Here is a list of quantum topological codes.

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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 3-colex in a 3-manifold. In the original colex realization, qubits sit on vertices, \(X\)-type stabilizers are attached to 3-cells, and \(Z\)-type stabilizers are attached to faces [2].
3D fermionic surface code A non-CSS variant of the 3D Kitaev surface code that realizes \(\mathbb{Z}_2\) gauge theory with an emergent fermion, i.e., the fermionic-charge bosonic-loop (FcBl) phase [3]. The model can be defined on a cubic lattice in several ways [4; Eq. (D45-46)]. Realizations on other lattices also exist [5], 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 cubic lattice. Qubits are placed on edges, \(Z\)-type stabilizer generators are placed on square plaquettes oriented in all three directions, and \(X\)-type stabilizers are placed on the six edges neighboring every vertex [6].
Abelian TQD code TQD code whose codewords realize a 2D Abelian twisted-quantum-double topological order. For Abelian TQDs, the corresponding anyon theory is defined by an Abelian group and a group cocycle built from Type-I, Type-II, or Type-III 3-cocycles [7–9]. Abelian TQDs with Type-I and -II cocycles account for all 2D Abelian topological orders that admit gapped boundaries [10]. Abelian TQDs with Type-III cocycles may admit non-Abelian topological orders.
Abelian TQD stabilizer code Modular-qudit stabilizer code whose codewords realize a 2D Abelian twisted-quantum-double topological order on composite-dimensional qudits. For every finite Abelian group \(G=\prod_i \mathbb{Z}_{N_i}\) and every product of Type-I and Type-II cocycles, there is a Pauli stabilizer Hamiltonian realizing the corresponding Abelian TQD [8]. Equivalently, these codes exhaust the 2D Abelian topological orders that admit gapped boundaries [8,11].
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. The \(G=\mathbb{Z}_2\) instance on a torus is the toric code, and cyclic-group instances reduce to modular-qudit surface codes. All such codes can be realized by a stack of modular-qudit surface codes because all finite Abelian groups are direct 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 [12]. Unless otherwise noted, the phases discussed are bosonic.
Brickwork \(XS\) stabilizer code An \(XS\) stabilizer code that realizes the topological order of the Type-III \(G=\mathbb{Z}^3_2\) TQD model [13,14], which is the same topological order as the \(G=D_4\) quantum double [15]. Its qubits are placed on a 2D square lattice, and the stabilizers are defined using two overlapping rectangular tilings.
Chen-Hsin invertible-order code A geometrically local commuting-projector code that realizes beyond-group-cohomology invertible topological phases in arbitrary dimensions. Its code Hamiltonian terms include Pauli-\(Z\) operators and products of Pauli-\(X\) operators and \(CZ\) gates [16; Eq. (3.25)]. 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 [3,17]; see Ref. [18] 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 [19,20]. The corresponding phase can also be realized via a non-stabilizer Hamiltonian [21].
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 translationally invariant 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.
Compactified \(\mathbb{R}\) gauge theory code An integer-homology bosonic CSS code realizing 2D \(U(1)\) gauge theory on bosonic modes. The code can be obtained from the analog surface code by condensing certain anyons [22]. This results in a pinning of each mode to the space of periodic functions, which is the Hilbert space of a physical rotor, and can be thought of as compactification of the 2D \(\mathbb{R}\) gauge theory phase realized by the analog surface code.
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 family defined on triangulations in arbitrary spatial dimensions. Its Hamiltonian contains Pauli-\(Z\) flux terms and non-Pauli Gauss-law terms built from products of Pauli-\(X\) operators and \(CZ\) gates. These commuting non-Pauli stabilizers realize higher-form \(\mathbb{Z}_2^3\) gauge theories with Abelian electric excitations and non-Abelian magnetic excitations.
Dihedral \(G=D_m\) quantum-double code Quantum-double code whose codewords realize topological order associated with the dihedral group \(D_m\) of order \(2m\). For \(m \geq 3\), these codes are non-Abelian, with the simplest case given by \(D_3=S_3\), the permutation group on three objects. On an oriented lattice, each edge hosts a \(2m\)-dimensional group qudit, and the codespace is the ground-state subspace of the corresponding quantum double Hamiltonian [23].
Dijkgraaf-Witten gauge theory code A code whose codewords realize \(D\)-dimensional lattice Dijkgraaf-Witten gauge theory [24,25] 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. There exist lattice-model formulations in arbitrary spatial dimension [26]. Boundaries and excitations have been studied for arbitrary dimension [27].
Double-semion stabilizer code A 2D lattice modular-qudit stabilizer code with qudit dimension \(q=4\) that realizes the 2D double semion topological phase. The code can be obtained from a \(\mathbb{Z}_4\) toric-code ground state by condensing the emergent boson \(e^2 m^2\); in the stabilizer construction this condensation is implemented by two-body measurements [8,28]. Its ground-state subspace can be mapped to that of the double-semion string-net model by a finite-depth quantum circuit with ancillas [8].
Double-semion string-net code An \(XS\) stabilizer code that realizes the 2D double semion topological phase. The model can be extended to other spatial dimensions [29].
