| 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. |
| Analog stabilizer code | Also known as a linear, symplectic, or Gaussian stabilizer code. Oscillator-into-oscillator stabilizer code encoding \(k\) logical modes into \(n\) physical modes. An \(((n,k,d))_{\mathbb{R}}\) analog stabilizer code is denoted as \([[n,k,d]]_{\mathbb{R}}\), where \(d\) is the code's distance. |
| Analog surface code | Also called a continuous-variable (CV) surface code. An analog CSS version of the Kitaev surface code. |
| Approximate secret-sharing code | A family of \( [[n,k,d]]_q \) CSS codes approximately correcting errors on up to \(\lfloor (n-1)/2 \rfloor\) qubits, i.e., with approximate distance approaching the no-cloning bound \(n/2\). Constructed using a non-degenerate CSS code, such as a polynomial quantum code, and a classical authentication scheme. The code can be viewed as an \(t\)-error tolerant secret sharing scheme. Since the code yields a small logical subspace using large registers that contain both classical and quantum information, it is not useful for practical error correction problems, but instead demonstrates the power of approximate quantum error correction. |
| Balanced product (BP) code | Family of CSS quantum codes based on products of two classical codes which share common symmetries. The balanced product can be understood as taking the usual tensor/hypergraph product and then factoring out the symmetries factored. This reduces the overall number of physical qubits \(n\), while, under certain circumstances, leaving the number of encoded qubits \(k\) and the code distance \(d\) invariant. This leads to a more favourable encoding rate \(k/n\) and normalized distance \(d/n\) compared to the tensor/hypergraph product. |
| Ball color code | A color code defined on a \(D\)-dimensional colex. This family includes hyperoctahedron color codes (color codes defined on balls constructed from hyperoctahedra) and 3D ball color codes (color codes defined on duals of certain Archimedean solids). |
| Binary quantum Goppa code | Also known as a quantum AG code. Binary quantum Goppa codes are a family of \( [[n,k,d]]_q \) CSS codes for \( q=2^m \), generated using classical Goppa codes. |
| Bivariate bicycle code | One of several Abelian 2BGA codes which admit time-optimal syndrome measurement circuits that can be implemented in a two-layer architecture, a generalization of the square-lattice architecture optimal for the surface codes. |
| Bosonic stabilizer code | Also known as a continuous-variable (CV) stabilizer code. Bosonic code whose codespace is defined as the common \(+1\) eigenspace of a group of mutually commuting displacement operators. Displacements form the stabilizers of the code, and have continuous eigenvalues, in contrast with the discrete set of eigenvalues of qubit stabilizers. As a result, exact codewords are non-normalizable, so approximate constructions have to be considered. |
| Braunstein five-mode code | A \([[5,1,3]]_{\mathbb{R}}\) analog stabilizer version of the five-qubit perfect code. |
| Bring's 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. |
| CPC code | A qubit stabilizer code for which two binary linear codes are used to directly construct encoding and decoding circuits against \(X\)- and \(Z\)-type errors, respectively, via ZX calculus [3,4]. |
| Calderbank-Shor-Steane (CSS) stabilizer code | A stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type operators. The two sets of stabilizer generators can often, but not always, be related to parts of a chain complex over the appropriate ring or field. |
| Classical-product code | A CSS code constructed by separately constructing the \(X\) and \(Z\) check matrices using product constructions from classical codes. A particular \([[512,174,8]]\) code performed well [5] against erasure and depolarizing noise when compared to other notable CSS codes, such as the asymptotically good quantum Tanner codes. |
| 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. |
| Cluster-state code | Code consisting of cluster states [6], which are stabilizer states defined on a graph. There is one stabilizer generator \(S_v\) per graph vertex \(v\) of the form \begin{align} S_v = X_{v} \prod_{w\in N(v)} Z_w~, \tag*{(1)}\end{align} where the neighborhood \(N(v)\) is the set of vertices which share an edge with \(v\). |
| Color code | A family of Abelian topological CSS stabilizer codes defined on a \(D\)-dimensional graph which satisfies two properties: The graph is (1) a homogeneous simplicial \(D\)-complex obtained as a triangulation of the interior of a \(D\)-simplex and (2) is \(D+1\)-colorable. |
| Crystalline-circuit qubit code | Code dynamically generated by unitary Clifford circuits defined on a lattice with some crystalline symmetry. A notable example is the circuit defined on a rotated square lattice with vertices corresponding to iSWAP gates and edges decorated by \(R_X[\pi/2]\), a single-qubit rotation by \(\pi/2\) around the \(X\)-axis. This circuit is invariant under space-time translations by a unit cell \((T, a)\) and all transformations of the square lattice point group \(D_4\). |
