Here is a list of quantum Hamiltonian-based codes.
| Code | Description |
|---|---|
| 2D bosonization code | A mapping between a 2D lattice of qubits and a 2D lattice quadratic Hamiltonian of Majorana modes. This family also includes a super-compact fermionic encoding with a qubit-to-fermion ratio of \(1.25\) [1; Table I]. |
| 2D color code | Color code defined on a two-dimensional planar graph. Each face hosts two stabilizer generators, a Pauli-\(X\) and a Pauli-\(Z\) string acting on all the qubits of the face. |
| 2D hyperbolic surface code | Hyperbolic surface codes based on a tessellation of a closed 2D manifold with a hyperbolic geometry (i.e., non-Euclidean geometry, e.g., saddle surfaces when defined on a 2D plane). |
| 2D lattice stabilizer code | Lattice stabilizer code in two spatial dimensions. |
| 3D Kitaev honeycomb code | 3D subsystem stabilizer code whose Hamiltonian is a 3D generalization of the Kitaev honeycomb model. One of the phases realized by the 3D Kitaev honeycomb Hamiltonian is that of the 3D fermionic surface code [2]. |
| 3D bosonization code | A mapping that maps a 3D lattice quadratic Hamiltonian of Majorana modes into a lattice of qubits which realize a \(\mathbb{Z}_2\) gauge theory with a particular Gauss law. |
| 3D color code | Color code defined on a four-valent four-colorable tiling of 3D space. Logical dimension is determined by the genus of the underlying surface (for closed surfaces) and types of boundaries (for open surfaces). |
| 3D fermionic surface code | A non-CSS 3D Kitaev surface code that realizes \(\mathbb{Z}_2\) gauge theory with an emergent fermion, i.e., the fermionic-charge bosonic-loop (FcBl) phase [3]. The model can be defined on a cubic lattice in several ways [4; Eq. (D45-46)]. Realizations on other lattices also exist [5], and the phase of this code also exists in the 3D Kitaev honeycomb model [2]. |
| 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 via a local constant-depth Clifford circuit [4]. |
| 3D surface code | A generalization of the Kitaev surface code defined on a 3D lattice. |
| Abelian LP code | An LP code for Abelian group \(G\). The case of \(G\) being a cyclic group is a GB code (a.k.a. a quasi-cyclic LP code) [6; Sec. III.E]. A particular family with \(G=\mathbb{Z}_{\ell}\) yields codes with constant rate and nearly constant distance. |
| 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 [7; Sec. IV.A]. Abelian TQDs realize all modular gapped Abelian topological orders [7]. Many Abelian TQD code Hamiltonians were originally formulated as commuting-projector models [8]. |
| Abelian quantum-double stabilizer code | Modular-qudit stabilizer code whose codewords realize 2D modular gapped Abelian topological order with trivial cocycle. The corresponding anyon theory is defined by an Abelian group. All such codes can be realized by a stack of modular-qudit surface codes because all Abelian groups are Kronecker products of cyclic groups. |
| Abelian topological code | Code whose codewords realize topological order associated with an Abelian anyon theory. In 2D, this is equivalent to a unitary braided fusion category which is also an Abelian group under fusion [9]. Unless otherwise noted, the phases discussed are bosonic. |
| Analog cluster-state code | A code based on a continuous-variable (CV), or analog, cluster state. Such a state can be used to perform MBQC of logical modes, which substitutes the temporal dimension necessary for decoding a conventional code with a spatial dimension. The exact analog cluster state is non-normalizable, so approximate constructions have to be considered. |
| Analog repetition code | An \([[n,1]]_{\mathbb{R}}\) analog stabilizer version of the quantum repetition code, encoding the position states of one mode into an odd number of \(n\) modes. |
| Analog stabilizer code | An 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. Any analog stabilizer state can be thought of as a pure Gaussian state that has been infinitely squeezed on all modes [10]. |
| Analog surface code | An analog CSS version of the Kitaev surface code realizing a phase of 2D \(\mathbb{R}\) gauge theory. |
| 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. |
| Auxiliary qubit mapping (AQM) code | A concatenation of the JW transformation code with a qubit stabilizer code. |
| BB5 code | A BB code defined by weight-five measurements designed for long chains of trapped ions. Examples include \([[30,4,5]]\) and \([[48,4,7]]\) codes, |
| 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 hypercube color codes (color codes defined on balls constructed from hyperoctahedra) and 3D ball color codes (color codes defined on duals of certain Archimedean solids). |
| Ball-Verstraete-Cirac (BVC) code | A 2D fermion-into-qubit encoding that builds upon the JW transformation encoding by eliminating the weight-\(O(n)\) \(X\)-type string at the expense introducing additional qubits. See [1; Sec. IV.B] for details. |
| Bicycle code | A CSS code whose stabilizer generator matrix blocks are \(H_{X}=H_{Z}=(A|A^T)\), where \(A\) is a circulant matrix. The fact that \(A\) commutes with its transpose ensures that the CSS condition is satisfied. Bicycle codes are the first QLDPC codes. |
| Bivariate bicycle (BB) 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 CSS code | Bosonic stabilizer code admitting a set of stabilizer generators that are either position or momentum displacements. |
| Bosonic 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. Stabilizer groups are any locally compact Abelian subgroups of \(\mathbb{R}^n\), can themselves contain discrete or continuous subgroups, and can admit logical qudit and/or oscillator logical subspaces. |
| Bosonization code | A mapping that maps a \(D\)-dimensional lattice quadratic Hamiltonian of Majorana modes into a lattice of qubits. The resulting qubit code can realize various topological phases, depending on the initial Majorana-mode Hamiltonian and its symmetries. |
| Branching MERA code | Qubit stabilizer code whose encoding circuit corresponds to a branching MERA tensor network [11]. |
| Bravyi-Kitaev superfast (BKSF) code | An single error-detecting fermion-into-qubit encoding defined on 2D qubit lattice whose stabilizers are associated with loops in the lattice. The code can be generalized to a single error-correcting code (i.e., with distance three) on graphs of degree \(\geq 6\) [12]. |
| Bravyi-Kitaev transformation (BKT) code | A fermion-into-qubit encoding that maps Majorana operators into Pauli strings of weight \(\lceil \log (n+1) \rceil\). The code can be reformulated in terms of Fenwick trees [13], and the Pauli-string weight can be further optimized to yield the segmented Bravyi-Kitaev (SBK) transformation code [14] (see also Ref. [15]). |
| Brown-Fawzi Clifford-circuit code | An \([[n,k]]\) stabilizer code whose encoder is a random Clifford circuit of depth of order \(O(\log^3 n)\). |
| CSS-T code | A CSS code for which a physical transversal \(T\) gate is either the identity (up to a global phase) or a logical gate. CSS-T codes are constructed from a pair of linear binary codes via the CSS construction, with the pair satisfying certain conditions [16]. |
| 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 [4; 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. |
| Chen-Hsin invertible-order code | A geometrically local commuting-projector code that realizes beyond-group-cohomology invertible topological phases in arbitrary dimensions. Instances of the code in 4D realize the 3D \(\mathbb{Z}_2\) gauge theory with fermionic charge and either bosonic (FcBl) or fermionic (FcFl) loop excitations at their boundaries [3,17]; see Ref. [18] for a different lattice-model formulation of the FcBl boundary code. |
| Chiral semion Walker-Wang model code | A 3D lattice modular-qudit stabilizer code with qudit dimension \(q=4\) whose low-energy excitations on boundaries realize the chiral semion topological order. The model admits 2D chiral semion topological order at one of its surfaces [19,20]. The corresponding phase can also be realized via a non-stabilizer Hamiltonian [21]. |
| Chiral semion subsystem code | Modular-qudit subsystem stabilizer code with qudit dimension \(q=4\) that is characterized by the chiral semion topological phase. Admits a set of geometrically local stabilizer generators on a torus. |
| 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. |
| Circuit-to-Hamiltonian approximate code | Approximate qubit block code that forms the ground-state space of a frustration-free Hamiltonian with non-commuting terms. Its distance and logical-qubit number are both of order \(\Omega(n/\log^5 n)\) [22; Thm. 3.1]. The code is an approximate non-stabilizer QLWC code since the Hamiltonian consists of non-commuting weight-ten non-Pauli projectors, with each qubit acted on by order \(O(\text{polylog}(n)\) projectors. |
| 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 [23] against erasure and depolarizing noise when compared to other notable CSS codes, such as the asymptotically good quantum Tanner codes. These codes have been generalized to the intersecting subset code family [24]. |
