Here is a list of quantum topological 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. |
| 3D Kitaev honeycomb code | 3D subsystem stabilizer code whose Hamiltonian is a 3D generalization of the Kitaev honeycomb model. One of the phases realized by the 3D Kitaev honeycomb Hamiltonian is that of the 3D fermionic surface code [1]. |
| 3D color code | Color code defined on a four-valent four-colorable tiling of 3D space. Logical dimension is determined by the genus of the underlying surface (for closed surfaces) and types of boundaries (for open surfaces). |
| 3D fermionic surface code | A non-CSS 3D Kitaev surface code that realizes \(\mathbb{Z}_2\) gauge theory with an emergent fermion, i.e., the fermionic-charge bosonic-loop (FcBl) phase [2]. The model can be defined on a cubic lattice in several ways [3; Eq. (D45-46)]. Realizations on other lattices also exist [4], and the phase of this code also exists in the 3D Kitaev honeycomb model [1]. |
| 3D surface code | A generalization of the Kitaev surface code defined on a 3D lattice. |
| 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 [5; Sec. IV.A]. Abelian TQDs realize all modular gapped Abelian topological orders [5]. Many Abelian TQD code Hamiltonians were originally formulated as commuting-projector models [6]. |
| 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. In 2D, this is equivalent to a unitary braided fusion category which is also an Abelian group under fusion [7]. Unless otherwise noted, the phases discussed are bosonic. |
| Chen-Hsin invertible-order code | A geometrically local commuting-projector code that realizes beyond-group-cohomology invertible topological phases in arbitrary dimensions. Instances of the code in 4D realize the 3D \(\mathbb{Z}_2\) gauge theory with fermionic charge and either bosonic (FcBl) or fermionic (FcFl) loop excitations at their boundaries [2,8]; see Ref. [9] for a different lattice-model formulation of the FcBl boundary code. |
| Chiral semion Walker-Wang model code | A 3D lattice modular-qudit stabilizer code with qudit dimension \(q=4\) whose low-energy excitations on boundaries realize the chiral semion topological order. The model admits 2D chiral semion topological order at one of its surfaces [10,11]. The corresponding phase can also be realized via a non-stabilizer Hamiltonian [12]. |
| 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. |
| Cubic honeycomb color code | 3D color code defined on a four-colorable bitruncated cubic honeycomb uniform tiling. |
| Cubic theory code | A geometrically local commuting-projector code defined on triangulations of lattices in arbitrary spatial dimensions. Its code Hamiltonian terms include Pauli-\(Z\) operators and products of Pauli-\(X\) operators and \(CZ\) gates. The Hamiltonian realizes higher-form \(\mathbb{Z}_2^3\) gauge theories whose excitations obey non-Abelian Ising-like fusion rules. |
| 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. The code can be realized as the ground-state subspace of the quantum double model, defined for \(D_m\)-valued qudits [13]. An alternative qubit-based formulation realizes the gauged \(G=\mathbb{Z}_3^2\) twisted quantum double phase [14], which is the same topological order as the \(G=D_4\) quantum double [15,16]. |
| Dijkgraaf-Witten gauge theory code | A code whose codewords realize \(D\)-dimensional lattice Dijkgraaf-Witten gauge theory [17,18] for a finite group \(G\) and a \(D+1\)-cocycle \(\omega\) in the cohomology class \(H^{D+1}(G,U(1))\). When the cocycle is non-trivial, the gauge theory is called a twisted gauge theory. For trivial cocycles in 3D, the model can be called a quantum triple model, in allusion to being a 3D version of the quantum double model. There exist lattice-model formulations in arbitrary spatial dimension [19] as well as explicitly in 3D [20,21]. |
| 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 \(e^2 m^2\) [22]. Originally formulated as the ground-state space of a Hamiltonian with non-commuting terms [23], which can be extended to other spatial dimensions [24], and later as a commuting-projector code [6,25]. |
| 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-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. |
| 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. |
| 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 \(G=\mathbb{Z}_2\times\mathbb{Z}_2\)). |
| Groupoid toric code | Extension of the Kitaev surface code from Abelian groups to groupoids, i.e., multi-fusion categories in which every morphism is an isomorphism [26]. Some models admit fracton-like features such as extensive ground-state degeneracy and excitations with restricted mobility. The robustness of these features has not yet been established. |