| Dinur-Hsieh-Lin-Vidick (DHLV) code | Stub. |
| Distance-balanced code | Galois-qudit CSS code constructed from a CSS code and a classical code using a distance-balancing procedure based on a generalized homological product. The initial code is said to be unbalanced, i.e., tailored to noise biased toward either bit- or phase-flip errors, and the procedure can result in a code that is treats both types of errors on a more equal footing. The original distance-balancing procedure [7], later generalized [8; Thm. 4.2], can yield QLDPC codes [7; Thm. 1]. |
| 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 \(e^2 m^2\) [9]. Originally formulated as a non-stabilizer qubit code [10], which can be extended to other spatial dimensions [11]. |
| Expander LP code | Family of \(G\)-lifted product codes constructed using two random classical Tanner codes defined on expander graphs [12]. For certain parameters, this construction yields the first asymptotically good QLDPC codes. Classical codes resulting from this construction are one of the first two families of \(c^3\)-LTCs. |
| Fiber-bundle code | Also called a twisted product code. CSS code constructed by combining one code as the base and another as the fiber of a fiber bundle. In particular, taking a random LDPC code as the base and a cyclic repetition code as the fiber yields, after distance balancing, a QLDPC code with distance \(\Omega(n^{3/5}\text{polylog}(n))\) and rate \(\Omega(n^{-2/5}\text{polylog}(n))\) is obtained. |
| Five-qubit perfect code | Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error. |
| Five-rotor code | Extension of the five-qubit stabilizer code to the integer alphabet, i.e., the angular momentum states of a planar rotor. The code is \(U(1)\)-covariant and ideal codewords are not normalizable. |
| Folded quantum Reed-Solomon (FQRS) code | CSS code on \(q^m\)-dimensional Galois-qudits that is constructed from folded Reed-Solomon (FRS) codes via the Galois-qudit CSS construction. This code is used to construct Singleton-bound approaching approximate quantum codes. |
| 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. |
| Fractal surface code | Kitaev surface code on a fractal geometry, which is obtained by removing qubits from the surface code on a cubic lattice. Stub. |
| Fracton code | A code whose codewords make up the ground-state space of a fracton-phase Hamiltonian. |
| Freedman-Meyer-Luo code | Hyperbolic surface code constructed using cellulation of a Riemannian Manifold \(M\) exhibiting systolic freedom [13]. 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 [14]. 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. |
| Frobenius code | Let \(C\) be a quantum cyclic code on \(n\) prime-dimensional qudits. \(C\) is a Frobenius code if there exists a positive integer \(t\) such that \(n\) divides \(p^t +1\). |
| Fusion-based quantum computing (FBQC) code | Code whose codewords are resource states used in an FBQC scheme. Related to a cluster state via Hadamard transformations. |
| GKP cluster-state code | Multi-mode code encoding logical qubits into a cluster-state stabilizer code concatenated with a single-mode GKP code. Provides a way to perform a continuous-variable (CV) analogue of fault-tolerant MBQC. |
| GKP-stabilizer code | Multimode GKP code with an infinite-dimensional logical space. Can be obtained by considering an \(n\)-mode GKP code with a finite-dimensional logical space, removing stabilizers such that the logical space becomes infinite dimensional, and applying a Gaussian circuit. |
| Galois-qudit BCH code | True Galois-qudit stabilizer code constructed from BCH codes via either the Hermitian construction or the Galois-qudit CSS construction. |
| Galois-qudit CSS code | An \([[n,k,d]]_q \) Galois-qudit true stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type Galois-qudit Pauli strings. Codes can be defined from chain complexes over \(GF(q)\) via an extension of qubit CSS-to-homology correspondence to Galois qudits. |
| Galois-qudit GRS code | True \(q\)-Galois-qudit stabilizer code constructed from generalized Reed-Solomon (GRS) codes via either the Hermitian construction [15–17] or the Galois-qudit CSS construction [18,19]. |
| Galois-qudit RS code | Also called a polynomial code (QPyC). An \([[n,k,n-k+1]]_q\) (with \(q>n\)) Galois-qudit CSS code constructed using two Reed-Solomon codes over \(GF(q)\). |
| Galois-qudit quantum RM code | True \(q\)-Galois-qudit stabilizer code constructed from generalized Reed-Muller (GRM) codes via either the Hermitian construction or the Galois-qudit CSS construction. |
| Galois-qudit stabilizer code | An \(((n,K,d))_q\) Galois-qudit code whose logical subspace is the joint eigenspace of commuting Galois-qudit Pauli operators forming the code's stabilizer group \(\mathsf{S}\). Traditionally, the logical subspace is the joint \(+1\) eigenspace, and the stabilizer group does not contain \(e^{i \phi} I\) for any \(\phi \neq 0\). The distance \(d\) is the minimum weight of a Galois-qudit Pauli string that implements a nontrivial logical operation in the code. |
| Galois-qudit topological code | Abelian topological code, such as a surface [20,21] or color [22] code, constructed on lattices of Galois qudits. |