| 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 | A code based on a cluster state and often used in measurement-based quantum computation (MBQC) [25,26] (a.k.a. one-way quantum processing), which substitutes the temporal dimension necessary for decoding a conventional code with a spatial dimension. This is done by encoding the computation into the features of the cluster state''s graph. |
| Coherent-parity-check (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 [27,28]. CPC codes can be obtained from numerical search [29]. |
| Color code | Member of a family of qubit CSS codes defined on particular \(D\)-dimensional graphs. |
| Commuting-projector Hamiltonian 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. |
| Compactified \(\mathbb{R}\) gauge theory code | An integer-homology bosonic CSS code realizing 2D \(U(1)\) gauge theory on bosonic modes. The code can be obtained from the analog surface code by condensing certain anyons [30]. This results in a pinning of each mode to the space of periodic functions, which make up a physical rotor, and can be thought of as compactification of the 2D \(\mathbb{R}\) gauge theory phase realized by the analog surface code. |
| Concatenated GKP code | A concatenated code whose outer code is a GKP code. In other words, a bosonic code that can be thought of as a concatenation of an arbitrary inner code and another bosonic outer code. Most examples encode physical qubits of an inner stabilizer code into the square-lattice GKP code. |
| Concatenated Steane code | A member of the family of \([[7^m,1,3^m]]\) CSS codes, each of which is a recursive level-\(m\) concatenation of the Steane code. This family is one of the first to admit a concatenated threshold [31–35]. |
| Concatenated qubit code | A concatenated code whose outer code is a qubit code. In other words, a qubit code that can be thought of as a concatenation of an arbitrary inner code and another qubit outer code. An inner \(C = ((n_1,K,d_1))\) and outer \(C^\prime = ((n_2,2,d_2))\) qubit code yield an \(((n_1 n_2, K, d \geq d_1d_2))\) concatenated qubit code. |
| Conformal-field theory (CFT) code | Approximate code whose codewords lie in the low-energy subspace of a conformal field theory, e.g., the quantum Ising model at its critical point [36,37]. Its encoding is argued to perform source coding (i.e., compression) as well as channel coding (i.e., error correction) [36]. |
| 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 (and, similarly, for fermion) 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 \(D_4\). |
| Cubic honeycomb color code | 3D color code defined on a four-colorable bitruncated cubic honeycomb uniform tiling. |
| Cubic theory code | A geometrically local commuting-projector code defined on triangulations of lattices in arbitrary spatial dimensions. Its code Hamiltonian terms include Pauli-\(Z\) operators and products of Pauli-\(X\) operators and \(CZ\) gates. The Hamiltonian realizes higher-form \(\mathbb{Z}_2^3\) gauge theories whose excitations obey non-Abelian Ising-like fusion rules. |
| Derby-Klassen (DK) code | A fermion-into-qubit code defined on regular tilings with maximum degree 4 whose stabilizers are associated with loops in the tiling. The code outperforms several other encodings in terms of encoding rate [38; Table I]. It has been extended for models with several modes per site [39]. |
| Dihedral \(G=D_m\) quantum-double code | Quantum-double code whose codewords realize \(G=D_m\) topological order associated with a \(2m\)-element dihedral group \(D_m\). Includes the simplest non-Abelian order \(D_3 = S_3\) associated with the permutation group of three objects. The code can be realized as the ground-state subspace of the quantum double model, defined for \(D_m\)-valued qudits [40]. An alternative qubit-based formulation realizes the gauged \(G=\mathbb{Z}_3^2\) twisted quantum double phase [41], which is the same topological order as the \(G=D_4\) quantum double [42,43]. |
| Dijkgraaf-Witten gauge theory code | A code whose codewords realize \(D\)-dimensional lattice Dijkgraaf-Witten gauge theory [44,45] for a finite group \(G\) and a \(D+1\)-cocycle \(\omega\) in the cohomology class \(H^{D+1}(G,U(1))\). When the cocycle is non-trivial, the gauge theory is called a twisted gauge theory. For trivial cocycles in 3D, the model can be called a quantum triple model, in allusion to being a 3D version of the quantum double model. There exist lattice-model formulations in arbitrary spatial dimension [46] as well as explicitly in 3D [47,48]. |
| Dinur-Hsieh-Lin-Vidick (DHLV) code | A family of asymptotically good QLDPC codes which are related to expander LP codes in that the roles of the check operators and physical qubits are exchanged. |
| 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 (\(t=2\)). |
| 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 [49], later generalized [50; Thm. 4.2], can yield QLDPC codes [49; Thm. 1]. |
| Double-semion stabilizer code | A 2D lattice modular-qudit stabilizer code with qudit dimension \(q=4\) that is characterized by the 2D double semion topological phase. The code can be obtained from the \(\mathbb{Z}_4\) surface code by condensing the anyon \(e^2 m^2\) [51]. Originally formulated as the ground-state space of a Hamiltonian with non-commuting terms [52], which can be extended to other spatial dimensions [53], and later as a commuting-projector code [8,54]. |
| Dual-rail quantum code | Two-mode bosonic 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 \(|01\rangle\). This encoding is often realized in temporal or spatial modes, corresponding to a time-bin or frequency-bin encoding. Two different types of photon polarization can also be used. |
| Eigenstate thermalization hypothesis (ETH) code | 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 or frustration-free systems such as Motzkin chains and 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 [55]. 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. |
| Fermion-into-qubit code | Qubit stabilizer code encoding a logical fermionic Hilbert space into a physical space of \(n\) qubits. Such codes are primarily intended for simulating fermionic systems on quantum computers, and some of them have error-detecting, correcting, and transmuting properties. |
| Fiber-bundle code | A 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 of order \(\Omega(n^{3/5}\text{polylog}(n))\) and rate of order \(\Omega(n^{-2/5}\text{polylog}(n))\) is obtained. |
| Fibonacci fractal spin-liquid code | A fractal type-I fracton CSS code defined on a cubic lattice [4; 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. |
| Finite-geometry (FG) QLDPC code | CSS code constructed from linear binary codes whose parity-check or generator matrices are incidence matrices of points, hyperplanes, or other structures in finite geometries. These codes can be interpreted as quantum versions of FG-LDPC codes, but some of them [56,57] are not strictly QLDPC. |
| 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 RS (FQRS) code | CSS code on \(q^m\)-dimensional Galois-qudits that is constructed from folded RS (FRS) codes (i.e., an RS code whose coordinates have been grouped together) via the Galois-qudit CSS construction. This code is used to construct Singleton-bound approaching approximate quantum codes. |
| Four-qubit single-deletion code | Four-qubit PI code that is the smallest qubit code to correct one deletion error. |
| Four-rotor code | \([[4,2,2]]_{\mathbb Z}\) CSS rotor code that is an extension of the four-qubit code to the integer alphabet, i.e., the angular momentum states of a planar rotor. |
| Fractal surface code | Kitaev surface code on a fractal geometry, which is obtained by removing qubits from the surface code on a cubic lattice. A related construction, the fractal product code, is a hypergraph product of two classical codes defined on a Sierpinski carpet graph [58]. The underlying classical codes form classical self-correcting memories [59–61]. |
| Fracton stabilizer code | A 3D translationally invariant modular-qudit stabilizer code whose codewords make up the ground-state space of a Hamiltonian in a fracton phase. Unlike topological phases, whose excitations can move in any direction, fracton phases are characterized by excitations whose movement is restricted. |
| Freedman-Meyer-Luo code | Hyperbolic surface code constructed using cellulation of a Riemannian Manifold \(M\) exhibiting systolic freedom [62]. 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 [63]. 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 | A cyclic prime-qudit stabilizer code whose length \(n\) divides \(p^t + 1\) for some positive integer \(t\). |
| Frustration-free Hamiltonian code | Hamiltonian-based code whose Hamiltonian is frustration free, i.e., whose ground states minimize the energy of each term. |
| 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 CV-cluster-state code | Cluster-state code can consists of a generalized analog cluster state that is initialized in GKP (resource) states for some of its physical modes. Alternatively, it can be thought of as an oscillator-into-oscillator GKP code whose encoding consists of initializing \(k\) modes in momentum states (or, in the normalizable case, squeezed vacua), \(n-k\) modes in (normalizable) GKP states, and applying a Gaussian circuit consisting of two-body \(e^{i V_{jk} \hat{x}_j \hat{x}_k }\) for some angles \(V_{jk}\). Provides a way to perform fault-tolerant MBQC, with the required number \(n-k\) of GKP-encoded physical modes determined by the particular protocol [64–67]. |
