Here is a list of quantum subsystem codes.

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

3D subsystem surface code | Subsystem generalization of the surface code on a 3D cubic lattice with stabilizer generators of weight at most three. |

Bacon-Casaccino subsystem code | Subsystem version of the generalized Shor code that has the same parameters as the subspace version, but requires fewer stabilizer measurements, resulting in a simpler error recovery routine. The code can be obtained from a generalized Shor code by removing certain stabilizers that do no affect the code distance. |

Bacon-Shor code | CSS subsystem stabilizer code defined on an \(m_1 \times m_2\) lattice of qubits. It is said to be symmetric when \(m_1=m_2\). The \(X\)-type and \(Z\)-type stabilizers defined as \(X\) and \(Z\) operators acting on all qubits on adjacent columns and rows, respectively. Let \(O_{i,j}\) denote an operator acting on the qubit at a position \((i,j)\) on the lattice, with \(i\in\{0,1,\ldots ,m_1-1\}\) and \(j\in\{0,1,\ldots,m_2-1\}\). The code's stabilizer group is \begin{align} \mathsf{S}=\langle X_{i,*}X_{i+1,*},Z_{*,j}Z_{*,j+1}\rangle~, \tag*{(1)}\end{align} with generators expressed as products of nearest-neightbour 2-qubit gauge operators, \begin{align} \begin{split} X_{i,*}X_{i+1,*}= \bigotimes_{k=0}^{m_2-1} X_{i,k}X_{i+1,k} \\ Z_{*,j}Z_{*,j+1}=\bigotimes_{k=0}^{m_1-1} Z_{k,j}Z_{k,j+1}~. \end{split} \tag*{(2)}\end{align} Syndrome extraction can be done by measuring these gauge operators, which are on fewer qubits and local. |

Bravyi-Bacon-Shor (BBS) code | An \([[n,k,d]]\) CSS subsystem stabilizer code generalizing Bacon-Shor codes to a larger set of qubit geometries. Defined through a binary matrix \(A\) such that physical qubits live on sites \((i,j)\) whenever \(A_{i,j}=1\). The gauge group is generated by 2-qubit operators, including \(XX\) interations between any two qubits sharing a column in \(A\), and \(ZZ\) interations between two qubits sharing a row. The code parameters are: \(n=\sum_{i,j}A_{i,j}\), \(k=\text{rank}(A)\), and the distance is the minimum weight of any row or column. |

Chiral semion subsystem code | Modular-qudit subsystem stabilizer code with qudit dimension \(q=4\) that is characterized by the chiral semion topological phase. Admits a set of geometrically local stabilizer generators on a torus. |

Doubled color code | Family of \([[2t^3+8t^2+6t-1,1,2t+1]]\) subsystem color codes (with \(t\geq 1\)), constructed using a generalization of the doubling transformation [1], that admit a Clifford + \(T\) transversal gate set using gauge fixing. |

Entanglement-assisted (EA) operator QECC | Subsystem QECC whose encoding and decoding utilizes pre-shared entanglement between sender and receiver. |

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. |

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 [2]. 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. |

Subsystem CSS code | Subsystem stabilizer code which admits a set of gauge-group generators which consist of either all-\(Z\) or all-\(X\) Pauli strings. This ensures that the code's stabilizer group is also CSS. |

Subsystem Galois-qudit CSS code | Galois-qudit subsystem stabilizer code which admits a set of gauge-group generators which consist of either all-\(Z\) or all-\(X\) Galois-qudit Pauli strings. |

Subsystem Galois-qudit code | Subsystem QECC encoding into a \(q^n\)-dimensional Hilbert space consisting of \(n\) Galois qudits. |

Subsystem Galois-qudit stabilizer code | Galois-qudit generalization of a subsystem qubit stabilizer code. Can be obtained by taking a Galois-qudit stabilizer code and assigning some of its logical qubits to be gauge qubits. |

Subsystem color code | Stub. |

Subsystem modular-qudit CSS code | Modular-qudit subsystem stabilizer code which admits a set of gauge-group generators which consist of either all-\(Z\) or all-\(X\) modular-qudit Pauli strings. This ensures that the code's stabilizer group is also CSS. |

Subsystem modular-qudit code | Subsystem QECC encoding into a \(q^n\)-dimensional Hilbert space consisting of \(n\) modular qudits. |