| Generalized Shor code | Code constructed by concatenating two classical codes in a way the generalizes the Shor and quantum parity codes. |
| Generalized bicycle (GB) code | A quasi-cyclic Galois-qudit CSS code constructed using a generalized version of the bicycle ansatz [23] from a pair of equivalent index-two quasi-cyclic linear codes. Various instances of qubit GB codes are constructed in Ref. [24] (only codes with \(k=2\)) and in Ref. [25]. |
| Generalized homological-product CSS code | CSS code whose properties are determined from an underlying chain complex, which often consists of some type of product of other chain complexes. |
| Generalized homological-product code | Stabilizer code whose properties are determined from an underlying chain complex, which often consists of some type of product of other chain complexes. The Qubit CSS-to-homology correspondence yields an interpretation of codes in terms of manifolds, thus allowing for the use of various products from topology in constructing codes. |
| Generalized homological-product qubit CSS code | Qubit CSS code whose properties are determined from an underlying chain complex, which often consists of some type of product of other chain complexes. |
| Generalized surface code | Also called the \(D\)-dimensional surface or \(D\)-dimensional toric code. CSS-type extenstion of the Kitaev surface code to arbitrary \(D\)-dimensional manifolds. The 4D surface code serves as a self-correcting quantum memory, while surface codes in higher dimensions can have distances not possible in lower dimensions. |
| Golden code | Variant of the Guth-Lubotzky hyperbolic surface code that uses regular tessellations for 4-dimensional hyperbolic space. |
| Good QLDPC code | Also called asymptotically good QLDPC codes. A family of QLDPC codes \([[n_i,k_i,d_i]]\) whose asymptotic rate \(\lim_{i\to\infty} k_i/n_i\) and asymptotic distance \(\lim_{i\to\infty} d_i/n_i\) are both positive. |
| Gottesman-Kitaev-Preskill (GKP) code | Quantum lattice code for a non-degenerate lattice, thereby admitting a finite-dimensional logical subspace. Codes on \(n\) modes can be constructed from lattices with \(2n\)-dimensional full-rank Gram matrices \(A\). |
| 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. |
| Haah cubic code | Class of stabilizer codes on a length-\(L\) cubic lattice with one or two qubits per site. We also require that the stabilizer group \(\mathsf{S}\) is translation invariant and generated by two types of operators with support on a cube. |
| Hemicubic code | Stub. |
| Hermitian-construction code | An \([[n,k,d]]_q\) true Galois-qudit stabilizer code constructed from a Hermitian self-orthogonal linear code over \(GF(q^2)\) using the one-to-one correspondence between the Galois-qudit Pauli matrices and elements of the Galois field \(GF(q^2)\). |
| Hermitian-construction qubit code | An \([[n,k,d]]\) stabilizer code constructed from a Hermitian self-orthogonal linear quaternary code using the one-to-one correspondence between the four Pauli matrices \(\{I,X,Y,Z\}\) and the four elements \(\{0,1,\alpha^2,\alpha\}\) of the quaternary field \(GF(4)\). |
| Hexagonal GKP code | Single-mode GKP qudit-into-oscillator code based on the hexagonal lattice. Offers the best error correction against displacement noise in a single mode due to the optimal packing of the underlying lattice. |
| Hierarchical code | Member of a family of \([[n,k,d]]\) qubit stabilizer codes resulting from a concatenation of a constant-rate QLDPC code with a rotated surface code. Concatenation allows for syndrome extraction to be performed on a 2D geometry while maintining a threshold at the expense of a logarithmically vanishing rate. The growing syndrome extraction circuit depth allows known bounds in the literature to be weakened [26,27]. |
| High-dimensional expander (HDX) code | CSS code constructed from a Ramanujan quantum code and an asymptotically good classical LDPC code using distance balancing. Ramanujan quantum codes are defined using Ramanujan complexes which are simplicial complexes that generalise Ramanujan graphs [28,29]. Combining the quantum code obtained from a Ramanujan complex and a good classical LDPC code, which can be thought of as coming from a 1-dimensional chain complex, yields a new quantum code that is defined on a 2-dimensional chain complex. This 2-dimensional chain complex is obtained by the co-complex of the product of the 2 co-complexes. The length, dimension and distance of the new quantum code depend on the input codes. |
| Homological bosonic code | An \([[n,1]]_{\mathbb{R}}\) analog CSS code defined using homological structres associated with an \(n-1\) simplex. Relevant to the study of spacetime replication of quantum information [30]. |
| Homological product code | CSS code formulated using the homological product of two chain complexes (see Qubit CSS-to-homology correspondence). Given two classical codes, \(C_i=[n_i,k_i,d_i]\) with \(i\in\{1,2\}\), whose parity-check matrices \(H_i\) satisfy \(H_i^2 = 0\), their homological product yields two classical codes with \(C_{X,Z}\) with parity-check matrices \begin{align} H_X = H_Z^T = H_1 \otimes I + I \otimes H_2~, \tag*{(3)}\end{align} where \(I\) is the identity. These two codes then yield a homological product code via the CSS construction. |