| Honeycomb (6.6.6) color code | 2D color code defined on a patch of the 6.6.6 (honeycomb) tiling. |
| Hopf-algebra quantum-double code | Code whose codewords realize 2D gapped topological order defined on qudits valued in a Hopf algebra \(H\). The code Hamiltonian is an generalization [27,28] of the quantum double model from group algebras to Hopf algebras, as anticipated by Kitaev [13]. Boundaries of these models have been examined [29,30]. |
| 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 unitarily 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). It can be thought of as the Majorana stabilizer analogue of the quantum repetition code, and it encodes a logical fermion because its logical Majorana operator has odd weight [31]. |
| Kitaev honeycomb code | Code whose logical subspace is labeled by different fusion outcomes of Ising anyons present in the Ising-anyon topological phase of the Kitaev honeycomb model [32]. Ising anyons also exist in other phases, such as the fractional quantum Hall phase [33]. |
| 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. |
| Klein-bottle surface code | A family of Kitaev surface codes on the non-orientable Klein bottle. |
| Layer code | Member of a family of 3D lattice CSS codes with stabilizer generator weights \(\leq 6\) that are obtained by coupling layers of 2D surface code according to the Tanner graph of a QLDPC code. Geometric locality is maintained because, instead of being concatenated, each pair of parallel surface-code squares are fused (or quasi-concatenated) with perpendicular surface-code squares via lattice surgery. |
| Loop toric code | A generalization of the Kitaev surface code defined on a 4D lattice. The code is called a \((2,2)\) toric code because it admits 2D membrane \(Z\)-type and \(X\)-type logical operators. Both types of operators create 1D (i.e., loop) excitations at their edges. The code serves as a self-correcting quantum memory [34,35]. |
| Majorana box qubit | A Majorana stabilizer code which forms a fixed-parity subspace of the ground-state subspace of one or more Kitaev Majorana chain Hamiltonians. The \([[n,1,2]]_{f}\) Majorana box qubit forms the even-fermion-parity ground-state subspace of two parallel Kitaev Majorana chains in their fermionic topological phase. Its \([[2,1,2]]_{f}\) version is called the tetron Majorana code. An \([[3,2,2]]_{f}\) extension using three Kitaev chains and housing two logical qubits of the same parity is called the hexon Majorana code. Similarly, octon, decon, and dodecon are codes defined by the positive-parity subspace of \(4\), \(5\), and \(6\) fermionic modes, respectively [36]. |
| 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 [13,37] and more general modular qudits [38]. 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. |
| Multi-fusion string-net code | Family of codes resulting from the string-net construction but whose input is a unitary multi-fusion category (as opposed to a unitary fusion category). |
| 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 | 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). |
| Qudit 3D surface code | A generalization of the 3D surface code to modular qudits. |
| Raussendorf-Bravyi-Harrington (RBH) cluster-state code | A three-dimensional cluster-state code defined on the bcc lattice (i.e., a cubic lattice with qubits on edges and faces). |
| 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 | 2D 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 [39]. |
| String-net 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 (multiplicity-free) 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. These models realize local topological order (LTO) [40]. |
| Symmetry-protected topological (SPT) code | A code whose codewords form the ground-state or low-energy subspace of a code Hamiltonian realizing symmetry-protected topological (SPT) order. |
| Tetrahedral color code | 3D color code defined on select tetrahedra of a 3D tiling. Qubits are placed on the vertices, edges, triangles, and in the center of each tetrahedron. The code has both string-like and sheet-like logical operators [41]. |
| Tetron code | A \([[2,1,2]]_{f}\) Majorana box qubit, encoding two fixed-fermion states into the four-dimensional ground-state space of two Kitaev chains, each of length two. The code encodes a logical qubit into four Majorana modes (i.e., two physical fermions), allowing it to be concatenated with various qubit codes such as surface codes and color codes. Embedding each physical qubit into two fermions via the tetron code is useful for exactly solving the Kitaev honeycomb model Hamiltonian [32] and other qubit Hamiltonians on certain graphs [42,43]. |