| GKP-surface code | A concatenated code whose outer code is a GKP code and whose inner code is a toric surface code [68], rotated surface code [66,69–72], or XZZX surface code [73]. |
| Galois-qudit BCH code | True Galois-qudit stabilizer code constructed from BCH codes via either the Hermitian construction or the Galois-qudit CSS construction. Parameters can be improved by applying Steane enlargement [74], e.g., as in Ref. [75]. |
| 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 GRS codes via either the Hermitian construction [76–78] or the Galois-qudit CSS construction [79,80]. |
| Galois-qudit HGP code | A member of a family of Galois-qudit CSS codes whose stabilizer generator matrix is obtained from a hypergraph product of two classical linear \(q\)-ary codes. |
| Galois-qudit RS code | An \([[n,k,n-k+1]]_q\) (with \(q>n\)) Galois-qudit CSS code constructed using two RS codes over \(GF(q)\). |
| Galois-qudit color code | Extension of the color code to 2D lattices of Galois qudits. |
| Galois-qudit expander code | Galois-qudit CSS code constructed from a hypergraph product of expander codes. |
| Galois-qudit quantum RM code | True \(q\)-Galois-qudit stabilizer code constructed from generalized Reed-Muller (GRM) codes via the Hermitian construction, the Galois-qudit CSS construction, or directly from their parity-check matrices [81; Sec. 4.2]. |
| 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 surface code | Extension of the surface code to 2D lattices of Galois qudits. |
| Generalized 2D color code | Member of a family of non-Abelian 2D topological codes, defined by a finite group \( G \), that serves as a generalization of the color code (for which \(G=\mathbb{Z}_2\times\mathbb{Z}_2\)). |
| Generalized Shor code | Qubit CSS 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 [82] from a pair of equivalent index-two quasi-cyclic linear codes. Equivalently, the codes can constructed via the lifted-product construction for \(G\) being a cyclic group [6; Sec. III.E]. |
| 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 Tanner code | An extension of quantum Tanner codes to codes constructed from two commuting regular graphs with the same vertex set. This allows for code construction using finite sets and Schreier graphs, yielding a broader family of square 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 \(t\) [83; Def. V.1]. Such codes admit gates at the \(\nu\)th level of the Clifford hierarchy. Such codes can also be level-lifted [83; 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 \(A\). Any GKP code can be generated from a Gram matrix in standard form via a Gaussian unitary transformation [84; Corr. 1]. |
| Graph quantum code | A stabilizer code on tensor products of \(G\)-valued qudits for Abelian \(G\) whose encoding isometry is defined using a graph [85; Eqs. (4-5)]. An analytical form of the codewords exists in terms of the adjacency matrix of the graph and bicharacters of the Abelian group [85]; see [86; Eq. (1)]. A graph quantum code for \(G=\mathbb{Z}_2\) contains a cluster state as one of its codewords and reduces to a cluster state when its logical dimension is one [87]. |
| 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 [88]. 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 (CC) | A 3D lattice stabilizer code on a length-\(L\) cubic lattice with one or two qubits per site. Admits two types of stabilizer generators with support on each cube of the lattice. In the non-CSS case, these two are related by spatial inversion. For CSS codes, we require that the product of all corner operators is the identity. We lastly require that there are no non-trival string operators, meaning that single-site operators are a phase, and any period one logical operator \(l \in \mathsf{S}^{\perp}\) is just a phase. |
| Hamiltonian-based code | Code whose codespace corresponds to a set of energy eigenstates of a quantum-mechanical Hamiltonian i.e., a Hermitian operator whose expectation value measures the energy of its underlying physical system. The codespace is typically a set of low-energy eigenstates or ground states, but can include subspaces of arbitrarily high energy. Hamiltonians whose eigenstates are the canonical basis elements are called classical; otherwise, a Hamiltonian is called quantum. |
| Hayden-Nezami-Salton-Sanders bosonic code | An \([[n,1]]_{\mathbb{R}}\) analog CSS code defined using homological structures associated with an \(n-1\) simplex. Relevant to the study of spacetime replication of quantum information [89]. |
| Hemicubic code | Homological code constructed out of cubes in high dimensions. The hemicubic code family has asymptotically diminishing soundness that scales as order \(\Omega(1/\log n)\), locality of stabilizer generators scaling as order \(O(\log n)\), and distance of order \(\Theta(\sqrt{n})\). |
| Heptagon holographic code | Holographic tensor-network code constructed out of a network of encoding isometries of the Steane code. Depending on how the isometry tensors are contracted, there is a zero-rate and a finite-rate code family. |
| 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 \(GF(4)\) representation. |
| Hexagonal GKP code | Single-mode GKP qudit-into-oscillator code based on the triangular 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 [90,91]. |
| 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 [92,93]. 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. |
| Holographic tensor-network code | Quantum Lego code whose encoding isometry forms a holographic tensor network, i.e., a tensor network associated with a tiling of 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 number of layers emanating form the central point of the tiling is the radius of the code. |
| Homological code | CSS-type extension 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 tensor product of two chain complexes of length one or greater (see Qubit CSS-to-homology correspondence). |
| 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. |
| Honeycomb (6.6.6) color code | 2D color code defined on a patch of the 6.6.6 (honeycomb) tiling. |
| Hopf-algebra quantum-double code | Code whose codewords realize 2D gapped topological order defined on qudits valued in a Hopf algebra \(H\). The code Hamiltonian is an generalization [94,95] of the quantum double model from group algebras to Hopf algebras, as anticipated by Kitaev [40]. Boundaries of these models have been examined [96,97]. |
| 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 [4]. |
| 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 [4; Eqs. (D42-D43)]. |
| Hurwitz surface code | Homological code constructed on triangulations of Hurwitz surfaces. |
| Hyperbolic color code | An extension of the color code construction to hyperbolic manifolds. As opposed to there being only three types of uniform three-valent and three-colorable lattice tilings in the 2D Euclidean plane, there is an infinite number of admissible hyperbolic tilings in the 2D hyperbolic plane [98]. Certain double covers of hyperbolic tilings also yield admissible tilings [99]. Other admissible hyperbolic tilings can be obtained via a fattening procedure [100]; see also a construction based on the more general quantum pin codes [101]. |
| 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 binary codes. Codes from hypergraph products in higher dimension are called higher-dimensional HGP codes [102]. |
| Hyperinvariant tensor-network (HTN) code | Holographic tensor-network error-detecting code constructed out of a hyperinvariant tensor network [103], i.e., a MERA-like network admitting a hyperbolic geometry. The network is defined using two layers A and B, with constituent tensors satisfying isometry conditions (a.k.a. multitensor constraints). |
| Hypersphere product code | Homological code based on products of hyperspheres. The hypersphere product code family has asymptotically diminishing soundness that scales as order \(O(1/\log (n)^2)\), locality of stabilizer generators scaling as order \(O(\log n/ \log\log n)\), and distance of order \(\Theta(\sqrt{n})\). |
| Integer-homology bosonic CSS code | A bosonic stabilizer code whose physical modes have been restricted, via a single GKP stabilizer, from the space of functions on the real line to the space of periodic functions. This restriction effectively realizes a rotor on each physical mode, allowing one to construct homological rotor codes out of displacement stabilizer groups. The stabilizer group is continuous, but contains discrete components in the form of the single-mode GKP stabilizers. 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. |
| Jordan-Wigner transformation code | A mapping between qubit Pauli strings and Majorana operators that can be thought of as a trivial \([[n,n,1]]\) code. The mapping is best described as converting a chain of \(n\) qubits into a chain of \(2n\) Majorana modes (i.e., \(n\) fermionic modes). It maps Majorana operators into Pauli strings of weight \(O(n)\). |