| Homological rotor code | A homological quantum rotor code is an extension of analog stabilizer codes to rotors. The code is stabilized by a continuous group of rotor \(X\)-type and \(Z\)-type generalized Pauli operators. Codes are formulated using an extension of the qubit CSS-to-homology correspondence to rotors. The homology group of the logical operators has a torsion component because the chain complexes are defined over the ring of integers, which yields codes with finite logical dimension, i.e., encoding logical qudits instead of only logical rotors. Such finite-dimensional encodings are not possible with analog stabilizer codes. |
| 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. |
| Hypergraph product (HGP) code | A family of \([[n,k,d]]\) CSS codes whose construction is based on two binary linear seed codes, \(C_i=[n_i,k_i,d_i]\) with \(i\in\{1,2\}\). Given the two seed parity-check matrices \(H_{1,2}\), the hypergraph product yields two classical codes \(C_{X,Z}\) with parity-check matrices \begin{align} H_{X}&=\begin{pmatrix}H_{1}\otimes I_{n_{2}} & \,\,I_{n_{1}-k_{1}}\otimes H_{2}^{T}\end{pmatrix}\tag*{(4)}\\ H_{Z}&=\begin{pmatrix}I_{n_{1}}\otimes H_{2} & \,\,H_{1}^{T}\otimes I_{n_{2}-k_{2}}\end{pmatrix}~, \tag*{(5)}\end{align} where \(I_m\) is the \(m\)-dimensional identity matrix. These two codes then yield a hypergraph product code via the CSS construction. |
| Hypersphere product code | Stub. |
| 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 current-mirror qubit code | Member of the family of \([[2n,(0,2),(2,n)]]_{\mathbb{Z}}\) homological rotor codes storing a logical qubit on a thin Möbius strip. The ideal code can be obtained from a Josephson-junction [31] system [32]. |
| 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 [33]. 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 [34–36]. Codewords correspond to ground states of the surface code Hamiltonian, and error operators create or annihilate pairs of anyonic charges or vortices. |
| Lifted-product (LP) code | Code that utilizes the notion of a lifted product in its construction. Lifted products of certain classical Tanner codes are the first (asymptotically) good QLDPC codes. |
| Lloyd-Slotine nine-mode code | A \([[9,1,3]]_{\mathbb{R}}\) analog CSS version of Shor's nine-qubit code. The nullifiers for this code are \begin{align} \begin{split} &\hat{x}_1 - \hat{x}_2~, \hat{x}_2 - \hat{x}_3~, \hat{x}_4 - \hat{x}_5~ , \hat{x}_5 - \hat{x}_6~ , \hat{x}_7 - \hat{x}_8, \hat{x}_8 - \hat{x}_9~,\\ &(\hat{p}_1 + \hat{p}_2 + \hat{p}_3) - (\hat{p}_4 + \hat{p}_5 + \hat{p}_6)~,\\ &(\hat{p}_4 + \hat{p}_5 + \hat{p}_6) - (\hat{p}_7 +\hat{p}_8 + \hat{p}_9)~. \end{split} \tag*{(6)}\end{align} Logical mode operators are generated by \begin{align} \begin{split} \bar q &=& \hat{q}_1 + \hat{q}_4 + \hat{q}_7~, \\ \bar p &=& \hat{p}_1 + \hat{p}_2 + \hat{p}_3~. \end{split} \tag*{(7)}\end{align} |
| Lossless expander balanced-product code | QLDPC code constructed by taking the balanced product of lossless expander graphs. Using one part of a quantum-code chain complex constructed with one-sided loss expanders [37] yields a \(c^3\)-LTC [38]. Using two-sided expanders, which are only conjectured to exist, yields an asymptotically good QLDPC code family [39]. |
| Low-depth random Clifford-circuit qubit code | An \([[n,k]]\) qubit stabilizer code whose encoder is an \(n\)-qubit unitary transformation that takes a \(k\)-qubit state as input (with \(k\leq n\), and the remaining \(n-k\) qubits initialized to \(|0\rangle^{\otimes n-k}\) ) to give a corresponding state in the codespace as the output. An \(n\)-qubit quantum circuit with random two-qubit Clifford gates can act as an encoder into a code with distance \(d\) with high probability, with a size (i.e. number of gates in the circuit) at most \(O(n^2 log n)\)). Noting that two gates acting on disjoint qubits could in fact be executed simultaneously, this is equivalent to the depth (number of time steps in the circuit) being at most \(O(log^3 n)\). |
| Majorana stabilizer code | A stabilizer code whose stabilizers are products of an even number of Majorana fermion operators, analogous to Pauli strings for a traditional stabilizer code and referred to as Majorana stabilizers. The codespace is the mutual \(+1\) eigenspace of all Majorana stabilizers. In such systems, Majorana fermions may either be considered individually or paired into creation and annihilation operators for fermionic modes. Codes can be denoted as \([[n,k,d]]_{f}\) [40], where \(n\) is the number of fermionic modes. |
| Matching code | Member of a class of qubit stabilizer codes based on the Abelian phase of the Kitaev honeycomb model. |
| Modular-qudit CSS code | An \(((n,K,d))_q\) modular-qudit stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type Pauli strings. Codes can be defined from two classical codes and/or chain complexes over the ring \(\mathbb{Z}_q\) via an extension of qubit CSS-to-homology correspondence to modular qudits. The homology group of the logical operators has a torsion component because the chain complexes are defined over a ring, which yields codes whose logical dimension is not a power of \(q\). |
