Here is a list of quantum Hamiltonian-based codes.

Code | Description |
---|---|

2D lattice stabilizer code | Lattice stabilizer code in two spatial dimensions. |

3D fermionic surface code | A 3D Kitaev surface code that realizes \(\mathbb{Z}_2\) gauge theory with an emergent fermion. The model can be defined on a cubic lattice in several ways [1; Eq. (D45-46)] |

3D lattice stabilizer code | Lattice stabilizer code in three spatial dimensions. Qubit codes are conjectured to admit either fracton phases or abelian topological phases that are equivalent to multiple copies of the 3D surface code and/or the 3D fermionic surface code [1]. |

3D surface code | A variant of the Kitaev surface code on a 3D lattice. The closely related solid code [2] 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 [3; 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. |

Abelian topological code | Code whose codewords realize topological order associated with an Abelian anyon theory, equivalently, a unitary braided fusion category which is also an Abelian group under fusion [4]. |

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. |

Bacon-Shor code | Subsystem CSS code defined on an \(m_1 \times m_2\) lattice of qubits that generalizes the \([[9,1,3]]\) (subspace) Shor code. It is said to be symmetric when \(m_1=m_2\), and asymmetric otherwise. |

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 [5,6]. |

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. |

Camara-Ollivier-Tillich code | A Hermitian qubit QLDPC code whose stabilizer generator matrix is constructed using two nested subgroups of \(GF(4)^n\). |

Chamon model code | A foliated type-I fracton non-CSS code defined on a cubic lattice using one weight-eight stabilizer generator acting on the eight vertices of each cube in the lattice [1; Eq. (D38)]. |

Checkerboard model code | A foliated type-I fracton code defined on a cubic lattice that admits weight-eight \(X\)- and \(Z\)-type stabilizer generators on the eight vertices of each cube in the lattice. |

Chuang-Leung-Yamamoto (CLY) code | Bosonic Fock-state code that encodes \(k\) qubits into \(n\) oscillators, with each oscillator restricted to having at most \(N\) excitations. Codewords are superpositions of oscillator Fock states which have exactly \(N\) total excitations, and are either uniform (i.e., balanced) superpositions or unbalanced superpositions. Codes can be denoted as \([[N,n,2^k,d]]\), which conflicts with stabilizer code notation. |

Classical fractal liquid code | Member of a family of \([L^D,O(L^{D-1}),O(L^{D-\epsilon})]_p\) linear codes on \(D\)-dimensional square lattices of side length \(L\) and for some prime \(p\) and \(\epsilon > 0\) that is based on \(p\)-ary generalizations of the Sierpinski triangle. |

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 [7] 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 [8], 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. |

Commuting-projector code | Hamiltonian-based code whose Hamiltonian terms can be expressed as orthogonal projectors (i.e., Hermitian operators with eigenvalues 0 or 1) that commute with each other. |

Constant-excitation (CE) code | Code whose codewords lie in an excited-state eigenspace of a Hamiltonian governing the total energy or total number of excitations of the underlying quantum system. For qubit codes, such a Hamiltonian is often the total spin Hamiltonian, \(H=\sum_i Z_i\). For spin-\(S\) codes, this generalizes to \(H=\sum_i J_z^{(i)}\), where \(J_z\) is the spin-\(S\) \(Z\)-operator. For bosonic codes, such as Fock-state codes, codewords are often in an eigenspace with eigenvalue \(N>0\) of the total excitation or energy Hamiltonian, \(H=\sum_i \hat{n}_i\). |

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 \(D_4\). |

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. |

Dinur-Hsieh-Lin-Vidick (DHLV) code | Stub. |

Dinur-Lin-Vidick (DLV) code | Member of a family of quantum locally testable codes constructed using cubical chain complexes, which are \(t\)-order extensions of the complexes underlying expander codes (\(t=1\)) and expander lifted-product codes (\(t=2\)). For \(t=4\), assuming a conjecture about random linear maps, there exists a family with linear dimension and inverse poly-logarithmic relative distance and soundness. |

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 [9], later generalized [10; Thm. 4.2], can yield QLDPC codes [9; 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 \(e^2 m^2\) [11]. Originally formulated as a non-stabilizer qubit code [12], which can be extended to other spatial dimensions [13]. |