| Jump code | A CE code designed to detect and correct AD errors. An \(((n,K))\) jump code is denoted as \(((n,K,t))_w\) (which conflicts with modular-qudit notation), where \(t\) is the maximum number of qubits that can be corrected after each one has undergone a jump error \(|0\rangle\langle 1|\), and where each codeword is a uniform superposition of qubit basis states with Hamming weight \(w\). |
| Kitaev chain code | An \([[n,1,1]]_{f}\) Majorana stabilizer code forming the ground-state of the Kitaev Majorana chain (a.k.a. Kitaev Majorana wire) in its fermionic topological phase, which is unitarily equivalent to the 1D quantum Ising model in the symmetry-breaking phase via the Jordan-Wigner transformation. The code is usually defined using the algebra of two anti-commuting Majorana operators called Majorana zero modes (MZMs) or Majorana edge modes (MEMs). It can be thought of as the Majorana stabilizer analogue of the quantum repetition code, and it encodes a logical fermion because its logical Majorana operator has odd weight [104]. |
| 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 [105] system [106]. |
| Kitaev honeycomb code | Code whose logical subspace is labeled by different fusion outcomes of Ising anyons present in the Ising-anyon topological phase of the Kitaev honeycomb model [107]. Ising anyons also exist in other phases, such as the fractional quantum Hall phase [108]. |
| Kitaev surface code | A family of Abelian topological CSS stabilizer codes whose generators are few-body \(X\)-type and \(Z\)-type Pauli strings associated to the stars and plaquettes, respectively, of a cellulation of a two-dimensional surface (with a qubit located at each edge of the cellulation). Codewords correspond to ground states of the surface code Hamiltonian, and error operators create or annihilate pairs of anyonic charges or vortices. |
| Klein-bottle surface code | A family of Kitaev surface codes on the non-orientable Klein bottle. |
| La-cross code | Code constructed using a hypergraph product of two copies of a cyclic LDPC code. The construction uses cyclic LDPC codes with generating polynomials \(1+x+x^k\) for some \(k\). Using a length-\(n\) seed code yields an \([[2n^2,2k^2]]\) family for periodic boundary conditions and an \([[(n-k)^2+n^2,k^2]]\) family for open boundary conditions. |
| Lattice stabilizer code | A geometrically local stabilizer code with sites 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-type 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 | Galois-qudit 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. |
| Log-depth geometrically local Clifford-circuit code | A random \([[n,k]]\) stabilizer code whose encoder is a random Clifford circuit of depth of order \(O(\log n)\) on a 1D Euclidean geometry. |
| Long-range enhanced surface code (LRESC) | Code constructed using a hypergraph product of two copies of a concatenated LDPC-repetition seed code. This family interpolates between surface codes and hypergraph codes since the hypergraph product of two repetition codes yields the planar surface code. The construction uses small \([3,2,2]\) and \([6,2,4]\) LDPC codes concatenated with \([4,1,4]\) and \([2,1,2]\) repetition codes, respectively. An example using a \([5,2,3]\) code is also presented. |
| Loop toric code | A generalization of the Kitaev surface code defined on a 4D lattice. The code is called a \((2,2)\) toric code because it admits 2D membrane \(Z\)-type and \(X\)-type logical operators. Both types of operators create 1D (i.e., loop) excitations at their edges. The code serves as a self-correcting quantum memory [109,110]. |
| 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 [111] yields a \(c^3\)-LTC [112]. Using two-sided expanders, which are only conjectured to exist, yields an asymptotically good QLDPC code family [113]. |
| Low-density parity-check (LDPC) code | A binary linear code with a sparse parity-check matrix. Alternatively, a member of an infinite family of \([n,k,d]\) codes for which the number of nonzero entries in each row and column of the parity-check matrix are both bounded above by a constant as \(n\to\infty\). |
| Magnon code | An \(n\)-spin approximate code whose codespace of \(k=\Omega(\log n)\) qubits is efficiently described in terms of particular matrix product states or Bethe ansatz tensor networks. Magnon codewords are low-energy excited states of the frustration-free Heisenberg-XXX model Hamiltonian [114]. |
| Majorana box qubit | A Majorana stabilizer code which forms a fixed-parity subspace of the ground-state subspace of one or more Kitaev Majorana chain Hamiltonians. The \([[n,1,2]]_{f}\) Majorana box qubit forms the even-fermion-parity ground-state subspace of two parallel Kitaev Majorana chains in their fermionic topological phase. Its \([[2,1,2]]_{f}\) version is called the tetron Majorana code. An \([[3,2,2]]_{f}\) extension using three Kitaev chains and housing two logical qubits of the same parity is called the hexon Majorana code. Similarly, octon, decon, and dodecon are codes defined by the positive-parity subspace of \(4\), \(5\), and \(6\) fermionic modes, respectively [115]. |
| Majorana checkerboard code | A Majorana analogue of the X-cube model defined on a cubic lattice. The code admits weight-eight Majorana stabilizer generators on the eight vertices of each cube of a checkerboard sublattice. |
| Majorana color code | Majorana analogue of the color code defined on a 2D tricolorable lattice and constructed out of Majorana box qubit codes or other small Majorana stabilizer codes placed on patches of the lattice. |
| Majorana loop stabilizer code (MLSC) | An single error-correcting fermion-into-qubit encoding defined on a 2D qubit lattice whose stabilizers are associated with loops in 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. |
| 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. |
| 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 [116]. 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 cluster-state code | A code based on a modular-qudit cluster state. |
| Modular-qudit color code | Extension of the color code to lattices of modular qudits. Codes are defined analogous to qubit color codes on suitable lattices of any spatial dimension, but a directionality is required in order to make the modular-qudit stabilizer commute. This can be done by puncturing a hyperspherical lattice [117] or constructing a star-bipartition; see [118; Sec. III]. Logical dimension is determined by the genus of the underlying surface (for closed surfaces), types of boundaries (for open surfaces), and/or any twist defects present. |
| Modular-qudit shift-resistant code | Monolithic code encoding a qubit into a single modular qudit and protecting against either \(Z\)-type or \(X\)-type modular-qudit Pauli shifts. |
| 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 | Extension of the surface code to prime-dimensional [40,119] and more general modular qudits [120]. 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 [121]. Randomized constructions yield constant-rate GKP code families whose largest decodable displacement length scales as \(O(\sqrt{n})\) with high probability. |
| 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. This code is useful for encoding and performing operations on qudits in multiple qubits [122–126]. |
| Oscillator-into-oscillator GKP 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. |
| Ouyang-Chao constant-excitation PI code | A constant-excitation PI Fock-state code whose construction is based on integer partitions. |
| 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 | Holographic code constructed out of a network of hexagonal perfect tensors that tesselates hyperbolic space. The code serves as a minimal model for several aspects of the AdS/CFT holographic duality [127] and potentially a dF/CFT duality [128]. It has been generalized to higher dimensions [129] and to include gauge-like degrees of freedom on the links of the tensor network [130,131]. All boundary global symmetries must be dual to bulk gauge symmetries, and vice versa [132]. |
| Prime-qudit RM code | Modular-qudit stabilizer code constructed from generalized Reed-Muller (GRM) codes or their duals via the modular-qudit CSS construction. An odd-prime-qudit CSS code family constructed from first-order punctured GRM codes transversally implements a diagonal gate at any level of the qudit Clifford hierarchy [133,134]. |
| Prime-qudit RS code | Prime-qudit CSS code constructed using two RS codes. |
| Prime-qudit triorthogonal code | An \(m \times n\) matrix over \(GF(p)=\mathbb{Z}_p\) is triorthogonal if its rows \(r_1, \ldots, r_m\) satisfy \(|r_i \cdot r_j| = 0\) and \(|r_i \cdot r_j \cdot r_k| = 0\) modulo \(p\), where addition and multiplication are done on \(GF(p)\). The triorthogonal prime-qudit CSS code associated with the matrix is constructed by mapping nonzero entries in self-orhogonal rows to \(X\) operators, and \(Z\) operators for each row in the orthogonal complement [135,136]. |
| 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. |
| Purity-testing stabilizer code | A qubit stabilizer code that is constructed from a normal rational curve and that is relevant to testing the purity of an entangled Bell state stabilized by two parties [137]. |
