Here is a list of quantum topological codes.
| Code | Description |
|---|---|
| 3D fermionic surface code | A 3D Kitaev surface code that realizes \(\mathbb{Z}_2\) gauge theory with an emergent fermion. |
| 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 TQD stabilizer code | Modular-qudit stabilizer code whose codewords realize 2D modular gapped Abelian topological order. The corresponding anyon theory is defined by an Abelian group and a Type-III group cocycle that can be decomposed as a product of Type-I and Type-II group cocycles; see [2; Sec. IV.A]. |
| Abelian quantum double stabilizer code | Modular-qudit stabilizer code whose codewords realize 2D modular gapped Abelian topological order with trivial cocycle. The corresponding anyon theory is defined by an Abelian group. All such codes can be realized by a stack of modular-qudit surface codes because all Abelian groups are Kronecker products of cyclic groups. |
| Abelian topological code | Code whose codewords realize topological order associated with an Abelian anyon theory, equivalently, a unitary braided fusion category which is also an Abelian group under fusion [3]. |
| 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. |
| 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. |
| Dihedral \(G=D_m\) quantum-double code | Quantum-double code whose codewords realize \(G=D_m\) topological order associated with a \(2m\)-element dihedral group \(D_m\). Includes the simplest non-Abelian order \(D_3 = S_3\) associated with the permutation group of three objects. |
| Double-semion stabilizer code | Modular-qudit stabilizer code with qudit dimension \(q=4\) that is characterized by the 2D double semion topological phase. The code can be obtained from the \(\mathbb{Z}_4\) surface code by condensing the anyon \(e^2 m^2\) [4]. Originally formulated as a non-stabilizer qubit code [5], which can be extended to other spatial dimensions [6]. |
| 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 cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error. |
| Four group-qudit code | \([[4,2,2]]_{G}\) group quantum code that is an extension of the four-qubit code to group-valued qudits. Ideal codewords are, for \(g_1 ,g_2 \in G\), \begin{align} |\overline{g_{1},g_{2}}\rangle=\frac{1}{\sqrt{|G|}}\sum_{g\in G}|g,gg_{1},gg_{2},gg_{1}g_{2}\rangle~. \tag*{(1)}\end{align} |
| Four-rotor code | \([[4,2,2]]_{\mathbb Z}\) CSS rotor code that is an extension of the four-qubit code to the integer alphabet, i.e., the angular momentum states of a planar rotor. The code is \(U(1)\)-covariant and its ideal logical-rotor codewords, \begin{align} |\overline{x,y}\rangle = \sum_{j,k,l\in\mathbb{Z}} \delta_{a,j+k}\delta_{b,l} \left| j,k,j+l,k+l \right\rangle~, \tag*{(2)}\end{align} where \(a,b\in\mathbb{Z}\), are not normalizable. |
| Galois-qudit topological code | Abelian topological code, such as a surface [7,8] or color [9] code, constructed on lattices of Galois qudits. |
| Generalized 2D color code | Member of a family of non-Abelian 2D topological codes, defined by a finite group \( G \), that serves as a generalization of the color code (for which \(G=\mathbb{Z}_2\times\mathbb{Z}_2\)). |
| Kitaev chain code | An \([[n,1,1]]_{f}\) Majorana stabilizer code forming the ground-state of the Kitaev Majorana chain (a.k.a. Kitaev Majorana wire) in its fermionic topological phase, which is equivalent to the 1D quantum Ising model in the symmetry-breaking phase via the Jordan-Wigner transformation. The code is usually defined using the algebra of two anti-commuting Majorana operators called Majorana zero modes (MZMs) or Majorana edge modes (MEMs). |
| 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 [10]. 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 [11–13]. Codewords correspond to ground states of the surface code Hamiltonian, and error operators create or annihilate pairs of anyonic charges or vortices. |
| Matching code | Member of a class of qubit stabilizer codes based on the Abelian phase of the Kitaev honeycomb model. |
| Modular-qudit surface code | Also known as the \(\mathbb{Z}_q\) surface code. Extension of the surface code to prime-dimensional [7,14] and more general modular qudits [15]. Stabilizer 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. Since qudits have more than one \(X\) and \(Z\)-type operator, various sets of stabilizer generators can be defined. Ground-state degeneracy and the associated phase depends on the qudit dimension and the stabilizer generators. |