Dual-rail quantum code | Two-mode code encoding a logical qubit in Fock states with one excitation. The logical-zero state is represented by \(|10\rangle\), while the logical-one state is represented by \(|01\rangle\). The two modes of the encoding can represent temporal or spatial modes, corresponding to a time-bin or frequency-bin encoding. |

Eigenstate thermalization hypothesis (ETH) code | Also called a thermodynamic code [14]. An \(n\)-qubit approximate code whose codespace is formed by eigenstates of a translationally-invariant quantum many-body system which satisfies the Eigenstate Thermalization Hypothesis (ETH). ETH ensures that codewords cannot be locally distinguished in the thermodynamic limit. Relevant many-body systems include 1D non-interacting spin chains, Motzkin chains, or Heisenberg models. |

Error-corrected sensing code | Code that can be obtained via an optimization procedure that ensures correction against a set \(\cal{E}\) of errors as well as guaranteeting optimal precision in locally estimating a parameter using a noiseless ancilla. For tensor-product spaces consisting of \(n\) subsystems (e.g., qubits, modular qudits, or Galois qudits), the procedure can yield a code whose parameter estimation precision satisfies Heisenberg scaling, i.e., scales quadratically with the number \(n\) of subsystems. |

Expander LP code | Family of \(G\)-lifted product codes constructed using two random classical Tanner codes defined on expander graphs [15]. 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. |

Fibonacci fractal spin-liquid code | A fractal type-I fracton CSS code defined on a cubic lattice [1; Eq. (D23)]. |

Fibonacci string-net code | Quantum error correcting code associated with the Levin-Wen string-net model with the Fibonacci input category, admitting two types of encodings. |

Five-qubit perfect code | Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error. |

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 group-qudit code | \([[4,2,2]]_{G}\) group quantum code that is an extension of the four-qubit code to group-valued qudits. Ideal codewords are, for \(g_1 ,g_2 \in G\), \begin{align} |\overline{g_{1},g_{2}}\rangle=\frac{1}{\sqrt{|G|}}\sum_{g\in G}|g,gg_{1},gg_{2},gg_{1}g_{2}\rangle~. \tag*{(2)}\end{align} |

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*{(3)}\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 stabilizer code | A 3D translationally invariant 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. |

Freedman-Meyer-Luo code | Hyperbolic surface code constructed using cellulation of a Riemannian Manifold \(M\) exhibiting systolic freedom [16]. 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 [17]. 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. |

GNU permutation-invariant code | Permutation-invariant code whose codewords can be expressed as superpositions of Dicke states with coefficients are square-roots of the binomial distribution. |

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 [18–20] or the Galois-qudit CSS construction [21,22]. |

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 [23,24] or color [25] code, constructed on 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 \(G=\mathbb{Z}_2\times\mathbb{Z}_2\)). |

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 [26] from a pair of equivalent index-two quasi-cyclic linear codes. Various instances of qubit GB codes are constructed in Ref. [27] (for \(k=2\)) and in Ref. [28]. |

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 quantum divisible code | A level-\(\nu\) generalized quantum divisible code is a CSS code whose \(X\)-type stabilizers, in the symplectic representation, have zero norm and form a \((\nu,t)\)-null matrix (defined below) with respect to some odd-integer vector \(t\) [29; Def. V.1]. Such codes admit gates at the \(\nu\)th level of the Clifford hierarchy. Such codes can also be level-lifted [29; Theorem V.6], \(\nu\to\nu+1\), which recursively yields towers of generalized divisible codes from a particular ground code. |

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 \(A\). |

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. |

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. |

Hamiltonian-based code | Code whose codespace corresponds to a set of energy eigenstates of a quantum-mechanical Hamiltonian. The codespace is typically a set of low-energy eigenstates or ground states, but can include subspaces of arbitrarily high energy. |

Hemicubic code | Stub. |

Hermitian Galois-qudit 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 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 [31,32]. |

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 [33,34]. 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 [35]. |

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. |

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*{(4)}\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. |

Hsieh-Halasz (HH) code | Member of one of two families of fracton codes, named HH-I and HH-II, defined on a cubic lattice with two qubits per site. HH-I (HH-II) is a CSS (non-CSS) stabilizer code family, with the former identified as a foliated type-I fracton code [1]. |

Hsieh-Halasz-Balents (HHB) code | Member of one of two families of fracton codes, named HHB model A and B, defined on a cubic lattice with two qubits per site. Both are expected to be foliated type-I fracton codes [1; Eqs. (D42-D43)]. |

