&1&1 \end{matrix}\right), \end{align} and the check matrix for the Steane code is therefore \begin{align} \left(\begin{matrix} 0&H\\ H&0 \end{matrix}\right). \end{align} The stabilizer group for the Steane code has six generators. Logical codewords are \begin{align} \begin{split} |\overline{0}\rangle&=\frac{1}{\sqrt{8}}\Big(|0000000\rangle+|1010101\rangle+|0110011\rangle+|1100110\rangle\\&\,\,\,\,\,\,\,\,+|0001111\rangle+|1011010\rangle+|0111100\rangle+|1101001\rangle\Big)\\|\overline{1}\rangle&=\frac{1}{\sqrt{8}}\Big(|1111111\rangle+|0101010\rangle+|1001100\rangle+|0011001\rangle\\&\,\,\,\,\,\,\,\,+|1110000\rangle+|0100101\rangle+|1000011\rangle+|0010110\rangle\Big)~. \end{split} \end{align} The automorphism group of the code is \(PGL(3,2)\) [11]. |

\([[8,3,2]]\) code | Smallest 3D color code whose physical qubits lie on vertices of a cube and which admits a transversal CCZ gate. Similar constructions exist on \(d\)-dimensional hypercubes and are called hyperoctahedron \([[2^d,d,2]]\) codes [12]. |

## References

- [1]
- K. P. Michnicki, “3D Topological Quantum Memory with a Power-Law Energy Barrier”, Physical Review Letters 113, (2014). DOI; 1406.4227
- [2]
- A. M. Kubica, The Abcs of the Color Code: A Study of Topological Quantum Codes as Toy Models for Fault-tolerant Quantum Computation and Quantum Phases of Matter, California Institute of Technology, 2018. DOI
- [3]
- H. Bombin, “An Introduction to Topological Quantum Codes”. 1311.0277
- [4]
- M. H. Freedman, “<b>Z</b><sub>2</sub>–Systolic-Freedom”, Proceedings of the Kirbyfest (1999). DOI
- [5]
- E. Fetaya, “Bounding the distance of quantum surface codes”, Journal of Mathematical Physics 53, 062202 (2012). DOI
- [6]
- S. S. Bullock and G. K. Brennen, “Qudit surface codes and gauge theory with finite cyclic groups”, Journal of Physics A: Mathematical and Theoretical 40, 3481 (2007). DOI; quant-ph/0609070
- [7]
- Iryna Andriyanova, Denise Maurice, and Jean-Pierre Tillich, “New constructions of CSS codes obtained by moving to higher alphabets”. 1202.3338
- [8]
- P. Sarvepalli, “Topological color codes over higher alphabet”, 2010 IEEE Information Theory Workshop (2010). DOI
- [9]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006). DOI; cond-mat/0506438
- [10]
- S. B. Bravyi and A. Yu. Kitaev, “Quantum codes on a lattice with boundary”. quant-ph/9811052
- [11]
- Hanson Hao, “Investigations on Automorphism Groups of Quantum Stabilizer Codes”. 2109.12735
- [12]
- M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022). DOI; 2112.01446