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.
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 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 \(G=\mathbb{Z}_2\times\mathbb{Z}_2\)). Hamiltonian terms are built from group-based right- and left-multiplication \(X\)-type operators together with \(Z\)-type operators.
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 [30]. 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.
Hexagonal \(CZ\) code A hexagonal-lattice realization of the \(2+1\)D \(l=m=n=1\) cubic theory / Type-III \(\mathbb{Z}_2^3\) twisted quantum double phase. Its stabilizers are products of Pauli-\(Z\) operators and \(CZ\) gates [13; Fig. 6][31; Fig. 3]. The ground-state subspace of the hexagonal \(CZ\) code realizes the topological order of the Type-III \(G=\mathbb{Z}^3_2\) Abelian TQD model [13,14], which is the same topological order as the \(G=D_4\) non-Abelian quantum double [15]. The stabilizers include \(CZ\) operators acting on hexagonal loops, but a reduced version exists where only two \(CZ\) gates act on each loop [31].
Honeycomb (6.6.6) color code 2D color code defined on a (typically triangular) patch of the 6.6.6 (honeycomb) tiling. The usual triangular patch has three differently colored boundaries, encodes one logical qubit, and is local-Clifford equivalent to a folded surface/toric code with two smooth and two rough boundaries [32].
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 a generalization [33,34] of the quantum double model from group algebras to Hopf algebras, as anticipated by Kitaev [23]. Boundaries of these models have been examined [35,36].
Kitaev chain code A Majorana stabilizer code obtained from the ground-state subspace of the Kitaev Majorana chain in its fermionic topological phase [37]. Its codespace is stabilized by nearest-neighbor Majorana bilinears, while two unpaired edge Majoranas furnish one logical fermionic mode. Under parity-preserving noise, it behaves as the Majorana analogue of the repetition code [38].
Kitaev honeycomb code Subsystem qubit stabilizer code underlying the Kitaev honeycomb model [39,40]. Its gauge generators are the two-qubit \(XX\), \(YY\), and \(ZZ\) link operators on the three edge types of the honeycomb lattice [40; Sec. 3.2]. Its stabilizer group is generated by loop operators, and syndrome extraction can be reduced to ordered measurements of the two-qubit link operators [40; Sec. 3.2]. This is the \(q=2\) instance of the \(\mathbb{Z}_q^{(1)}\) subsystem code and does not encode any logical qubits [40][28; Sec. 7.3].
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.
Majorana box qubit A family of Majorana stabilizer codes obtained by fixing the total fermion parity of \(n\) fermionic modes, equivalently \(2n\) Majorana zero modes, within the ground-state subspace of \(n\) Kitaev Majorana chain Hamiltonians. The resulting positive-parity subspace encodes \(n-1\) logical qubits and has Majorana distance \(2\).
Matching code Member of a class of qubit stabilizer codes based on the Abelian phase of the Kitaev honeycomb model.
Modular-qudit 3D surface code A generalization of the 3D surface code to modular qudits. Qudits are placed on edges, \(Z\)-type stabilizer generators are placed on square plaquettes oriented in all three directions, and \(X\)-type stabilizers are placed on the six edges neighboring every vertex [6].
Modular-qudit surface code Extension of the surface code to prime-dimensional [23,41] and more general modular qudits. 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.
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).
Non-Abelian Kitaev honeycomb code Code whose logical subspace in the gapped non-Abelian phase of the Kitaev honeycomb model with a magnetic field is labeled by different fusion outcomes of Ising anyons [39].
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-based 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 tessellation). The original Hamiltonian can be re-expressed via group-based right- and left-multiplication \(X\)-type as well as \(Z\)-type error operators [42; Sec. 3.3].
Quantum-triple code Group-based code whose codewords realize 3D topological order defined by a finite group \(G\).
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 [2]. Among the three semiregular triangular 2D color-code families, the 4.8.8 family uses the fewest physical qubits for a given distance and is the only one of the three with transversal implementations of the full Clifford group [43].
String-net code A non-stabilizer commuting-projector 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) [44].
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 A 3D color code defined on a colored tetrahedron cut from a suitably colored BCC lattice [45]. Qubits are placed on tetrahedra, on the triangles covering the tetrahedron faces, on the edges along the tetrahedron edges, and on the tetrahedron vertices. The code has both string-like and sheet-like logical operators [46].
Tetron code A \([[2,1,2]]_{f}\) Majorana box qubit encoding a logical qubit into four Majorana modes, equivalently into the fixed-total-parity sector of two physical fermionic modes. Four Majorana zero modes are the smallest aggregate that supports a qubit in a fixed fermion-parity sector [47]. This code can be concatenated with various qubit codes such as surface codes and color codes. Four-boundary Majorana surface-code patches are logical tetrons, i.e., higher-distance analogues of this physical tetron block [48].
Three-fermion (3F) Walker-Wang model code A 3D lattice stabilizer code whose bulk realizes a 3D time-reversal SPT order [49] and whose gapped boundary supports the 2D three-fermion (3F) topological order. The code can be used as a resource state for fault-tolerant MBQC [50].