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 member of a family of CSS codes whose stabilizer generator matrix is obtained from a hypergraph product of two classical linear codes. Codes from hypergraph products in higher dimension are called higher-dimensional HGP codes [36]. |

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 [37] system [38]. |

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 [39]. 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. Ising anyons also exist in other phases, such as the fractional quantum Hall phase [40]. |

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. |

Lattice 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 \(D\). On an infinite lattice, its stabilizer group is generated by few-site Pauli operators and their translations, in which case the code is called translationally invariant stabilizer code. Boundary conditions have to be imposed on the lattice in order to obtain finite-dimensional versions. Lattice defects and boundaries between different codes can also be introduced. |

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. |

Lift-connected surface (LCS) code | Member of one of several families of lifted-product codes that consist of sparsely interconnected stacks of surface codes. |

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. |

Linear binary code | An \((n,2^k,d)\) linear code is denoted as \([n,k]\) or \([n,k,d]\), where \(d\) is the code's distance. Its codewords form a linear subspace, i.e., for any codewords \(x,y\), \(x+y\) is also a codeword. A code that is not linear is called nonlinear. |

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*{(5)}\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*{(6)}\end{align} |

Long-range enhanced surface code (LRESC) | Code constructed using the hypergraph product of two copies of a concatanated LDPC-repetition code. This family interpolates between surface codes and hypergraph codes since the hypergraph product of two repetition codes yields the planar surface code. |

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 [41] yields a \(c^3\)-LTC [42]. Using two-sided expanders, which are only conjectured to exist, yields an asymptotically good QLDPC code family [43]. |

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 box qubit | An \([[n,1,2]]_{f}\) Majorana stabilizer code forming the even-fermion-parity ground-state subspace of two parallel Kitaev Majorana chains in their fermionic topological phase. The \([[2,1,2]]_{f}\) version is called the tetron Majorana code. An \([[3,2,2]]_{f}\) extension using three Kitaev chains and housing two logical qubits of the same parity is called the hexon Majorana code. Similarly, octon, decon, and dodecon are codes defined by the positive-parity subspace of \(4\), \(5\), and \(6\) fermionic modes, respectively [44]. |

Majorana color code | Majorana analogue of the color code defined on a 2D tricolorable lattice and constructed out of Majorana box qubit codes placed on patches of the lattice. |

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 \([[n,k,d]]_{f}\) [45], where \(n\) is the number of fermionic modes (equivalently, \(2n\) Majorana modes). |

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. |

Matching code | Member of a class of qubit stabilizer codes based on the Abelian phase of the Kitaev honeycomb model. |

Matrix-model code | Multimode-mode Fock-state bosonic approximate code derived from a matrix model, i.e., a non-Abelian bosonic gauge theory with a large gauge group. The model's degrees of freedom are matrix-valued bosons \(a\), each consisting of \(N^2\) harmonic oscillator modes and subject to an \(SU(N)\) gauge symmetry. |

Matrix-product state (MPS) code | Also called a magnon code. An \(n\)-qubit approximate code whose codespace of \(k=\Omega(\log n)\) qubits is efficiently described in terms of matrix product states (MPS) or Bethe ansatz tensor networks. A no-go theorem states that open-boundary MPS that form a degenerate ground-state space of a gapped local Hamiltonian yield codes with distance that is only constant in the number of qubits \(n\), so MPS excitation ansatze have to be used to achieve a distance scaling nontrivially with \(n\). |

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 [23,46] and more general modular qudits [47]. 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. |

Movassagh-Ouyang Hamiltonian code | This is a family of codes derived via an algorithm that takes as input any binary classical code and outputs a quantum code (note that this framework can be extended to \(q\)-ary codes). The algorithm is probabalistic but succeeds almost surely if the classical code is random. An explicit code construction does exist for linear distance codes encoding one logical qubit. For finite rate codes, there is no rigorous proof that the construction algorithm succeeds, and approximate constructions are described instead. |

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). |

NTRU-GKP code | Multi-mode GKP code whose underlying lattice is utilized in variations of the NTRU cryptosystem [48]. Randomized constructions yield asymptotically good GKP code families. |

Newman-Moore code | Member of a family of \([L^2,O(L),O(L^{\frac{\log 3}{\log 2}})]\) binary linear codes on \(L\times L\) square lattices that form the ground-state subspace of a class of exactly solvable spin-glass models with three-body interactions. The codewords resemble the Sierpinski triangle on a square lattice, which can be generated by a cellular automaton [49]. |