| Modular-qudit GKP code | Modular-qudit analogue of the GKP code. Encodes a qudit into a larger qudit and protects against Pauli shifts up to some maximum value. |
| Modular-qudit stabilizer code | An \(((n,K,d))_q\) modular-qudit code whose logical subspace is the joint eigenspace of commuting qudit Pauli operators forming the code's stabilizer group \(\mathsf{S}\). Traditionally, the logical subspace is the joint \(+1\) eigenspace, and the stabilizer group does not contain \(e^{i \phi} I\) for any \(\phi \neq 0\). The distance \(d\) is the minimum weight of a qudit Pauli string that implements a nontrivial logical operation in the code. |
| Modular-qudit surface code | Also known as the \(\mathbb{Z}_q\) surface code. Extension of the surface code to prime-dimensional [20,41] and more general modular qudits [42]. 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. |
| NTRU-GKP code | Multi-mode GKP code whose underlying lattice is utilized in variations of the NTRU cryptosystem [43]. Randomized constructions yield asymptotically good GKP code families. |
| Pastawski-Yoshida-Harlow-Preskill (HaPPY) code | Also known as a hyperbolic pentagon code (HyPeC). Holographic code constructed out of a network of perfect tensors that tesselates hyperbolic space. Physical qubits are associated with uncontracted tensor legs at the boundary of the tesselation, while logical qubits are associated with uncontracted legs in the bulk. The code serves as a minimal model for several aspects of the AdS/CFT holographic duality and potentially a dF/CFT duality [44]. The construction below is described for qubits, but straightforward generalizations exist to modular qudits, oscillators, and rotors [45]. |
| 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 Reed-Muller code | A CSS code formed from a classical Reed-Muller code (or its punctured versions) in which polynomials over finite fields encode data. This is done by transforming these polynomials into the stabilizer generator matrices. |
| Quantum Reed-Solomon code | Also called prime-qudit polynomial code (QPyC). Prime-qudit CSS code constructed using two Reed-Solomon codes. |
| Quantum Tanner code | Member of a family of QLDPC codes based on two compatible classical Tanner codes defined on a two-dimensional Cayley complex. For certain choices of codes and complex, the resulting codes have asymptotically good parameters. |
| Quantum check-product code | Stub. |
| Quantum convolutional code | One-dimensional translationally invariant qubit stabilizer code whose whose stabilizer group can be partitioned into constant-size subsets of constant support and of constant overlap between neighboring sets. Initially formulated as a quantum analogue of convolutional codes, which were designed to protect a continuous and never-ending stream of information. Precise formulations sometimes begin with a finite-dimensional lattice, with the intent to take the thermodynamic limit; logical dimension can be infinite as well. |
| Quantum divisible code | Consider a CSS code whose \(Z\)-stabilizers are determined by the dual of a classical \([n, k_1]\) linear binary code \(C_1\), and whose \(X\)-stabilizers are determined by a classical \([n, k_2]\) binary code \(C_2 \subset C_1\). This code is quantum divisible if all weights in \(C_2\) share a common divisor \(\Delta > 1\), and all weights in each coset of \(C_2\) in \(C_1\) are congruent to \(\Delta\). |
| Quantum expander code | CSS codes constructed from a hypergraph product of bipartite expander graphs [12] with bounded left and right vertex degrees. For every bipartite graph there is an associated matrix (the parity check matrix) with columns indexed by the left vertices, rows indexed by the right vertices, and 1 entries whenever a left and right vertex are connected. This matrix can serve as the parity check matrix of a classical code. Two bipartite expander graphs can be used to construct a quantum CSS code (the quantum expander code) by using the parity check matrix of one as \(X\) checks, and the parity check matrix of the other as \(Z\) checks. |
| Quantum lattice code | Bosonic stabilizer code on \(n\) bosonic modes whose stabilizer group is an infinite countable group of oscillator displacement operators which implement lattice translations in phase space. |
| Quantum low-density parity-check (QLDPC) code | Member of a family of \([[n,k,d]]\) modular-qudit or Galois-qudit stabilizer codes for which the number of sites participating in each stabilizer generator and the number of stabilizer generators that each site participates in are both bounded by a constant as \(n\to\infty\). A geometrically local stabilizer code is a QLDPC code where the sites involved in any syndrome bit are contained in a fixed volume that does not scale with \(n\). As opposed to general stabilizer codes, syndrome extraction of the constant-weight check operators of a QLDPC codes can be done using a constant-depth circuit. |