| Quantum AG code | A Galois-qudit CSS code constructed using two linear AG codes. |
| Quantum Gabidulin code | A Galois-qudit stabilizer code over \(n\) Galois qudits of dimension \(q = 2^n \) that is useful in protecting against faults in qubit Clifford circuits acting on stacked quantum memories. This code can be treated as a code on an \(n\times n\) qubit stacked memory by decomposing each Galois qudit into a Kronecker product of \(n\) qubits; see [138,140–142][139; Sec. 5.3]. |
| Quantum Goppa code | A Galois-qudit CSS code constructed using two Goppa codes. |
| Quantum LDPC (QLDPC) code | Member of a family of \([[n,k,d]]\) 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 \(w\) as \(n\to\infty\). The code can be denoted by \([[n,k,d,w]]\) or, for \(q\)-dimensional Galois-qudit QLDPC codes, \([[n,k,d]]_{(w,q)}\). Sometimes, the two parameters are explicitly stated: each site of an an \((l,w)\)-regular QLDPC code is acted on by \(\leq l\) generators of weight \(\leq w\). For finite size codes, the notation QLDPC codes can correct many stochastic errors far beyond the distance, which may not scale as favorably. Together with more accurate, faster, and easier-to-parallelize measurements than those of general stabilizer codes, this property makes QLDPC codes interesting in practice. |
| Quantum Reed-Muller code | A CSS code formed from a classical Reed-Muller (RM) code or its punctured/shortened versions. Such codes often admit transversal logical gates in the Clifford hierarchy. |
| Quantum Tamo-Barg (QTB) code | A member of a family of Galois-qudit CSS codes whose underlying classical codes consist of Tamo-Barg codes together with specific low-weight codewords. Folded versions of QTB codes, or FQTB codes, defined on qudits whose dimension depends on \(n\) yield explicit examples of QLRCs of arbitrary locality \(r\) [143; Thm. 2]. |
| Quantum Tanner code | Member of a family of QLDPC codes based on two compatible classical Tanner codes defined on a two-dimensional Cayley complex, a complex constructed from Cayley graphs of groups. For certain choices of codes and complex, the resulting codes have asymptotically good parameters. This construction has been generalized to Schreier graphs [144]. |
| Quantum check-product code | CSS code constructed from an extension of check product (between two classical codes) to a product between a classical and a quantum code. |
| 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 data-syndrome (QDS) code | Stabilizer code designed to correct both data qubit errors and syndrome measurement errors simultaneously due to extra redundancy in its stabilizer generators. |
| 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 stabilizers have weight divisible by four (eight), i.e., if they form a doubly even (triply even) linear binary code. |
| Quantum duadic code | True Galois-qudit stabilizer code constructed from \(q\)-ary duadic codes via the Hermitian construction or the Galois-qudit CSS construction. |
| Quantum expander code | CSS code constructed from a hypergraph product of bipartite expander graphs [55] 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 irregular convolutional code (QIRCC) | Quantum convolutional code whose stabilizer group consists of different constant-size subsets. |
| 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-weight check (QLWC) code | Member of a family of \([[n,k,d]]\) stabilizer codes for which the number of sites participating in each stabilizer generator is bounded by a constant as \(n\to\infty\). |
| 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. |
| Quantum pin code | Member of a family of CSS codes that encompasses both quantum Reed-Muller and color codes and that is defined using intersections of pinned sets. |
| Quantum quadratic-residue (QR) code | Galois-qudit \([[n,1]]_q\) pure self-dual CSS code constructed from a dual-containing QR code via the Galois-qudit CSS construction. For \(q\) not divisible by \(n\), its distance satisfies \(d^2-d+1 \geq n\) when \(n \equiv 3\) modulo 4 [145; Thm. 40] and \(d \geq \sqrt{n}\) when \(n\equiv 1\) modulo 4 [145; Thm. 41]. |
| Quantum rainbow code | A CSS code whose qubits are associated with vertices of a simplex graph with \(m+1\) colors. |
| Quantum repetition code | Encodes \(1\) qubit into \(n\) qubits according to \(|0\rangle\to|\phi_0\rangle^{\otimes n}\) and \(|1\rangle\to|\phi_1\rangle^{\otimes n}\). The code is called 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 [146] and stabilizer constructions [147]. 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 synchronizable code | A qubit stabilizer code designed to protect against synchronization errors (a.k.a. misalignment), which are errors that misalign the code block in a larger block by one or more locations. |
| Quantum tensor-product code | CSS code constructed from a tensor code. In some cases, only one of the classical codes forming the tensor code needs to be self-orthogonal. |
| Quantum turbo code | A quantum version of the turbo code, obtained from an interleaved serial quantum concatenation [148; Def. 30] of quantum convolutional codes. |
| 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). |
| Quasi-cyclic QLDPC (QC-QLDPC) code | A QLDPC code such that cyclic shifts of the subsystems by \(\ell\geq 1\) leave the codespace invariant. Such codes have circulant stabilizer generator matrices [149,150]. |
| Quasi-hyperbolic color code | An extension of the color code construction to quasi-hyperbolic manifolds, e.g., a product of a 2D hyperbolic surface and a circle. |
| 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. |
| Qubit 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 3D surface code | A generalization of the 3D surface code to modular qudits. |
| Qudit X-cube model code | Generalization of the X-cube model code to modular qudits. |
| Qudit cubic code | Generalization of the Haah cubic code to modular qudits. |
| Random stabilizer code | An \(n\)-qubit, modular-qudit, or Galois-qudit stabilizer code whose construction is non-deterministic. Since stabilizer encoders are Clifford circuits, such codes can be thought of as arising from random Clifford circuits. |
| Raussendorf-Bravyi-Harrington (RBH) cluster-state code | A three-dimensional cluster-state code defined on the bcc lattice (i.e., a cubic lattice with qubits on edges and faces). |
| Rotated surface code | Variant of the surface code defined on a square lattice that has been rotated 45 degrees such that qubits are on vertices, and both \(X\)- and \(Z\)-type check operators occupy plaquettes in an alternating checkerboard pattern. |
| Rotor GKP code | GKP code protecting against small angular position and momentum shifts of a planar rotor. |
| Rotor cluster-state code | Rotor analogue of the qubit and analog cluster-state codes. The exact rotor cluster state is non-normalizable, so approximate constructions have to be considered. Defined from an real-valued weighed adjacency matrix of a graph [10]. |
| 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 [151,152] or other low-rank SYK models [153,154]. |
| Self-correcting quantum code | A block quantum code that forms the ground-state subspace of an \(n\)-body geometrically local Hamiltonian whose logical information is recoverable for arbitrary long times in the \(n\to\infty\) limit after interaction with a sufficiently cold thermal environment. Typically, one also requires a decoder whose decoding time scales polynomially with \(n\) and a finite energy density. The original criteria for a self-correcting quantum memory, informally known as the Caltech rules [58,155], also required finite-spin Hamiltonians. |
| Sierpinsky fractal spin-liquid (SFSL) code | A fractal type-I fracton CSS code defined on a cubic lattice [4; Eq. (D22)]. The code admits an excitation-moving operator shaped like a Sierpinski triangle [4; 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 [156,157]. Sampling a description of this code can be done with an efficient randomized algorithm with \(2^{-\Omega(n)}\) failure probability. |
| Six-qubit-tensor holographic code | Holographic tensor-network code constructed out of a network of encoding isometries of the \([[6,1,3]]\) six-qubit stabilizer code. The structure of the isometry is similar to that of the heptagon holographic code since both isometries are rank-six tensors, but the isometry in this case is neither a perfect tensor nor a planar-perfect tensor. |
| Skew-cyclic CSS code | A member of a family of Galois-qudit CSS codes constructed from skew-cyclic classical codes over rings [158; Thm. 5.8]. See related study [159] that uses cyclic codes over rings. |
| Small-distance qubit stabilizer code | A qubit stabilizer code that either detects or corrects errors on at most two subsystems, i.e., have distance \(\leq 5\). |
| 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. |
| Sphere packing | An analog code whose points can be thought of as forming centers of (real or complex) spheres that pack (real or complex) space. Sphere packings provide ways of encoding digital or analog information into the frequency, amplitude, and phase of one or more analog waveforms for transmission through, e.g., an optical fiber or free space. This is due to Kotelnikov's [160] and Shannon's [161] fundamental observation that a discretized electromagnetic signal of finite bandwidth and average power \(P\) can be represented as a vector in \(\mathbb{R}^n\) with norm \(nP\). Questions of capacity of electromagnetic communication channels then translate to packing problems in \(\mathbb{R}^n\) [162]. |
