Here is a list of codes related to and generalizing the Kitaev surface code.

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Code Description Relation
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. The 2D color code is equivalent to multiple decoupled copies of the 2D surface code via a local constant-depth Clifford circuit [1–3] and has the same topological entanglement entropy [4]. The convertion process can be viewed as an ungauging [5–7,7] of certain symmetries. Conversely, the 2D color code can condense to form the 2D surface code in nine different ways, i.e., by adding two body hopping terms along one of its three triangular directions to the stabilizer group and then taking the center of the resulting nonabelian group [8]. Both the surface and 2D color codes can be constructed from two distinct types of lattices, namely, 4-valent and 3-valent 3-colorable lattices, respectively [9].
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. Translation-invariant 2D qubit lattice stabilizer codes are equivalent to several copies of the Kitaev surface code via a local constant-depth Clifford circuit [10–12]. There exists an algorithm with which one can determine the fusion and braiding rules of a 2D translationally invariant qubit code, and decompose the given code into copies of the surface code [13].
3D subsystem surface code Subsystem generalization of the surface code on a 3D cubic lattice with gauge-group generators of weight at most three.
3D surface code A generalization of the Kitaev surface code defined on a 3D lattice.
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.
Analog surface code An analog CSS version of the Kitaev surface code realizing a phase of 2D \(\mathbb{R}\) gauge theory.
Asymmetric quantum code Quantum systems can be roughly characterized by two types of noise, a bit-flip noise that maps canonical basis states into each other, and a phase-flip noise that induces relative phases between superpositions of such basis states. A code cannot protect against both types of noise arbitrarily well, and there is a tradeoff between the two types of protection. An asymmetric quantum code is one that performs much better against one type of noise than the other type. Such codes typically have tunable distances against each noise type and include CSS codes, GKP codes, and QSCs. The surface code on the honeycomb tiling is an asymmetric CSS code [14].
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. CDSC codes are deformations of the surface code via constant-depth Clifford circuits that may not be CSS.
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 [15]. An alternative qubit-based formulation realizes the gauged \(G=\mathbb{Z}_3^2\) twisted quantum double phase [16], which is the same topological order as the \(G=D_4\) quantum double [17,18].
Dynamical code Dynamically generated stabilizer-based code whose (not necessarily periodic) sequence of few-body measurements implements state initialization, logical gates and error detection. One of the instantaneous stabilizer codes of the 2D DA color code are stacks of surface codes
Five-qubit perfect code Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error.
Floquet color code 2D Floquet code on a trivalent 2D lattice whose parent topological phase is the \(\mathbb{Z}_2\times\mathbb{Z}_2\) 2D color-code phase and whose measurements cycle logical quantum information between the nine \(\mathbb{Z}_2\) surface-code condensed phases of the parent phase. The code's ISG is the stabilizer group of one of the nine surface codes. The ISG of the Floquet color code is the stabilizer group of one of nine realizations of the \(\mathbb{Z}_2\) 2D surface code.
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.
Fractal surface code Kitaev surface code on a fractal geometry, which is obtained by removing qubits from the surface code on a cubic lattice. A related construction, the fractal product code, is a hypergraph product of two classical codes defined on a Sierpinski carpet graph [19]. The underlying classical codes form classical self-correcting memories [20–22].
Fracton stabilizer code A 3D translationally invariant modular-qudit stabilizer code whose codewords make up the ground-state space of a Hamiltonian in a fracton phase. Unlike topological phases, whose excitations can move in any direction, fracton phases are characterized by excitations whose movement is restricted. Foliated type-I fracton phase codes can be grown by foliation, i.e., stacking copies of the 2D surface code; see [23; Eq. (D32)].
Freedman-Meyer-Luo code Hyperbolic surface code constructed using cellulation of a Riemannian Manifold \(M\) exhibiting systolic freedom [24]. Codes derived from such manifolds can achieve distances scaling better than \(\sqrt{n}\), something that is impossible using closed 2D surfaces or 2D surfaces with boundaries [25]. Improved codes are obtained by studying a weak family of Riemann metrics on closed 4-dimensional manifolds \(S^2\otimes S^2\) with the \(Z_2\)-homology.