| Three-fermion (3F) Walker-Wang model code | A 3D lattice stabilizer code whose low-energy excitations on boundaries realize a 3D time-reversal SPT order [44] and that can be used as a resource state for fault-tolerant MBQC [45]. The anyons at the boundary of the lattice are described by the 3F anyon theory. |
| Three-fermion (3F) subsystem code | 2D subsystem stabilizer code whose low-energy excitations realize the three-fermion anyon theory [46–48]. One version uses two qubits at each site [22], while other manifestations utilize a single qubit per site and only weight-two (two-body) interactions [47,49]. All are expected to be equivalent to each other via a local constant-depth Clifford circuit. |
| Topological code | A code whose codewords form the ground-state or low-energy subspace of a (typically geometrically local) 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. |
| 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 [50; Fig. 8] refers to the construction on a torus with twisted (a.k.a. shifted) boundary conditions. |
| Truncated trihexagonal (4.6.12) color code | 2D color code defined on a patch of the 4.6.12 (truncated trihexagonal or square-hexagon-dodecagon) tiling. |
| 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 \(XZIZXII\) [51]. Codes encode either one or two logical qubits, depending on qubit geometry, and perform well against biased noise [52]. |
| 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 \(\omega\), but can also be described in a category theoretic way [53]. |
| Two-gauge theory code | A code whose codewords realize lattice two-gauge theory [54–62] for a finite two-group (a.k.a. a crossed module) in arbitrary spatial dimension. There exist several lattice-model formulations in arbitrary spatial dimension [63,64] as well as explicitly in 3D [65–68] and 4D [68], with the 3D case realizing the Yetter model [69–72]. |
| Union-Jack color code | 2D color code defined on a patch of the Tetrakis square tiling (a.k.a. the Union Jack lattice). |
| 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 [73] and realizes the Crane-Yetter model [74–76]. A single-state version of the code provides a resource state for MBQC [45]. |
| 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\(^2\) hexagonal stabilizer code | An instance of the matching code based on the Kitaev honeycomb model. It is described on a honeycomb tiling 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). |
| \((1,3)\) 4D toric code | A generalization of the Kitaev surface code defined on a 4D lattice. The code is called a \((1,3)\) toric code because it admits 1D \(Z\)-type and 3D \(X\)-type logical operators. |
| \(G\)-enriched Walker-Wang model code | A 3D topological code defined by a unitary \(G\)-crossed braided fusion category \( \mathcal{C} \) [77,78], where \(G\) is a finite group. The model realizes TQFTs that include two-gauge theories, those behind Walker-Wang models, as well as the Kashaev TQFT [79,80]. It has been generalized to include domain walls [81]. |
| \([[13,1,5]]\) 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 [82; Exam. 11 and Fig. 3]. |
| \([[15,1,3]]\) quantum Reed-Muller code | \([[15,1,3]]\) CSS code that is most easily thought of as a tetrahedral 3D color code. |
| \([[17,1,5]]\) 4.8.8 color code | Seventeen-qubit doubly even 2D color code that admits a transversal implementation of the logical Clifford group. |
| \([[4,2,2]]\) Four-qubit code | Four-qubit CSS stabilizer code is the smallest qubit stabilizer code to detect a single-qubit error. |
| \([[4,2,2]]_{G}\) four group-qudit code | \([[4,2,2]]_{G}\) group quantum code that is an extension of the four-qubit code to group-valued qudits. |
| \([[5,1,2]]\) 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 [83; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [84]. |
| \([[7,1,3]]\) Steane code | A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [85]. The code is constructed using the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors. |
| \([[8,3,2]]\) Smallest interesting color code | Smallest 3D color code whose physical qubits lie on vertices of a cube and which admits a (weakly) transversal \(CCZ\) gate. |
| \([[9,1,3]]\) Shor code | Nine-qubit CSS code that is the first quantum error-correcting code. |
| \([[9,1,3]]\) 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. |
| \([[9,1,3]]_{\mathbb{Z}_q}\) 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 [32] and its generalization [86], that is characterized by the \(\mathbb{Z}_q^{(1)}\) anyon theory [87], 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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