| Square homological product code | Homological product code whose underlying quantum-code boundary operators are square matrices (see Qubit CSS-to-homology correspondence). |
| Square-lattice GKP code | Single-mode GKP qudit-into-oscillator CSS 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. |
| Square-lattice cluster-state code | A code based on the cluster state on a square lattice that was used in the the first proposal for MBQC [25,26]. |
| Square-octagon (4.8.8) color code | 2D color code defined on a patch of the 4.8.8 (square-octagon) tiling, which itself is obtained by applying a fattening procedure to the square lattice [100]. |
| Stabilizer code | A code whose logical subspace is the joint eigenspace (usually with eigenvalue \(+1\)) of a set of commuting unitary Pauli-type 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. |
| Stellated color code | A non-CSS color code on a lattice patch with a single twist defect at the center of the patch. |
| String-net code | Code whose codewords realize a 2D topological order rendered by a Turaev-Viro topological field theory. The corresponding anyon theory is defined by a (multiplicity-free) unitary fusion category \( \mathcal{C} \). The code is defined on a cell decomposition dual to a triangulation of a two-dimensional surface, with a qudit of dimension \( |\mathcal{C}| \) located at each edge of the decomposition. These models realize local topological order (LTO) [163]. |
| Surface-code-fragment (SCF) holographic code | Holographic tensor-network code constructed out of a network of encoding isometries of the \([[5,1,2]]\) rotated surface code. The structure of the isometry is similar to that of the HaPPY code since both isometries are rank-six tensors. In the case of the SCF holographic code, the isometry is only a planar-perfect tensor (as opposed to a perfect tensor). |
| Symmetry-protected self-correcting quantum code | A code which admits a 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. |
| Symmetry-protected topological (SPT) code | A code whose codewords form the ground-state or low-energy subspace of a code Hamiltonian realizing symmetry-protected topological (SPT) order. |
| Tensor-product HDX code | Code constructed in a similar way as the HDX code, but utilizing tensor products of multiple Ramanujan complexes and then applying distance balancing. These improve the asymptotic code distance over the HDX codes 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. |
| Ternary-tree fermion-into-qubit code | A fermion-into-qubit encoding defined on ternary trees that maps Majorana operators into Pauli strings of weight \(\lceil \log_3 (2n+1) \rceil\). |
| Tetrahedral color code | 3D color code defined on select tetrahedra of a 3D tiling. Qubits are placed on the vertices, edges, triangles, and in the center of each tetrahedron. The code has both string-like and sheet-like logical operators [164]. |
| Tetron code | A \([[2,1,2]]_{f}\) Majorana box qubit, encoding two fixed-fermion states into the four-dimensional ground-state space of two Kitaev chains, each of length two. The code encodes a logical qubit into four Majorana modes (i.e., two physical fermions), allowing it to be concatenated with various qubit codes such as surface codes and color codes. Embedding each physical qubit into two fermions via the tetron code is useful for exactly solving the Kitaev honeycomb model Hamiltonian [107] and other qubit Hamiltonians on certain graphs [165,166]. |
| Three-fermion (3F) Walker-Wang model code | A 3D lattice stabilizer code whose low-energy excitations on boundaries realize a 3D time-reversal SPT order [167] and that can be used as a resource state for fault-tolerant MBQC [168]. The anyons at the boundary of the lattice are described by the 3F anyon theory. |
| Three-fermion (3F) subsystem code | 2D subsystem stabilizer code whose low-energy excitations realize the three-fermion anyon theory [169–171]. One version uses two qubits at each site [51], while other manifestations utilize a single qubit per site and only weight-two (two-body) interactions [170,172]. All are expected to be equivalent to each other via a local constant-depth Clifford circuit. |
| 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. |
| Topological code | A code whose codewords form the ground-state or low-energy subspace of a (typically geometrically local) code Hamiltonian realizing a topological phase. A topological phase may be bosonic or fermionic, i.e., constructed out of underlying subsystems whose operators commute or anti-commute with each other, respectively. Unless otherwise noted, the phases discussed are bosonic. |
| Toric code | Version of the Kitaev surface code on the two-dimensional torus, encoding two logical qubits. Being the first manifestation of the surface code, "toric code" is often an alternative name for the general construction. Twisted toric code [173; Fig. 8] refers to the construction on a torus with twisted (a.k.a. shifted) boundary conditions. |
| 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 (e.g., a star graph [174,175]) that has been proposed in the context of quantum repeater and MBQC architectures. |
| Triangular surface code | A surface code with weight-four stabilizer generators defined on a honeycomb tiling that are examples of twist-defect surface code with a single twist defect at the center of the patch. The codes use about \(25\%\) fewer physical per logical qubit for a given distance compared to the surface code. |
| Triorthogonal code | Qubit CSS code whose \(X\)-type logicals and stabilizer generators form a triorthogonal matrix (defined below) in the symplectic representation. |
| True Galois-qudit stabilizer code | A \([[n,k,d]]_q\) stabilizer code whose stabilizer's Galois 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. |
| Truncated trihexagonal (4.6.12) color code | 2D color code defined on a patch of the 4.6.12 (truncated trihexagonal or square-hexagon-dodecagon) tiling. |
| Twist-defect color code | A non-CSS extension of the 2D color code whose non-CSS stabilizer generators are associated with twist defects of the associated lattice. |
| Twist-defect surface code | A non-CSS extension of the 2D surface-code construction whose non-CSS stabilizer generators are associated with twist defects of the associated lattice. A related construction [176] doubles the number of qubits in the lattice via symplectic doubling. |
| Twisted XZZX toric code | A cyclic code that can be thought of as the XZZX toric code with shifted (a.k.a twisted) boundary conditions. Admits a set of stabilizer generators that are cyclic shifts of a particular weight-four \(XZZX\) Pauli string. For example, a seven-qubit \([[7,1,3]]\) variant has stabilizers generated by cyclic shifts of \(XZIZXII\) [177]. Codes encode either one or two logical qubits, depending on qubit geometry, and perform well against biased noise [178]. |
| Twisted quantum double (TQD) code | Code whose codewords realize a 2D topological order rendered by a Chern-Simons topological field theory. The corresponding anyon theory is defined by a finite group \(G\) and a Type-III group cocycle \(\omega\), but can also be described in a category theoretic way [179]. |
| Two-block CSS code | Galois-qudit CSS code whose stabilizer generator matrices \(H_X=(A_1,B_1)\) and \(H_Z=(B^T_2,-A^T_2)\), are constructed from four matrices satisfying \(A_1 B_2 - B_1 A_2 = 0\). In the case the two pairs are equal, we have \(H_X=(A,B)\) and \(H_Z=(B^T,-A^T)\), constructed from a pair of square commuting matrices \(A\) and \(B\). |
| 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. A construction of such codes in terms of Kronecker products of circulant matrices was introduced in [180]. |
| Two-component cat code | Code whose codespace is spanned by two coherent states \(\left|\pm\alpha\right\rangle\) for nonzero complex \(\alpha\). |
| Two-foliated fracton code | A type-I fracton code obtained by gauging [181–183,183] a topological phase with subsystem symmetry. |
| Two-gauge theory code | A code whose codewords realize lattice two-gauge theory [184–192] for a finite two-group (a.k.a. a crossed module) in arbitrary spatial dimension. There exist several lattice-model formulations in arbitrary spatial dimension [193,194] as well as explicitly in 3D [195–198] and 4D [198], with the 3D case realizing the Yetter model [199–202]. |
| 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 [4; Eqs. (D9-D10)]. |
| Union-Jack color code | 2D color code defined on a patch of the Tetrakis square tiling (a.k.a. the Union Jack lattice). |
| Valence-bond-solid (VBS) code | An \(n\)-qubit approximate \(q\)-dimensional spin code family whose codespace is described in terms of \(SU(q)\) valence-bond-solid (VBS) [203] matrix product states with various boundary conditions. The codes become exact when either \(n\) or \(q\) go to infinity. The original work on these codes studied the \(q=2\) case [204]. |