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. The Galois-qudit surface code for \(q=2\) reduces to the surface code.
Generalized five-squares code Member of a family of subsystem codes that are generalizations [26,27] of a code defined on a three-valent hypergraph associated with the five-squares lattice [28]. Decoding of five-squares codes leads to a mapping of these codes to two copies of the surface code [27,28].
Golden code Variant of the Guth-Lubotzky hyperbolic surface code that uses regular tessellations for 4-dimensional hyperbolic space.
Guth-Lubotzky code Hyperbolic surface code based on cellulations of certain four-dimensional manifolds. The manifolds are shown to have good homology and systolic properties for the purposes of code construction, with corresponding codes exhibiting linear rate.
Hamiltonian-based code Code whose codespace corresponds to a set of energy eigenstates of a quantum-mechanical Hamiltonian i.e., a Hermitian operator whose expectation value measures the energy of its underlying physical system. The codespace is typically a set of low-energy eigenstates or ground states, but can include subspaces of arbitrarily high energy. Hamiltonians whose eigenstates are the canonical basis elements are called classical; otherwise, a Hamiltonian is called quantum. While codewords of the surface code form ground states of the code's stabilizer Hamiltonian, they can also be ground states of other gapless Hamiltonians [29].
Heavy-hexagon code Subsystem stabilizer code on the heavy-hexagonal point set 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 point set 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. The code can be split into a surface and a Bacon-Shor code, with the idling qubits of one code serving as the physical qubits of the other [30]. Surface code stabilizers are used to measure the Z-type stabilizers of the code. There are various ways to embed the surface code into the heavy-hex lattice [31].
Hemicubic code Homological code constructed out of cubes in high dimensions. The hemicubic code family has asymptotically diminishing soundness that scales as order \(\Omega(1/\log n)\), locality of stabilizer generators scaling as order \(O(\log n)\), and distance of order \(\Theta(\sqrt{n})\).
Homological code CSS-type extenstion of the Kitaev surface code to arbitrary manifolds. The version on a Euclidean manifold of some fixed dimension is called the \(D\)-dimensional "surface" or \(D\)-dimensional toric code. The surface-code CSS stabilizer generator prescription is extendable to higher-dimensional manifolds.
Honeycomb (6.6.6) color code 2D color code defined on a patch of the 6.6.6 (honeycomb) tiling.
Honeycomb Floquet code 2D Floquet code based on the Kitaev honeycomb model [32] whose logical qubits are generated through a particular sequence of measurements. A CSS version of the code has been proposed which loosens the restriction of which sequences to use [33]. The code has also been generalized to arbitrary non-chiral, Abelian topological order [34]. Measurement of each check operator of the honeycomb Floquet code involves two qubits and projects the state of the two qubits to a two-dimensional subspace, which we regard as an effective qubit. These effective qubits form a surface code on an enlarged honeycomb tiling [35; Fig. 2]. Electric and magnetic operators on the embedded surface code correspond to outer logical operators of the Floquet code. In fact, outer logical operators transition back and forth from magnetic to electric surface code operators under the measurement dynamics. Inspired by the honeycomb Floquet code, various weight-two measurement schemes have been designed [36–38], with the scheme in Ref. [37] being a special case of DWR. Numerical comparisons have been performed [39].
Hyperbolic surface code An extension of the Kitaev surface code construction to hyperbolic manifolds. Given a cellulation of a manifold, qubits are put on \(i\)-dimensional faces, \(X\)-type stabilizers are associated with \((i-1)\)-faces, while \(Z\)-type stabilizers are associated with \(i+1\)-faces.
Hypersphere product code Homological code based on products of hyperspheres. The hypersphere product code family has asymptotically diminishing soundness that scales as order \(O(1/\log (n)^2)\), locality of stabilizer generators scaling as order \(O(\log n/ \log\log n)\), and distance of order \(\Theta(\sqrt{n})\).