| Quantum multi-dimensional parity-check (QMDPC) code | High-rate low-distance CSS code whose qubits lie on a \(D\)-dimensional rectangle, with \(X\)-type stabilizer generators defined on each \(D-1\)-dimensional rectangle. For example, the \(D=2\) square geometry corresponds to a \([[n^2,n^2-4n+2,4]]\) code, with \(X\)-type stabilizer generators defined on rows and columns. The \(Z\)-type stabilizer generators are defined via permutations in order to commute with the \(X\)-type generators. |
| Quantum parity code (QPC) | A \([[m_1 m_2,1,\min(m_1,m_2)]]\) CSS code family obtained from concatenating an \(m_1\)-qubit phase-flip repetition code with an \(m_2\)-qubit bit-flip repetition code. Logical codewords are \begin{align} \begin{split} |\overline{0}\rangle&=\frac{1}{2^{m_2/2}}\left(|0\rangle^{\otimes m_1}+|1\rangle^{\otimes m_1}\right)^{\otimes m_2}\\ |\overline{1}\rangle&=\frac{1}{2^{m_2/2}}\left(|0\rangle^{\otimes m_1}-|1\rangle^{\otimes m_1}\right)^{\otimes m_2}~. \end{split} \tag*{(8)}\end{align} |
| Quantum polar code | Entanglement-assisted CSS code utilized in a quantum polar coding scheme producing entangled pairs of qubits between sender and receiver. In such a scheme, the amplitude and phase information of a quantum state is handled in complementary fashion [46] using an encoding based on classical polar codes. Variants of the initial scheme have been developed for degradable channels [47] and extended to arbitrary channels [48]. |
| Quantum repetition code | Encodes \(1\) qubit into \(n\) qubits according to \(|0\rangle\to|\phi_0\rangle^{\otimes n}\) and \(|1\rangle\to|\phi_1\rangle^{\otimes n}\). Also known as a bit-flip code when \(|\phi_i\rangle = |i\rangle\), and a phase-flip code when \(|\phi_0\rangle = |+\rangle\) and \(|\phi_1\rangle = |-\rangle\). |
| Quantum spatially coupled (SC-QLDPC) code | QLDPC code whose stabilizer generator matrix resembles the parity-check matrix of SC-LDPC codes. There exist CSS [49] and stabilizer constructions [50]. In either case, the stabilizer generator matrix is constructed by "spatially" coupling sub-matrix blocks in chain-like fashion (or, more generally, in grid-like fashion) to yield a band matrix. The sub-matrix blocks have to satisfy certain conditions amongst themselves so that the resulting band matrix is a stabilizer generator matrix. Matrices corresponding to translationally invariant chains are called time-variant, and otherwise are called time-invariant. |
| Qubit BCH code | Qubit stabilizer code constructed from a self-orthogonal binary BCH code via the CSS construction, from a Hermitian self-orthogonal quaternary BCH code via the Hermitian construction, or by taking a Euclidean self-orthogonal BCH code over \(GF(2^m)\), converting it to a binary code, and applying the CSS construction. |
| Qubit CSS code | An \([[n,k,d]]\) stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type Pauli strings. Codes can be defined from two classical codes and/or chain complexes over \(\mathbb{Z}_2\) per the qubit CSS-to-homology correspondence below. Strong CSS codes are codes for which there exists a set of \(X\) and \(Z\) stabilizer generators of equal weight. |
| Qubit stabilizer code | Also called a Pauli stabilizer code. An \(((n,2^k,d))\) qubit stabilizer code is denoted as \([[n,k]]\) or \([[n,k,d]]\), where \(d\) is the code's distance. Logical subspace is the joint eigenspace of commuting Pauli operators forming the code's stabilizer group \(\mathsf{S}\). Traditionally, the logical subspace is the joint \(+1\) eigenspace of a set of \(2^{n-k}\) commuting Pauli operators which do not contain \(-I\). The distance is the minimum weight of a Pauli string that implements a nontrivial logical operation in the code. |
| 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 [51] 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. |
| Rotor GKP code | GKP code protecting against small angular position and momentum shifts of a planar rotor. |
| Rotor stabilizer code | Rotor code whose codespace is defined as the common \(+1\) eigenspace of a group of mutually commuting rotor generalized Pauli operators. The stabilizer group can be either discrete or continuous, corresponding to modular or linear constraints on angular positions and momenta. Both cases can yield finite or infinite logical dimension. Exact codewords are non-normalizable, so approximate constructions have to be considered. |
| Singleton-bound approaching AQECC | Approximate quantum code of rate \(R\) that can tolerate adversarial errors nearly saturating the quantum Singleton bound of \((1-R)/2\). The formulation of such codes relies on a notion of quantum list decoding. Sampling a description of this code can be done with an efficient randomized algorithm with \(2^{-\Omega(n)}\) failure probability. |
| Skew-cyclic CSS code | Stub. |
| 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. |
| Square-lattice GKP code | Single-mode GKP qudit-into-oscillator code based on the rectangular lattice. Its stabilizer generators are oscillator displacement operators \(\hat{S}_q(2\alpha)=e^{-2i\alpha \hat{p}}\) and \(\hat{S}_p(2\beta)=e^{2i\beta \hat{x}}\). To ensure \(\hat{S}_q(2\alpha)\) and \(\hat{S}_p(2\beta)\) generate a stabilizer group that is Abelian, there is a constraint that \(\alpha\beta=2q\pi\) where \(q\) is an integer denoting the logical dimension. |