Three-fermion (3F) subsystem code 2D subsystem stabilizer code whose low-energy excitations realize the three-fermion anyon theory [51–53]. One version uses two qubits at each site [28], while other manifestations utilize a single qubit per site and only weight-two (two-body) interactions [52,54]. 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 a square lattice with periodic boundary conditions, 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 [55; Fig. 8] refers to the construction on a torus with twisted (a.k.a. shifted) boundary conditions. In the original Hamiltonian construction, open Pauli-\(Z\) and Pauli-\(X\) strings create pairs of electric charges and magnetic vortices, and braiding one type around the other yields the nontrivial Abelian anyonic phase [23].
Truncated trihexagonal (4.6.12) color code 2D color code defined on a (typically triangular) 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 \(XZIZXII\) [56]. Codes encode either one or two logical qubits, depending on qubit geometry, and perform well against biased noise [57]. See Ref. [58] for a table of some of these for small instances, where they are called genus-one genon codes.
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 3-cocycle \(\omega\in H^3(G,U(1))\) [7,8,59]. Canonical TQD models [7] are defined on group-valued qudits.
Twisted quantum triple (TQT) code Group-based code realizing a 3D topological order rendered by a Dijkgraaf-Witten gauge theory. The corresponding anyon theory is defined by a finite group \(G\) and a Type-IV four-cocycle \(\omega\). Canonical TQT models [26,60] and other formulations whose ground states are in the same phase are all defined on group-valued qudits.
Two-gauge theory code A code whose codewords realize lattice two-gauge theory [61–69] 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 [70,71] as well as explicitly in 3D [72–75] and 4D [75], with the 3D case realizing the Yetter model [76–79].
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 non-stabilizer commuting-projector 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 [80] and realizes the Crane-Yetter model [81–83]. A single-state version of the code provides a resource state for MBQC [50].
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. In the hypercubic lattice version, qubits are placed on edges, each \(Z\)-type stabilizer generator is supported on cubes on the boundary of a hypercube, and \(X\)-type stabilizers are placed on the edges neighboring every vertex [84].
\((2,2)\) 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 [85,86].
\(G\)-enriched Walker-Wang model code A 3D topological code defined by a unitary \(G\)-crossed braided fusion category \( \mathcal{C} \) [87,88], 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 [89,90]. It has been generalized to include domain walls [91].
\([[13,1,5]]\) twisted toric code Thirteen-qubit twisted toric 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 [92; Exam. 11 and Fig. 3].
\([[15,1,3]]\) quantum RM code A \([[15,1,3]]\) quantum Reed-Muller 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. The smallest distance-five CSS code has \(n=17\) [93]. It is also a normal self-dual CSS code whose transversal Hadamard acts logically, making it suitable as an inner code for fifth-order magic-state distillation [94].
\([[4,1,2]]\) Leung-Nielsen-Chuang-Yamamoto (LNCY) code A four-qubit CSS stabilizer code that is the only qubit CSS code with such parameters [95; ID 6].
\([[4,2,2]]\) Four-qubit code A four-qubit hyperbolic self-dual CSS stabilizer code that is the smallest two-logical-qubit stabilizer code to detect a single-qubit error. It is unique for its parameters [96; Thm. 8][95; ID 9].
\([[4,2,2]]_{G}\) 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.
\([[5,1,3]]\) Five-qubit perfect code Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error.
\([[6,2,2]]\) \(C_6\) code Error-detecting normal self-dual CSS code on three qubit pairs that encodes a logical qubit pair and detects any error acting on one pair [97]. In Knill’s \(C_4/C_6\) architecture, this code is used at the second and higher concatenation levels. A choice of check operators used in that construction is \(XIIXXX\), \(XXXIIX\), \(ZIIZZZ\), and \(ZZZIIZ\), with logical operators \(X_L = IXXIII\), \(Z_L = IIZZIZ\), \(X_S = XIXXII\), and \(Z_S = IIIZZI\) [97][95; ID 126].
\([[6,4,2]]\) 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 [98; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [99].
\([[7,1,3]]\) Steane code A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [100][95; ID 226]. 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. In encoded IQP sampling, the final measurement outcomes determine both the logical sample and stabilizer checks, enabling end-of-circuit error detection or postselected decoding directly from the classical samples [101].
\([[9,1,3]]\) Shor code Nine-qubit CSS code that is the first quantum error-correcting code. Among indecomposable \([[9,1,3]]\) CSS codes, the Shor code has the largest automorphism group [95].
\([[9,1,3]]\) Surface-17 code A \([[9,1,3]]\) rotated surface code named for the sum of its 9 data qubits and 8 syndrome qubits. It is one of the four inequivalent CSS gauge fixings of the nine-qubit Bacon-Shor code [95]. It uses the smallest number of qubits to perform fault-tolerant error correction on a surface code with parallel syndrome extraction.
\(\mathbb{Z}_3\times\mathbb{Z}_9\)-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 [39] and its generalization [102], that is characterized by the \(\mathbb{Z}_q^{(1)}\) anyon theory [103], 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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