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 [40]. The Kitaev honeycomb model can be formulated as a qubit subsystem stabilizer code. This code can be obtained from the square-lattice surface code by gauging out the anyon \(em\) [41; Sec. 7.3]. During this process, the square lattice is effectively expanded to a honeycomb tiling [41; Fig. 12].
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.
La-cross code Code constructed using a hypergraph product of two copies of a cyclic LDPC code. The construction uses cyclic LDPC codes with generating polynomials \(1+x+x^k\) for some \(k\). Using a length-\(n\) seed code yields an \([[2n^2,2k^2]]\) family for periodic boundary conditions and an \([[(n-k)^2+n^2,k^2]]\) family for open boundary conditions. La-cross codes at \(k=1\) yield the toric (planar surface) code and periodic (open) boundary conditions.
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. Layer codes are combinations of constant-rate QLDPC codes with surface codes built using lattice surgery.
Long-range enhanced surface code (LRESC) Code constructed using a hypergraph product of two copies of a concatenated LDPC-repetition seed code. This family interpolates between surface codes and hypergraph codes since the hypergraph product of two repetition codes yields the planar surface code. The construction uses small \([3,2,2]\) and \([6,2,4]\) LDPC codes concatenated with \([4,1,4]\) and \([2,1,2]\) repetition codes, respectively. An example using a \([5,2,3]\) code is also presented. LRESCs reduce to planar surface codes when a trivial LDPC code is used in the hypergraph product.
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 [42,43].
Majorana surface code Majorana analogue of the surface code defined on a 2D lattice and constructed out of Majorana box qubit codes placed on patches of the lattice. The Majorana surface code is a Majorana stabilizer analogue of the surface code.
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 [15,44] and more general modular qudits [45]. 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. The modular-qudit surface code for \(q=2\) reduces to the surface code.
Neural network quantum code An approximate qubit code obtained from a numerical optimization involving a reinforcement learning agent. Reinforcement learners can be used to optimize the geometry of the surface code to be more suited to a noise channel [46].
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). A quantum-double model with \(G=\mathbb{Z}_2\) is the surface code. Non-stabilizer surface-code states can be prepared by augmenting the surface code with a quantum double model [47].
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). The RBH state encodes the temporal gate operations on the surface code into a third spatial dimension [48,49]. In addition, one of possible 2D boundaries of the RBH code is effectively a 2D toric code.
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. The lattice of the rotated surface code can be obtained by taking the medial graph of the surface code lattice (treated as a graph) and applying a similar procedure to construct the check operators [50,51][52; Fig. 8]. Applying the quantum Tanner transformation to the surface code yields the rotated surface code [53,54]. The rotated surface code presents certain savings over the original surface code [55].
Self-correcting quantum code A block quantum code that forms the ground-state subspace of an \(n\)-body geometrically local Hamiltonian whose logical information is recoverable for arbitrary long times in the \(n\to\infty\) limit after interaction with a sufficiently cold thermal environment. Typically, one also requires a decoder whose decoding time scales polynomially with \(n\) and a finite energy density. The original criteria for a self-correcting quantum memory, informally known as the Caltech rules [19,56], also required finite-spin Hamiltonians. Various candidates for self-correcting quantum memories have been constructed by coupling neighboring anyons so as to prevent them from spreading [57–61]
Spacetime circuit code Qubit stabilizer code used to correct faults in Clifford circuits, i.e., circuits up made of Clifford gates and Pauli measurements. The code utilizes redundancy in the measurement outcomes of a circuit to correct circuit faults, which correspond to Pauli errors of the code. Stabilizer generators of a spacetime code are called detectors in Refs. [62,63].
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 [64].
Subsystem hyperbolic surface code Subsystem generalization of the surface code on a 2D hyperbolic tesselation with gauge-group generators of weight at most three. An \(\{r,s\}\) hyperbolic tesselation with \(E\) edges yields a \([[3E/2,(1/2-2/r)E+2,(1-2/r)E,d]]\) subsystem code.
