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

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 [1]. 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 [2]. |

Double-semion code | Stub. |

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 [3]. 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 [4]. 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 [5] or color [6] code, constructed on lattices of Galois qudits. |

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. |

Haah cubic code | Class of stabilizer codes on a length-\(L\) cubic lattice with one or two qubits per site. We also require that the stabilizer group \(\mathsf{S}\) is translation invariant and generated by two types of operators with support on a cube. In the non-CSS case, these two are related by spatial inversion. For CSS codes, we require that the product of all corner operators is the identity. We lastly require that there are no non-trival ''string operators'', meaning that single-site operators are a phase, and any period one logical operator \(l \in \mathsf{S}^{\perp}\) is just a phase. Haah showed in his original construction that there is exactly one non-CSS code of this form, and 17 CSS codes [7]. The non-CSS code is labeled code 0, and the rest are numbered from 1 - 17. Codes 1-4, 7, 8, and 10 do not have string logical operators [7][8]. |

Heavy-hexagon code | Subsystem stabilizer code on the heavy-hexagonal lattice that combines Bacon-Shor and surface-code stabilizers. Encodes one logical qubit into \(n=(5d^2-2d-1)/2\) physical qubits with distance \(d\). The heavy-hexagonal lattice allows for low degree (at most 3) connectivity between all the data and ancilla qubits, which is suitable for fixed-frequency transom qubits subject to frequency collision errors. |

Hemicubic code | Stub. |

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

Honeycomb code | Floquet code inspired by the Kitaev honeycomb model [9] whose logical qubits are generated through a particular sequence of measurements. |

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. |

Hypergraph product code | A family of \([[n,k,d]]\) CSS codes whose construction is based on two binary linear seed codes \(C_1\) and \(C_2\). |

Hypersphere product code | Stub. |

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]. |

Lifted-product (LP) code | Also called a Panteleev-Kalachev (PK) code. Code that utilizes the notion of a lifted product in its construction. Lifted products of certain classical Tanner codes are the first (asymptotically) good QLDPC codes. |

Majorana stabilizer code | Majorana fermion stabilizer codes are stabilizer codes whose stabilizers are products of an even number of Majorana fermion operators, analogous to Pauli strings for a traditional stabilizer code and referred to as Majorana stabilizers. The codespace is the mutual \(+1\) eigenspace of all Majorana stabilizers. In such systems, Majorana fermions may either be considered individually or paired into creation and annihilation operators for fermionic modes. Codes can be denoted as \([[n,k,d]]_{f}\) [11], where \(n\) is the number of fermionic modes. |

Modular-qudit surface code | A family of stabilizer codes whose generators are few-body \(X\)-type and \(Z\)-type Pauli strings associated to the stars and plaquettes, respectively, of a tessellation of a two-dimensional surface (with a qudit located at each edge of the tesselation). The code has \( n=E \) many physical qudits, where \( E \) is the number of edges of the tesselation, and \( k=2g \) many logical qudits, where \( g \) is the genus of the surface. |

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). |

Raussendorf-Bravyi-Harrington (RBH) code | Stub. (see Sec. III E of [12]) |

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. |

Solid code | A variant of Kitaev's surface code on a 3D lattice. |

String-net code | Also called a Turaev-Viro or Levin-Wen model code. A family of topological codes, defined by a finite unitary spherical 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. |

Translationally-invariant stabilizer code | A geometrically local qubit or qudit stabilizer code with qudits organized on a lattice modeled by the additive group \(\mathbb{Z}^D\) for spatial dimension \(D\) such that each lattice point, referred to as a site, contains \(m\) qudits of dimension \(q\). The stabilizer group of the translationally invariant code is generated by site-local Pauli operators and their translations. |

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

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). |

\([[4,2,2]]\) 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. |

## References

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