Subsystem modular-qudit stabilizer code | Also called a gauge stabilizer code. Modular-qudit generalization of a subsystem qubit stabilizer code. Can be obtained by taking a modular qudit stabilizer code and assigning some of its logical qubits to be gauge qubits. For composite qudit dimensions, such codes need not encode an integer number of qudits. |

Subsystem quantum error-correcting code | Quantum code encoding information in a subsystem \(\mathsf{A}\) of the code space \(\mathsf{C}\), which is part of the system Hilbert space \(\mathsf{H}\), as \begin{align} \mathsf{H}=\mathsf{C} \oplus \mathsf{C}^{\perp} = \mathsf{A} \otimes \mathsf{B} \oplus \mathsf{C}^{\perp}~. \tag*{(3)}\end{align} Following an error, it is sufficient to revert back to the original state modulo a transformation on the auxiliary or gauge subsystem \(\mathsf{B}\). The subsystem \(\mathsf{B}\) therefore gives additional freedom to the error correction process, and is said to encode gauge qubits when its dimension is a power of two. While strictly speaking all operator QECCs are also ordinary QECCs, the attachment of a subsystem to a code allows for a wider variety of encoding procedures, fault-tolerant logical operations, and efficient error-correction protocols. |

Subsystem qubit code | Subsystem QECC encoding into a \(2^n\)-dimensional (i.e., \(n\)-qubit) Hilbert space. |

Subsystem qubit stabilizer code | A stabilizer code with some of its logical qubits denoted as gauge qubits and not used for storage of logical information. Note that this doesnt lead to new codes but does lead to new error correction and fault tolerance procedures. Subsystem codes are denoted by \([[n,k,r,d]]\), similar to stabilizer codes, but with an extra parameter \(r\) denoting the number of gauge qubits. |

Subsystem rotated surface code | Subsystem version of the rotated surface code. |

Subsystem surface code | Subsystem version of the surface code defined on a square lattice with qubits placed at every vertex and center of everry edge. |

Three-fermion (3F) subsystem code | 2D subsystem stabilizer code whose low-energy excitations realize the three-fermion anyon theory [3–5]. One version uses two qubits at each site [6], while other manifestations utilize a single qubit per site and only two-body interactions [4,7]. All are expected to be equivalent to each other under local Clifford transformations. |

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

\(\mathbb{Z}_3\times\mathbb{Z}_9\)-fusion subsystem code | Modular-qudit 2D subsystem stabilizer code whose low-energy excitations realize a non-modular anyon theory with \(\mathbb{Z}_3\times\mathbb{Z}_9\) fusion rules. Encodes two qutrits when put on a torus. |

\(\mathbb{Z}_q^{(1)}\) subsystem code | Modular-qudit subsystem code, based on the Kitaev honeycomb model [2] and its generalization [8], that is characterized by the \(\mathbb{Z}_q^{(1)}\) anyon theory [9], which is modular for odd prime \(q\) and non-modular otherwise. Encodes a single \(q\)-dimensional qudit when put on a torus for odd \(q\), and a \(q/2\)-dimensional qudit for even \(q\). This code can be constructed using geometrically local gauge generators, but does not admit geometrically local stabilizer generators. For \(q=2\), the code reduces to the subsystem code underlying the Kitaev honeycomb model code as well as the honeycomb Floquet code. |

## References

- [1]
- K. Betsumiya and A. Munemasa, “On triply even binary codes”, Journal of the London Mathematical Society 86, 1 (2012) arXiv:1012.4134 DOI
- [2]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
- [3]
- E. Rowell, R. Stong, and Z. Wang, “On classification of modular tensor categories”, (2009) arXiv:0712.1377
- [4]
- H. Bombin, M. Kargarian, and M. A. Martin-Delgado, “Interacting anyonic fermions in a two-body color code model”, Physical Review B 80, (2009) arXiv:0811.0911 DOI
- [5]
- H. Bombin, G. Duclos-Cianci, and D. Poulin, “Universal topological phase of two-dimensional stabilizer codes”, New Journal of Physics 14, 073048 (2012) arXiv:1103.4606 DOI
- [6]
- T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
- [7]
- H. Bombin, “Topological subsystem codes”, Physical Review A 81, (2010) arXiv:0908.4246 DOI
- [8]
- M. Barkeshli et al., “Generalized Kitaev Models and Extrinsic Non-Abelian Twist Defects”, Physical Review Letters 114, (2015) arXiv:1405.1780 DOI
- [9]
- P. H. Bonderson, Non-Abelian Anyons and Interferometry, California Institute of Technology, 2007 DOI