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 [10,65,66]. One version uses two qubits at each site [41], while other manifestations utilize a single qubit per site and only weight-two (two-body) interactions [66,67]. All are expected to be equivalent to each other via a local constant-depth Clifford circuit. One version of the 3F subsystem code can be obtained from two copies of the square-lattice surface code by gauging out the anyons \(e_1m_1e_2\) and \(e_2m_2\) [41; Sec. 7.4].
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 [68; Fig. 8] refers to the construction on a torus with twisted (a.k.a. shifted) boundary conditions. The toric code is the surface code on a 2D torus.
Triangular surface code A surface code with weight-four stabilizer generators defined on a honeycomb tiling 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 2D color code defined on a patch of the 4.6.12 (truncated trihexagonal or square-hexagon-dodecagon) tiling.
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 [69] doubles the number of qubits in the lattice via symplectic doubling. Twist-defect surface codes reduce to surface codes when there are no defects.
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 \(XZIZXII\) [70]. Codes encode either one or two logical qubits, depending on qubit geometry, and perform well against biased noise [71].
Union-Jack color code 2D color code defined on a patch of the Tetrakis square tiling (a.k.a. the Union Jack lattice).
X-cube Floquet code 3D Floquet code whose qubits are placed on vertices of a truncated cubic honeycomb. Its weight-two check operators are placed on various edges. Its ISG can be that of the X-cube model code or that of several decoupled surface codes. A rewinding of the original measurement sequence yields a period-six sequence whose ISGs are those of the X-cube model, some 3D surface codes, and some decoupled surface codes [72]. The ISG of the X-cube Floquet code can be that of the X-cube model code or that of several decoupled surface codes. A rewinding of the original measurement sequence yields a period-six sequence whose ISGs are those of the X-cube model, some 3D surface codes, and some decoupled surface codes [72].
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.
\(D\)-dimensional twisted toric code Extension of the Kitaev toric code to higher-dimensional lattices with regular or shifted (a.k.a. twisted) boundary conditions. Such boundary conditions yields quibit geometries that are tori \(\mathbb{R}^D/\Lambda\), where \(\Lambda\) is an arbitrary \(D\)-dimensional lattice. Picking a hypercubic lattice yields the ordinary \(D\)-dimensional toric code. It is conjectured that appropriate twisted boundary conditions yield multi-dimensional toric code families with linear distance and logarithmic-weight stabilizer generators [73].
\([[13,1,5]]\) 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 [74; Exam. 11 and Fig. 3].
\([[14,3,3]]\) 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 [75] stemming from the geometry of the polytope.
\([[144,12,12]]\) 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 [76]. The name stems from the fact that a gross is a dozen dozen. The gross code 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 [76]. An architecture combining the surface and gross codes was proposed in [77].
\([[17,1,5]]\) 4.8.8 color code Seventeen-qubit doubly even 2D color code that admits a transversal implementation of the logical Clifford group.
\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code Member of a family of self-dual CCS codes constructed from \([2^r-1,2^r-r-1,3]=C_X=C_Z\) Hamming codes and their duals the simplex codes. The code's stabilizer generator matrix blocks \(H_{X}\) and \(H_{Z}\) are both the generator matrix for a simplex code. The weight of each stabilizer generator is \(2^{r-1}\). Quantum Hamming codes can be concatenated with surface codes [78].
\([[30,8,3]]\) 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. Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code [79].
\([[4,2,2]]_{G}\) 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 [80; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [81].
\([[7,1,3]]\) Steane code A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [82]. 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.
\([[9,1,3]]\) 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 \(\mathbb{Z}_q\).
\(t\)-design A code whose codewords are uniformly distributed in a way that is useful for determining averages of polynomials over the code's underlying space \(X\). In that way, the codewords form an approximation of the space. A code is a design on \(X\) of strength \(t\), i.e., a \(t\)-design on \(X\), if the average of any polynomial of degree up to \(t\) over its codewords is equal to the uniform average over all of \(X\). Unitary \(t\)-designs can be generated via coherent errors, syndrome extraction, and correction [83].

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