| Stabilizer code | A code whose logical subspace is the joint eigenspace (usually with eigenvalue \(+1\)) of a set of commuting unitary operators forming the code's stabilizer group. They can be block codes defined of tensor-product spaces of qubits or qudits, or non-block codes defined on single sufficiently large Hilbert spaces such as bosonic modes or group spaces. |
| 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. |
| Tensor-product HDX code | Code constructed in a similar way as the HDX code, but utilizing tensor products of Ramanujan complexes in order to improve code distance from \(\sqrt{n}\log n\) to \(\sqrt{n}~\text{polylog}(n)\). The utility of such tensor products comes from the fact that one of the Ramanujan complexes is a collective cosystolic expander as opposed to just a cosystolic expander. |
| 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 [52–54] and that can be used as a resource state for fault-tolerant MBQC [55]. |
| Three-qutrit code | A \([[3,1,2]]_3\) prime-qudit CSS code that is the smallest qutrit stabilizer code to detect a single-qutrit error. with stabilizer generators \(ZZZ\) and \(XXX\). The code defines a quantum secret-sharing scheme and serves as a minimal model for the AdS/CFT holographic duality. It is also the smallest non-trivial instance of a quantum maximum distance separable code (QMDS), saturating the quantum Singleton bound. The codewords are \begin{align} \begin{split} | \overline{0} \rangle &= \frac{1}{\sqrt{3}} (| 000 \rangle + | 111 \rangle + | 222 \rangle) \\ | \overline{1} \rangle &= \frac{1}{\sqrt{3}} (| 012 \rangle + | 120 \rangle + | 201 \rangle) \\ | \overline{2} \rangle &= \frac{1}{\sqrt{3}} (| 021 \rangle + | 102 \rangle + | 210 \rangle)~. \end{split} \tag*{(9)}\end{align} The elements in the superposition of each logical codeword are related to each other via cyclic permutations. |
| Three-rotor code | \([[3,1,2]]_{\mathbb Z}\) rotor code that is an extension of the \([[3,1,2]]_3\) qutrit CSS code to the integer alphabet, i.e., the angular momentum states of a planar rotor. The code is \(U(1)\)-covariant and its ideal codewords, \begin{align} |\overline{x}\rangle = \sum_{y\in\mathbb{Z}} \left| -3y,y-x,2(y+x) \right\rangle~, \tag*{(10)}\end{align} where \(x\in\mathbb{Z}\), are not normalizable. |
| Translationally invariant stabilizer code | A geometrically local qubit, modular-qudit, or Galois-qudit stabilizer code with qudits organized on a lattice modeled by the additive group \(\mathbb{Z}^D\) for spatial dimension \(D\) such that each lattice point, referred to as a site, contains \(m\) qudits of dimension \(q\). The stabilizer group of the translationally invariant code is generated by site-local Pauli operators and their translations. Boundary conditions have to be imposed on the lattice in order to obtain finite-dimensional codes. Infinite dimensional formulations are also possible, with the 1D lattice version reducing to quantum convolutional codes. |
| Transverse-field Ising model (TFIM) code | A 1D translationally invariant stabilizer code whose encoding is a constant-depth circuit of nearest-neighbor gates on alternating even and odd bonds that consist of transverse-field Ising Hamiltonian interactions. The code allows for perfect state transfer of arbitrary distance using local operations and classical communications (LOCC). |
| 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. |
| Triorthogonal code | A triorthogonal \(m \times n\) binary matrix is triorthogonal if its rows \(r_1, \ldots, r_m\) satisfy \(|r_i \cdot r_j| = 0\) and \(|r_i \cdot r_j \cdot r_k| = 0\), where addition and multiplication are done on \(\mathbb{Z}_2\). The triorthogonal code associated with the matrix is constructed by mapping non-zero entries in even-weight rows to \(X\) operators, and \(Z\) operators for each row in the orthogonal complement. |
| True Galois-qudit stabilizer code | Also called a linear stabilizer code. A \([[n,k,d]]_q\) stabilizer code whose stabilizer's symplectic representation forms a linear subspace. In other words, the set of \(q\)-ary vectors representing the stabilizer group is closed under both addition and multiplication by elements of \(GF(q)\). In contrast, Galois-qudit stabilizer codes admit sets of vectors that are closed under addition only. |
| Two-block group-algebra (2BGA) codes | 2BGA codes are the smallest LP codes LP\((a,b)\), constructed from a pair of group algebra elements \(a,b\in \mathbb{F}_q[G]\), where \(G\) is a finite group, and \(\mathbb{F}_q\) is a Galois field. For a group of order \(\ell\), we get a 2BGA code of length \(n=2\ell\). A 2BGA code for an Abelian group is called an Abelian 2BGA code. |
| Two-block quantum code | Galois-qudit CSS code whose stabilizer generator matrices \(H_X=(A,B)\) and \(H_Z=(B^T,-A^T)\), are constructed from a pair of square commuting matrices \(A\) and \(B\). |
| 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. |