One-hot quantum code | Encoding of a \(q\)-dimensional qudit into the single-excitation subspace of \(q\) modes. The \(j\)th logical state is the multi-mode Fock state with one photon in mode \(j\) and zero photons in the other modes. |

Ouyang-Chao constant-excitation permutation-invariant code | Stub. |

Pair-cat code | Two- or higher-mode extension of cat codes whose codewords are right eigenstates of powers of products of the modes' lowering operators. Many gadgets for cat codes have two-mode pair-cat analogues, with the advantage being that such gates can be done in parallel with a dissipative error-correction process. |

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 [50]. The construction below is described for qubits, but straightforward generalizations exist to modular qudits, oscillators, and rotors [14]. |

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 | A level-\(\nu\) quantum divisible code is a CSS code whose \(X\)-type stabilizers form a \(\nu\)-even linear binary code in the symplectic representation and which admits a transversal gate at the \(\nu\)th level of the Clifford hierarchy. A CSS code is doubly-even (triply-even) if all \(X\)-type stabilizer generators have weight divisible by four (eight), i.e., if they form a doubly-even (triply-even) linear binary code. Doubly-even codes can yield a transversal \(S\) gate, while triply-even codes yield a transversal \(T\) gate for odd \(n\) [51]. |

Quantum expander code | CSS codes constructed from a hypergraph product of bipartite expander graphs [15] 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 locally testable code (QLTC) | A local commuting-projector Hamiltonian-based block quantum code which has a nonzero average-energy penalty for creating large errors. Informally, QLTC error states that are far away from the codespace have to be excited states by a number of the code's local projectors that scales linearly with \(n\). |

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. 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*{(7)}\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 [52] using an encoding based on classical polar codes. Variants of the initial scheme have been developed for degradable channels [53] and extended to arbitrary channels [54]. |

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 [55] and stabilizer constructions [56]. 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. |

Quantum twisted code | Hermitian code arising constructed from twisted BCH codes. |

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). |

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. |

Qudit cubic code | Generalization of the Haah cubic code to modular qudits. |

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 [57] 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. |

SYK code | Approximate \(n\)-fermionic code whose codewords are low-energy states of the Sachdev-Ye-Kitaev (SYK) Hamiltonian [58,59] or other low-rank SYK models [60,61]. |

Self-correcting quantum code | Also called a self-correcting quantum memory or thermally stable encoding. 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 [62,63], also required finite-spin Hamiltonians. |

Sierpinsky fractal spin-liquid (SFSL) code | A fractal type-I fracton CSS code defined on a cubic lattice [1; Eq. (D22)]. The code admits an excitation-moving operator shaped like a Sierpinski triangle [1; Fig. 2]. |

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. |

String-net code | Also called a Turaev-Viro or Levin-Wen model code. Code whose codewords realize a 2D topological order rendered by a Turaev-Viro topological field theory. The corresponding anyon theory is defined by a 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. |

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. |

Symmetry-protected self-correcting quantum code | Also called a symmetry-protected self-correcting memory. An restricted notion of thermal stability against symmetric perturbations, i.e., perturbations that commute with a set of operators forming a group \(G\) called the symmetry group. |

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. |

Three-dimensional color code | Three-dimensional version of the color code. |

Three-fermion (3F) model code | A 3D lattice stabilizer code whose low-energy excitations realize the three-fermion anyon theory [64–66] and that can be used as a resource state for fault-tolerant MBQC [67]. |

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*{(8)}\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*{(9)}\end{align} where \(x\in\mathbb{Z}\), are not normalizable. |

Topological code | A code whose codewords form the ground-state or low-energy subspace of a 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. |

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). |

Tree cluster-state code | Code obtained from a cluster state on a tree graph that has been proposed in the context of quanutm repeater and MBQC architectures. |

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. |

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 topological phase. |

Twisted quantum double (TQD) code | Code whose codewords realize a 2D topological order rendered by a Chern-Simons topological field theory. The corresponding anyon theory is defined by a finite group \(G\) and a Type-III group cocycle. |

