Here is a list of 2D stabilizer codes.
| 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. |
| 2D 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). |
| 2D lattice stabilizer code | Lattice stabilizer code in two spatial dimensions. |
| 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 [1; Sec. IV.A]. Abelian TQDs realize all modular gapped Abelian topological orders [1]. Many Abelian TQD code Hamiltonians were originally formulated as commuting-projector models [2]. |
| 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. |
| 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. |
| Double-semion stabilizer code | A 2D lattice 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 \(e^2 m^2\) [3]. Originally formulated as the ground-state space of a Hamiltonian with non-commuting terms [4], which can be extended to other spatial dimensions [5], and later as a commuting-projector code [2,6]. |
| Five-qubit perfect code | Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error. |
| Galois-qudit color code | Extension of the color code to 2D lattices of Galois qudits. |
| Galois-qudit surface code | Extension of the surface code to 2D lattices of Galois qudits. |
| Honeycomb (6.6.6) color code | Triangular color code defined on a patch of the 6.6.6 (honeycomb) tiling. |
| 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). 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 | Extension of the surface code to prime-dimensional [7,8] and more general modular qudits [9]. 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. |
| Rhombic dodecahedron surface code | A \([[14,3,3]]\) twist-defect surface code whose qubits lie on the vertices of a rhombic dodecahedron. Its non-CSS nature is due to twist defects [10] 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. |
| 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 [11]. |
| Stellated color code | A non-CSS color code on a lattice patch with a single twist defect at the center of the patch. |
| 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 fault-tolerant error correction on a surface code with parallel syndrome extraction. |
| Toric code | Version of the Kitaev surface code on the two-dimensional torus, encoding two logical qubits. Being the first manifestation of the surface code, "toric code" is often an alternative name for the general construction. Twisted toric code [12; Fig. 8] refers to the construction on a torus with twisted (a.k.a. shifted) boundary conditions. |
| Triangular surface code | A surface code with weight-four stabilizer generators defined on a triangular lattice patch that are examples of twist-defect surface code with a single twist defect at the center of the patch. The codes use about \(25\%\) fewer physical per logical qubit for a given distance compared to the surface code. |
| 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. |
| Twist-defect surface code | A non-CSS extension of the 2D surface-code construction whose non-CSS stabilizer generators are associated with twist defects of the associated lattice. A related construction [13] doubles the number of qubits in the lattice via symplectic doubling. |
| Twisted XZZX toric code | A cyclic code that can be thought of as the XZZX toric code with shifted (a.k.a twisted) boundary conditions. Admits a set of stabilizer generators that are cyclic shifts of a particular weight-four \(XZZX\) Pauli string. For example, a seven-qubit \([[7,1,3]]\) variant has stabilizers generated by cyclic shifts of \(XZIZXII\) [14]. Codes encode either one or two logical qubits, depending on qubit geometry, and perform well against biased noise [15]. |
| Union-Jack color code | Triangular color code defined on a patch of the Tetrakis square tiling (a.k.a. the Union Jack lattice). |
| 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. |
| 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. A further variation called the domain wall color code admits generators of the form \(XXXZZZ\) and \(ZZZXXX\) [16]. |
| 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 \(XZZX\) 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). |
| \([[13,1,5]]\) cyclic code | Thirteen-qubit twisted surface code for which there is a set of stabilizer generators consisting of cyclic permutations of the \(XZZX\)-type Pauli string \(XIZZIXIIIIIII\). The code can be thought of as a small twisted XZZX code [17; Ex. 11 and Fig. 3]. |
| \([[144,12,12]]\) gross code | A BB code which requires less physical and ancilla qubits (for syndrome extraction) than the surface code with the same number of logical qubits and distance. The gross code is equivalent to 8 copies of the surface code via a constant-depth Clifford circuit, and is an element of a larger family of 2D stabilizer codes [18]. The name stems from the fact that a gross is a dozen dozen. |
| \([[30,8,3]]\) Bring code | A \([[30,8,3]]\) hyperbolic surface code on a quotient of the \(\{5,5\}\) hyperbolic tiling called Bring's curve. Its qubits and stabilizer generators lie on the vertices of the small stellated dodecahedron. Admits a set of weight-five stabilizer generators. |
| \([[4,2,2]]\) Four-qubit code | Four-qubit CSS stabilizer code is the smallest qubit stabilizer code to detect a single-qubit error. |
| \([[5,1,2]]\) rotated surface code | Rotated surface code on one rung of a ladder, with one qubit on the rung, and four qubits surrounding it. |
| \([[6,4,2]]\) 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 equivalence [19; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [20]. |
| \([[7,1,3]]\) Steane code | A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [21]. The code is constructed using the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors. |
| \([[7,1,3]]\) twist-defect surface code | A \([[7,1,3]]\) code (different from the Steane code) that is a small example of a twist-defect surface code. |
| \([[9,1,3]]\) 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 \(\mathbb{Z}_q\). |
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