| Quantum-double code | Group-GKP stabilizer code whose codewords realize 2D modular gapped topological order defined by a finite group \(G\). The code's 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) cluster-state code | Also called an RHG (Raussendorf-Harrington-Goyal) cluster-state code. A three-dimensional cluster-state code defined on the bcc lattice (equivalently, a cubic lattice with qubits on edges and faces). |
| Rhombic dodecahedron surface code | A \([[14,3,3]]\) non-CSS surface code whose qubits lie on the vertices of a rhombic dodecahedron. Its non-CSS nature is due to twist defects [16] stemming from the geometry of the polytope. |
| Rotated surface code | 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. Code whose codewords realize a 2D topological order rendered by a Turaev-Viro topological field theory. The corresponding anyon theory is defined by a unitary fusion category \( \mathcal{C} \). The code is defined 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. |
| Tetron Majorana code | Also called a Majorana box qubit or Majorana qubit. An \([[n,2,2]]_{f}\) Majorana stabilizer code forming the even-fermion-parity ground-state subspace of two parallel Kitaev Majorana chains in their fermionic topological phase. An extension using three Kitaev chains and housing two logical qubits of the same parity is called the hexon Majorana code. |
| Three-dimensional color code | Three-dimensional version of the color code. |
| Three-fermion (3F) model code | A 3D topological code whose low-energy excitations realize the three-fermion anyon theory [17–19] and that can be used as a resource state for fault-tolerant MBQC [20]. |
| Three-fermion (3F) subsystem code | 2D subsystem stabilizer code whose low-energy excitations realize the three-fermion anyon theory [17–19]. One version uses two qubits at each site [4], while other manifestations utilize a single qubit per site and only two-body interactions [18,21]. All are expected to be equivalent to each other under local Clifford transformations. |
| Topological code | A code whose codewords form the ground-state or low-energy subspace of a code Hamiltonian realizing a topological phase. A topological phase may be bosonic or fermionic, i.e., constructed out of underlying subsystems whose operators commute or anti-commute with each other, respectively. Unless otherwise noted, the phases discussed are bosonic. |
| Triangular color code | A planar color code defined on a trivalent lattice, typically the honeycomb or 4-8-8 (square octagon) lattice. Each boundary of the triangle intersects the lattice such that it only touches faces of two colors. The color of the boundary is then the other third color. |
| Twisted quantum double (TQD) code | Code whose codewords realize a 2D topological order rendered by a Chern-Simons topological field theory. The corresponding anyon theory is defined by a finite group \(G\) and a Type-III group cocycle. |
| Two-dimensional color code | Two-dimensional version of the color code, defined on a two-dimensional trivalent planar graph with 3-colorable faces. Each face hosts two stabilizer generators, a Pauli-\(X\) and a Pauli-\(Z\) string acting on all the qubits of the face. |
| Walker-Wang model code | A 3D topological code defined by a unitary braided fusion category \( \mathcal{C} \) (also known as a unitary premodular category). The code is defined on a cubic lattice that is resolved to be trivalent, with a qudit of dimension \( |\mathcal{C}| \) located at each edge. The codespace is the ground-state subspace of the Walker-Wang model Hamiltonian [22]. A single-state version of the code provides a resource state for MBQC [20]. |
| 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 | Four-qubit CSS stabilizer code is the smallest qubit stabilizer code to detect a single-qubit error. Admits 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} \tag*{(3)}\end{align} This code is 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 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. |
| \([[8,3,2]]\) CSS code | Smallest 3D color code whose physical qubits lie on vertices of a cube and which admits a transversal CCZ gate. |
| \([[9,1,3]]\) Shor code | Nine-qubit CSS code that is the first quantum error-correcting code. |
| \([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code | Modular-qudit CSS code that generalizes the \([[9,1,3]]\) Shor code using properties of the multiplicative group \(\mathbb{Z}_q\). |
| \(\mathbb{Z}_3\times\mathbb{Z}_9\)-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 [10] and its generalization [23], that is characterized by the \(\mathbb{Z}_q^{(1)}\) anyon theory [24], 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. |
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