| 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. |
| Walker-Wang model code | A 3D topological code defined by a unitary braided fusion category \( \mathcal{C} \) (also known as a unitary premodular category). The code is defined on a cubic lattice that is resolved to be trivalent, with a qudit of dimension \( |\mathcal{C}| \) located at each edge. The codespace is the ground-state subspace of the Walker-Wang model Hamiltonian [205] and realizes the Crane-Yetter model [206–208]. A single-state version of the code provides a resource state for MBQC [168]. |
| Wasilewski-Banaszek code | Three-oscillator constant-excitation Fock-state code encoding a single logical qubit. |
| X-cube model code | A foliated type-I fracton code supporting a subextensive number of logical qubits. Variants include the membrane-coupled [209], twice-foliated [210], and several generalized [211] X-cube models. |
| XY surface code | Non-CSS derivative of the surface code whose generators are \(XXXX\) and \(YYYY\), obtained by mapping \(Z \to Y\) in the surface code. |
| XYZ color code | Non-CSS variant of the 6.6.6 color code whose generators are \(XZXZXZ\) and \(ZYZYZY\) Pauli strings associated to each hexagonal in the hexagonal (6.6.6) tiling. A further variation called the domain wall color code admits generators of the form \(XXXZZZ\) and \(ZZZXXX\) [212]. |
| XYZ product code | A non-CSS QLDPC code constructed from a particular hypergraph product of three linear binary 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. When the underlying classical codes are 1D (e.g., repetition codes), the XYZ product yields a 3D code. Higher dimensional versions have been constructed [213]. |
| XYZ\(^2\) hexagonal stabilizer code | An instance of the matching code based on the Kitaev honeycomb model. It is described on a honeycomb tiling with \(XYZXYZ\) stabilizers on each hexagonal plaquette. Each vertical pair of qubits has an \(XX\), \(YY\), or \(ZZ\) link stabilizer depending on the orientation of the plaquette stabilizers. |
| XZZX surface code | Non-CSS variant of the rotated surface code whose generators are \(XZZX\) Pauli strings associated, clock-wise, to the vertices of each face of a two-dimensional lattice (with a qubit located at each vertex of the tessellation). |
| 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 does not impose additional connectivity constraints and can as much as triple the number of logical qubits per physical qubit when compared to the original surface code. Concatenation with 1D (2D) QMDPC yields codes with twice (four times) the distance. The stabilizer generators of the outer QMDPC code are referred to as yokes in this context. |
| 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 [105] system after a choice of grounding [106]. |
| \(((8,1,3))\) Plenio-Vedral-Knight CE code | An eight-qubit single error-correcting code that is the first CE code. Each logical state is a superposition of computational basis states with four excitations. |
| \(((n,1,2))\) Bravyi-Lee-Li-Yoshida PI code | PI distance-two code on \(n\geq4\) qubits whose degree of entanglement vanishes asymptotically with \(n\) [214; Appx. D] (cf. [215]). |
| \((1,3)\) 4D toric code | A generalization of the Kitaev surface code defined on a 4D lattice. The code is called a \((1,3)\) toric code because it admits 1D \(Z\)-type and 3D \(X\)-type logical operators. |
| \((5,1,2)\)-convolutional code | Family of quantum convolutional codes that are 1D lattice generalizations of the five-qubit perfect code, with the former''s lattice-translation symmetry being the extension of the latter''s cyclic permutation symmetry. |
| \(D\)-dimensional twisted toric code | Extension of the Kitaev toric code to higher-dimensional lattices with regular or shifted (a.k.a. twisted) boundary conditions. Such boundary conditions yields quibit geometries that are tori \(\mathbb{R}^D/\Lambda\), where \(\Lambda\) is an arbitrary \(D\)-dimensional lattice. Picking a hypercubic lattice yields the ordinary \(D\)-dimensional toric code. It is conjectured that appropriate twisted boundary conditions yield multi-dimensional toric code families with linear distance and logarithmic-weight stabilizer generators [216]. |
| \(D_4\) hyper-diamond GKP code | Two-mode GKP qudit-into-oscillator code based on the \(D_4\) hyper-diamond lattice. |
| \(G\)-enriched Walker-Wang model code | A 3D topological code defined by a unitary \(G\)-crossed braided fusion category \( \mathcal{C} \) [217,218], where \(G\) is a finite group. The model realizes TQFTs that include two-gauge theories, those behind Walker-Wang models, as well as the Kashaev TQFT [219,220]. It has been generalized to include domain walls [221]. |
| \(U(1)_{2n} \times U(1)_{-2m}\) Chern-Simons GKP code | A non-CSS multimode GKP code defined on a 2D mode lattice that encodes a qudit logical space and whose excitations are characterized by the \(U(1)_{2n} \times U(1)_{-2m}\) Chern-Simons theory. The code can be obtained from the analog surface code by condensing certain anyons [30]. |
| \([[10,1,2]]\) CSS code | Smallest stabilizer code to implement a logical \(T\) gate via application of physical \(T\), \(T^{\dagger}\), and \(CCZ\) gates. |
| \([[10,1,4]]_{G}\) tenfold code | A \([[10,1,4]]_{G}\) group code for finite Abelian \(G\). The code is defined using a graph that is closely related to the \([[5,1,3]]\) code. |
| \([[11,1,5]]\) quantum dodecacode | Eleven-qubit pure stabilizer code that is the smallest qubit stabilizer code to correct two-qubit errors. |
| \([[11,1,5]]_3\) qutrit Golay code | An \([[11,1,5]]_3\) constructed from the ternary Golay code via the CSS construction. The code's stabilizer generator matrix blocks \(H_{X}\) and \(H_{Z}\) are both the generator matrix of the ternary Golay code. |
| \([[12,2,2]]\) CSS code | CSS code that admits a logical \(CS\) gate via application of physical \(T\) and \(T^{\dagger}\) gates. |
| \([[12,2,4]]\) carbon code | Twelve-qubit CSS code resulting from the concatenation of the \([[4,2,2]]\) code and the \(C_6\) code that has simplified fault-tolerant state preparation circuits for the \(00\) and \(++\) states. |
| \([[13,1,5]]\) cyclic code | Thirteen-qubit twisted surface code for which there is a set of stabilizer generators consisting of cyclic permutations of the \(XZZX\)-type Pauli string \(XIZZIXIIIIIII\). The code can be thought of as a small twisted XZZX code [222; Exam. 11 and Fig. 3]. |
| \([[14,3,3]]\) Rhombic dodecahedron surface code | A \([[14,3,3]]\) twist-defect surface code whose qubits lie on the vertices of a rhombic dodecahedron. Its non-CSS nature is due to twist defects [223] stemming from the geometry of the polytope. |
| \([[144,12,12]]\) gross code | A BB code which requires less physical and ancilla qubits (for syndrome extraction) than the surface code with the same number of logical qubits and distance. The gross code is equivalent to 8 copies of the surface code via a constant-depth Clifford circuit, and is an element of a larger family of 2D stabilizer codes [224]. The name stems from the fact that a gross is a dozen dozen. |
| \([[15, 7, 3]]\) quantum Hamming code | Self-dual quantum Hamming code that admits permutation-based CZ logical gates. The code is constructed using the CSS construction from the \([15,11,3]\) Hamming code and its \([15,4,8]\) dual code. |
| \([[15,1,3]]\) quantum Reed-Muller code | \([[15,1,3]]\) CSS code that is most easily thought of as a tetrahedral 3D color code. |
| \([[16,4,3]]\) dodecahedral code | A \([[16,4,3]]\) qubit stabilizer code defined whose encoder-respecting form is the graph of vertices of a dodecahedron [225]. |
| \([[16,6,4]]\) Tesseract color code | A (self-dual CSS) 4D color code defined on a tesseract, with stabilizer generators of both types supported on each cube. A \([[16,4,2,4]]\) tesseract subsystem code can be obtained from this code by using two logical qubits as gauge qubits [226]. |
| \([[17,1,5]]\) 4.8.8 color code | Seventeen-qubit doubly even 2D color code that admits a transversal implementation of the logical Clifford group. |
| \([[23, 1, 7]]\) Quantum Golay code | A \([[23, 1, 7]]\) self-dual CSS code with eleven stabilizer generators of each type, and with each generator being weight eight. |
| \([[2^D,D,2]]\) hypercube quantum code | Member of a family of codes defined by placing qubits on a \(D\)-dimensional hypercube, \(Z\)-stabilizers on all two-dimensional faces, and an \(X\)-stabilizer on all vertices. These codes realize gates at the \((D-1)\)-st level of the Clifford hierarchy. It can be generalized to a \([[4^D,D,4]]\) error-correcting family [227]. Various other concatenations give families with increasing distance (see cousins). |
| \([[2^r+r, 2^r-r-2, 3]]\) Ring CPC code | A family of \([[2^r+r, 2^r-r-2, 3]]\) CPC codes for \(r \geq 3\) whose matrices are based on the shortened version of the \([2^r-1,2^r-r-1,3]\) Hamming code. See [228; Thm. 4] for their stabilizer generator matrix. |
| \([[2^r, 2^r-r-2, 3]]\) Gottesman code | A family of non-CSS stabilizer codes of distance \(3\) that saturate the asymptotic quantum Hamming bound. |