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-component cat code | Code whose codespace is spanned by two coherent states \(\left|\pm\alpha\right\rangle\) for nonzero complex \(\alpha\). An orthonormal basis for the codespace consists of the bosonic cat states \begin{align} |\overline{\pm}\rangle=\frac{\left|\alpha\right\rangle \pm\left|-\alpha\right\rangle }{\sqrt{2\left(1\pm e^{-2|\alpha|^{2}}\right)}} \tag*{(10)}\end{align} for any complex \(\alpha\). |

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). |

Two-mode binomial code | Two-mode constant-energy CLY code whose coefficients are square-roots of binomial coefficients. |

Type-II fractal spin-liquid code | A type-II fracton prime-qudit CSS code defined on a cubic lattice [1; Eqs. (D9-D10)]. |

Very small logical qubit (VSLQ) code | The two logical codewords are \(|\pm\rangle \propto (|0\rangle\pm|2\rangle)(|0\rangle\pm|2\rangle)\), where the total Hilbert space is the tensor product of two transmon qudits (whose ground states \(|0\rangle\) and second excited states \(|2\rangle\) are used in the codewords). Since the code is intended to protect against losses, the qutrits can equivalently be thought of as oscillator Fock-state subspaces. |

Wasilewski-Banaszek code | Three-oscillator Fock-state code encoding a single logical qubit using codewords \begin{align} \begin{split} |\overline{0}\rangle &= \frac{1}{\sqrt{3}}(|003\rangle+|030\rangle+|300\rangle)\\ |\overline{1}\rangle &= |111\rangle \end{split}. \tag*{(11)}\end{align} |

X-cube model code | A foliated type-I fracton code supporting a subextensive number of logical qubits. Variants include the membrane-coupled [68], twice-foliated [69], and several generalized [70] X-cube models. |

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 [37] system after a choice of grounding [38]. |

\((5,1,2)\)-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*{(12)}\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. |

\([[15, 7, 3]]\) quantum Hamming code | Self-dual quantum Hamming 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. |

\([[2^r, 2^r-r-2, 3]]\) Gottesman code | A family of non-CSS stabilizer codes of distance \(3\) that saturate the asymptotic quantum Hamming bound. |

\([[2^r-1, 1, 3]]\) quantum Reed-Muller code | Member of CSS code family constructed with a first-order punctured RM\((1,m=r)\) \([2^r-1,r+1,2^{r-1}-1]\) code and its even subcode for \(r \geq 3\). Each code transversally implements a diagonal gate at the \((r-1)\)st level of the Clifford hierarchy [71]. |

\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming 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\((r-2,r)\). |

\([[2^r-1, 2^r-2r-1, 3]]_p\) quantum Hamming code | A family of CSS codes extending quantum Hamming codes to prime qudits of dimension \(p\) by expressing the qubit code stabilizers in local-dimension-invariant (LDI) form [72]. |

\([[2^{2r-1}-1,1,2^r-1]]\) 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 [73]. |

\([[3k + 8, k, 2]]\) triorthogonal code | Member of the \([[3k + 8, k, 2]]\) family (for even \(k\)) of triorthogonal and quantum divisible codes that admit a transversal \(T\) gate and are relevant for magic-state distillation. |

\([[4,2,2]]\) CSS code | Four-qubit CSS stabilizer code is the smallest qubit stabilizer code to detect a single-qubit error. |

\([[49,1,5]]\) triorthogonal code | Triorthogonal and quantum divisible code which is the smallest distance-five stabilizer code to admit a transversal \(T\) gate [74,75]. Its \(X\)-type stabilizers form a triply-even linear binary code in the symplectic representation. |

\([[5,1,3]]_q\) 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 \(\mathbb{Z}_q\) [76]; see also [77; 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 [14; 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 that can be obtained from a modified CSS construction using a first-order \([8,4,4]\) RM code and a \([8,7,2]\) even-weight code [78]. The modification introduces signs between the codewords. |

\([[8,3,2]]\) 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 \(\mathbb{Z}_q\). |

\([[9,1,5]]_3\) quantum Glynn code | Nine-qutrit pure Hermitian code that is the smallest qutrit stabilizer code to correct two-qutrit errors. |

\([[k+4,k,2]]\) H code | Family of \([[k+4,k,2]]\) CSS codes (for even \(k\)) with transversal Hadamard gates that are 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 \(Z_1Z_2Z_5Z_6...Z_{k+4}\).' |

\(k\)-orthogonal code | Stabilizer code whose \(X\)-type generators form a \(k\)-orthogonal matrix (defined below) in the symplectic representation. |

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