| Two-dimensional 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). |
| X-cube model code | A Type-I fracton code supporting a subextensive number of logical qubits. |
| 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 product code | A non-CSS QLDPC code constructed from three classical codes. The construction of an XYZ product code is similar to that of a hypergraph product code and related codes. The idea is that rather than taking a product of only two classical codes to produce a CSS code, a third classical code is considered, acting with Pauli-\(Y\) operators. |
| 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). |
| Yoked surface code | Member of a family of \([[n,k,d]]\) qubit CSS codes resulting from a concatenation of a QMDPC code with a rotated surface code. Concatenation can as much as triple the number of logical qubits per physical qubit of the original surface code and does not impose additional connectivity constraints. |
| Zero-pi qubit code | A \([[2,(0,2),(2,1)]]_{\mathbb{Z}}\) homological rotor code on the smallest tiling of the projective plane \(\mathbb{R}P^2\). The ideal code can be obtained from a four-rotor Josephson-junction [31] system after a choice of grounding [32]. |
| \((5,1,2)\)-convolutional code | Quantum convolutional code with the stabilizer generators \begin{align} \begin{array}{cccccccc} X & Z & I & I & I & I & I & \cdots\\ Z & X & X & Z & I & I & I & \cdots\\ I & Z & X & X & Z & I & I & \cdots\\ I & I & Z & X & X & Z & I & \cdots\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{array} \tag*{(11)}\end{align} |
| \(D_4\) hyper-diamond GKP code | Two-mode GKP qudit-into-oscillator code based on the \(D_4\) hyper-diamond lattice. |
| \([[11,1,5]]\) quantum dodecacode | Eleven-qubit pure stabilizer code that is the smallest qubit stabilizer code to correct two-qubit errors. |
| \([[15, 7, 3]]\) Hamming-based CSS code | Self-dual Hamming-based CSS code that admits permutation-based CZ logical gates. |
| \([[15,1,3]]\) 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. |
| \([[2^r, 2^r-r-2, 3]]\) quantum Hamming code | A family of stabilizer codes of distance \(3\) that saturate the asymptotic quantum Hamming bound. Can be obtained from the CSS construction using a first-order \([2^r,r+1,2^{r-1}]\) RM code and a \([2^r,2^r-1,2]\) even-weight code [56]. |
| \([[2^r-1, 1, 3]]\) quantum Reed-Muller code | Member of CSS code family constructed with a first-order punctured RM\((1,r)\) \([2^r-1,r+1,2^{r-1}-1]\) code and its even subcode for \(r \geq 3\). Each code transversally implements a member of an infinite family of diagonal gates from the Clifford hierarchy [57]. |
| \([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS code | CCS code constructed with a classical Hamming code \([2^r-1,2^r-1-r,3]=C_X=C_Z\) a.k.a. a first-order punctured Reed-Muller code RM\((r-2,r)\). |
| \([[2^r-1, 2^r-2r-1, 3]]_p\) prime-qudit CSS code | A family of CSS codes extending Hamming-based CSS codes to prime qudits of dimension \(p\) by expressing the qubit code stabilizers in local-dimension-invariant (LDI) form [58]. |
| \([[2^{2r-1}-1,1,2^r-1]]\) quantum punctured Reed-Muller code | Member of CSS code family constructed with a punctured self-dual RM \([2^r-1,2^{r-1},\sqrt{2}^{r-1}-1]\) code and its even subcode for \(r \geq 2\). |
| \([[2m,2m-2,2]]\) error-detecting code | CSS stabilizer code for \(m\geq 2\) with generators \(\{XX\cdots X, ZZ\cdots Z\} \) acting on all \(2m\) physical qubits. This is the highest-rate distance-two code when an even number of qubits is used [59]. |
| \([[4,2,2]]\) 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*{(12)}\end{align} This code is the smallest instance of the toric code, and its various single-qubit subcodes are small planar surface codes. |
| \([[5,1,3]]_q\) Galois-qudit code | True stabilizer code that generalizes the five-qubit perfect code to Galois qudits of prime-power dimension \(q=p^m\). It has \(4(m-1)\) stabilizer generators expressed as \(X_{\gamma} Z_{\gamma} Z_{-\gamma} X_{-\gamma} I\) and its cyclic permutations, with \(\gamma\) iterating over basis elements of \(GF(q)\) over \(GF(p)\). |
| \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code | Modular-qudit stabilizer code that generalizes the five-qubit perfect code using properties of the multiplicative group \(\mathbb{Z}_q\) [60]; see also [61; Thm. 13]. It has four stabilizer generators consisting of \(X Z Z^\dagger X^\dagger I\) and its cyclic permutations. A concise expression for a set of codewords can be found in [45; Sec. VI.B]. |
| \([[7,1,3]]\) 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. |
| \([[8, 3, 3]]\) code | Eight-qubit code obtained from the CSS construction using a first-order \([8,4,4]\) RM code and a \([8,7,2]\) even-weight code [56]. |
| \([[8,3,2]]\) 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 \(\mathbb{Z}_q\). |
| \([[k+4,k,2]]\) H code | Family of \([[k+4,k,2]]\) CSS codes with transversal Hadamard gates; relevant to magic state distillation. The four stablizer generators are \(X_1X_2X_3X_4\), \(Z_1Z_2Z_3Z_4\), \(X_1X_2X_5X_6...X_{k+4}\), and \(Z_1Z_2Z_5Z_6...Z_{k+4}\).' |