| \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code | Member of a family of self-dual CCS codes constructed from \([2^r-1,2^r-r-1,3]=C_X=C_Z\) Hamming codes and their duals the simplex codes. The code's stabilizer generator matrix blocks \(H_{X}\) and \(H_{Z}\) are both the generator matrix for a simplex code. The weight of each stabilizer generator is \(2^{r-1}\). |
| \([[2^r-1, 2^r-2r-1, 3]]_p\) 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 [229]. |
| \([[2^r-1,1,3]]\) simplex code | Member of color-code code family constructed with a punctured first-order 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 [230,231]. Each code is a color code defined on a simplex in \(r-1\) dimensions [117,232], where qubits are placed on the vertices, edges, and faces as well as on the simplex itself. |
| \([[2^{2r-1}-1,1,2^r-1]]\) 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 \(r \geq 2\). |
| \([[2^{m-1},2^{m-1}-m-1,4]]_{f}\) Hamming Majorana code | A member of the \([[2^{m-1},2^{m-1}-m-1,4]]_{f}\) family of Majorana stabilizer codes for \(m \geq 3\) that is closely related to the Hamming code. The logical subspace of the \([[8,3,4]]_{f}\) Hamming Majorana code is a Cartan subspace of the \(E_8\) Lie algebra [233]. |
| \([[2m,2m-2,2]]\) error-detecting code | Self-complementary CSS code for \(m\geq 2\) with generators \(\{XX\cdots X, ZZ\cdots Z\} \) acting on all \(2m\) physical qubits. The code is constructed via the CSS construction from an SPC code and a repetition code [234; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [235]. |
| \([[3,1,2]]_3\) 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. |
| \([[30,8,3]]\) Bring code | A \([[30,8,3]]\) hyperbolic surface code on a quotient of the \(\{5,5\}\) hyperbolic tiling called Bring's curve. Its qubits and stabilizer generators lie on the vertices of the small stellated dodecahedron. Admits a set of weight-five stabilizer generators. |
| \([[3k + 8, k, 2]]\) 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. |
| \([[4,2,2]]\) Four-qubit code | Four-qubit CSS stabilizer code is the smallest qubit stabilizer code to detect a single-qubit error. |
| \([[4,2,2]]_{G}\) four group-qudit code | \([[4,2,2]]_{G}\) group quantum code that is an extension of the four-qubit code to group-valued qudits. |
| \([[49,1,5]]\) triorthogonal code | Triorthogonal and quantum divisible code which is the smallest distance-five stabilizer code to admit a transversal \(T\) gate [236–238]. Its \(X\)-type stabilizers form a triply even linear binary code in the symplectic representation. |
| \([[5,1,2]]\) rotated surface code | Rotated surface code on one rung of a ladder, with one qubit on the rung, and four qubits surrounding it. |
| \([[5,1,3]]_q\) 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 \(GF(p)\). |
| \([[5,1,3]]_{\mathbb{R}}\) Braunstein five-mode code | An analog stabilizer version of the five-qubit perfect code, encoding one mode into five and correcting arbitrary errors on any one mode. |
| \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code | Modular-qudit stabilizer code that generalizes the five-qubit perfect code using properties of the multiplicative group \(\mathbb{Z}_q\) [239]; see also [240; Thm. 13]. It has four stabilizer generators consisting of \(X Z Z^\dagger X^\dagger I\) and its cyclic permutations. |
| \([[54,6,5]]\) five-covered icosahedral code | A \([[54,6,5]]\) qubit stabilizer code defined whose encoder-respecting form is the graph of a five-cover of the icosahedron [225]. |
| \([[6,1,3]]\) Six-qubit stabilizer code | One of two six-qubit distance-three codes that are unique up to equivalence [235], with the other code a trivial extension of the five-qubit code [241]. The code admits fault-tolerant syndrome extraction using only one ancilla per stabilizer generator measurement. |
| \([[6,1,3]]_{f}\) Vijay-Fu Majorana code | A Majorana stabilizer code encoding a logical fermion into six physical fermions. This code is the shortest code correcting single fermion-parity flips [242]. |
| \([[6,2,2]]\) \(C_6\) code | Error-detecting self-dual CSS code used in concatenation schemes for fault-tolerant quantum computation. A set of stabilizer generators is \(IIXXXX\) and \(XXIIXX\), together with the same two \(Z\)-type generators. |
| \([[6,2,3]]_{q}\) code | Six-qudit MDS error-detecting code defined for Galois-qudit dimension \(q=3\) [243], \(q=2^2\) [244], and \(q \geq 5\) [145,243]. This code cannot exist for qubits (\(q=2\)). |
| \([[6,4,2]]\) error-detecting code | Error-detecting six-qubit code with rate \(2/3\) whose codewords are cat/GHZ states. A set of stabilizer generators is \(XXXXXX\) and \(ZZZZZZ\). It is the unique code for its parameters, up to equivalence [235; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [245]. |
| \([[6k+2,3k,2]]\) Campbell-Howard code | Family of \([[6k+2,3k,2]]\) qubit stabilizer codes with quasi-transversal \(CZZ^{\otimes k}\) gates that are relevant to magic-state distillation. |
| \([[6r,2r,2]]\) Ganti-Onunkwo-Young code | A member of the family of \([[6r,2r,2]]\) CSS codes designed to suppress errors in adiabatic quantum computation. All but two of its stabilizer generators are weight-two (two-body), and the remaining two are weight-\(4r\). |
| \([[7,1,3]]\) Steane code | A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [241]. The code is constructed using the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors. |
| \([[7,1,3]]\) bare code | A \([[7,1,3]]\) code that admits fault-tolerant syndrome extraction using only one ancilla per stabilizer generator measurement. |
| \([[7,1,3]]\) twist-defect surface code | A \([[7,1,3]]\) code (different from the Steane code) that is a small example of a twist-defect surface code. |
| \([[7,3,3]]_{q}\) code | Seven-qudit MDS error-detecting code defined for Galois-qudit dimension \(q=3\) [243] and \(q \geq 7\) [145,243]. This code cannot exist for qubits (\(q=2\)). |
| \([[8, 3, 3]]\) Eight-qubit Gottesman code | Eight-qubit non-degenerate code that can be obtained from a modified CSS construction using the \([8,4,4]\) extended Hamming code and a \([8,7,2]\) even-weight code [246]. The modification introduces signs between the codewords. |
| \([[8,2,2]]\) hyperbolic color code | An \([[8,2,2]]\) hyperbolic color code defined on the projective plane. |
| \([[8,3,2]]\) Smallest interesting color code | Smallest 3D color code whose physical qubits lie on vertices of a cube and which admits a (weakly) transversal \(CCZ\) gate. |
| \([[9,1,3]]\) Shor code | Nine-qubit CSS code that is the first quantum error-correcting code. |
| \([[9,1,3]]\) Surface-17 code | A \([[9,1,3]]\) rotated surface code named for the sum of its 9 data qubits and 8 syndrome qubits. It uses the smallest number of qubits to perform fault-tolerant error correction on a surface code with parallel syndrome extraction. |
| \([[9,1,3]]_{\mathbb{R}}\) Lloyd-Slotine code | An analog stabilizer version of Shor's nine-qubit code, encoding one mode into nine and correcting arbitrary errors on any one mode. |
| \([[9,1,3]]_{\mathbb{Z}_q}\) 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. |
| \([[9m-k,k,2]]_3\) triorthogonal code | Member of the \([[9m-k,k,2]]_3\) family of triorthogonal qutrit codes (for \(k\leq 3m-2\)) that admit a transversal diagonal gate in the third level of the qudit Clifford hierarchy and that are relevant for magic-state distillation. |
| \([[k+4,k,2]]\) H code | Family of \([[k+4,k,2]]\) self-dual 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 \(Z_1Z_2Z_5Z_6...Z_{k+4}\).' |
| \([[m 2^m / (m+1), 2^m / (m+1), 2\lfloor m/4 \rfloor + 1]]\) Khesin-Lu-Shor code | A family of \([[m 2^m / (m+1), 2^m / (m+1),]]\) qubit CSS codes derived from the Hamming code. Their encoder-respecting form is the graph of a hypercube in \(m = 2^r - 1\) dimensions, and input nodes in the graph are codewords of the \([2^r-1,2^r-r-1,3]\) Hamming code [225]. |
| \(\mathbb{Z}_3\times\mathbb{Z}_9\)-fusion subsystem code | Modular-qudit 2D subsystem stabilizer code whose low-energy excitations realize a non-modular anyon theory with \(\mathbb{Z}_3\times\mathbb{Z}_9\) fusion rules. Encodes two qutrits when put on a torus. |
| \(\mathbb{Z}_q^{(1)}\) subsystem code | Modular-qudit subsystem code, based on the Kitaev honeycomb model [107] and its generalization [247], that is characterized by the \(\mathbb{Z}_q^{(1)}\) anyon theory [248], which is modular for odd prime \(q\) and non-modular otherwise. Encodes a single \(q\)-dimensional qudit when put on a torus for odd \(q\), and a \(q/2\)-dimensional qudit for even \(q\). This code can be constructed using geometrically local gauge generators, but does not admit geometrically local stabilizer generators. For \(q=2\), the code reduces to the subsystem code underlying the Kitaev honeycomb model code as well as the honeycomb Floquet code. |
| \(k\)-orthogonal code | Qubit stabilizer code whose \(X\)-type logicals and generators form a \(k\)-orthogonal matrix (defined below) in the symplectic representation. In other words, the overlap between any \(k\) \(X\)-type code-preserving Paulis (including the identity) is even. The original definition is for qubit CSS codes [118], but it can be extended to more general qubit stabilizer codes [249; Def. 1]. This entry is formulated for qubits, but an extension exists for modular qudits [118]. |
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