Here is a list of qubit QLDPC codes.
| Code | Description |
|---|---|
| 2D bosonization code | A mapping between a 2D lattice of qubits and a 2D lattice quadratic Hamiltonian of Majorana modes. The original 2D bosonization code [1] is a stabilizer code whose generators are products of plaquettes and stars of the surface code. This family also includes a super-compact fermionic encoding with a qubit-to-fermion ratio of \(1.25\) [2; 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). |
| 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 variant of the 3D Kitaev surface code that realizes \(\mathbb{Z}_2\) gauge theory with an emergent fermion, i.e., the fermionic-charge bosonic-loop (FcBl) phase [3]. The model can be defined on a cubic lattice in several ways [4; Eq. (D45-46)]. Realizations on other lattices also exist [5], and the phase of this code also exists in the 3D Kitaev honeycomb model [6]. |
| 3D surface code | A generalization of the Kitaev surface code defined on a 3D cubic lattice. Qubits are placed on edges, \(Z\)-type stabilizer generators are placed on square plaquettes oriented in all three directions, and \(X\)-type stabilizers are placed on the six edges neighboring every vertex [7]. |
| BB5 code | A BB code with weight-five stabilizer generators (contrasting with the weight-six checks of standard BB codes), designed and benchmarked for long chains of trapped ions [8]. |
| Ball code | A distance-two “morphed” color code defined on a \(D\)-dimensional colex [9; Appx. A]. This family includes hypercube codes (defined on balls constructed from hyperoctahedra) and 3D ball codes (defined on duals of certain Archimedean solids). |
| 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. Codes can be classified by the weight of their checks, e.g., by BB\(w\) where \(w\) is the check weight. |
| 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 \(\geq 6\) [10]. |
| Campbell double homological product code | A multi-dimensional HGP code derived from two applications of the hypergraph product to a classical code, resulting in a length-\(4\) chain complex. The construction method allows for the use of two different classical codes as inputs, with Ref. [11] assuming identical input codes for simplicity. |
| 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. A tetrahedral Ising model can be used to obtain the checkerboard model by gauging [12–21] its subsystem symmetry [14]. |
| Classical-product code | A QLDPC qubit 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 [22] 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 [23]. |
| 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. |
| Color code | Member of a family of qubit CSS codes defined on particular \(D\)-dimensional graphs. |
| Crystalline-circuit qubit code | Code dynamically generated by constant-depth 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. |
| 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 codes constructed using cubical chain complexes, which are \(t\)-order extensions of the complexes underlying expander codes (\(t=1\)) and expander lifted-product codes (\(t=2\)). |
| 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 \(\Omega(n^{-2/5}\text{polylog}(n))\). |
| Fibonacci fractal spin-liquid code | A fractal type-I fracton CSS code defined on a cubic lattice [4; Eq. (D23)]. |
| Finite-geometry (FG) qubit 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 [24,25] are not strictly QLDPC. |
| Four Color Cube (FCC) fracton model code | A fracton code obtained from four coupled X-cube models using p-membrane condensation. A modular-qudit generalization has been proposed [26]. |
| 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 [27]. The underlying classical codes form classical self-correcting memories [28–30]. |
| Freedman-Meyer-Luo code | Hyperbolic surface code constructed using cellulation of a Riemannian Manifold \(M\) exhibiting systolic freedom [31]. 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 [32]. 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. |
| Generalized homological-product qubit CSS code | A qubit CSS code whose properties are determined from an underlying chain complex via the qubit CSS-to-homology correspondence. This complex 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. |
| Golden code | Variant of the Guth-Lubotzky hyperbolic surface code that uses regular tessellations for 4-dimensional hyperbolic space. |
| Guth-Lubotzky code | Homological linear-rate code based on cellulations of certain 4D hyperbolic manifolds with particular homology and systolic properties. |
| 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-trivial 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. |
| 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 \(\Theta(\sqrt{n})\). |
| 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 maintaining 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 [33,34]. |
| 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 [35,36]. 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. |
| Higher-dimensional homological product code | A qubit CSS code formulated using a tensor product of two or more chain complexes, each of length one or greater. The number of chain complexes participating in the product is the dimension of the code. When all chain complexes are length-one, meaning that they correspond to classical codes, the code is called a higher-dimensional HGP code (a.k.a. multi-sector HGP code or iterative HGP code). |
| Homological code | A CSS 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). |
| Honeycomb (6.6.6) color code | 2D color code defined on a patch of the 6.6.6 (honeycomb) tiling. |
| 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 [37]. Certain double covers of hyperbolic tilings also yield admissible tilings [38]. Other admissible hyperbolic tilings can be obtained via a fattening procedure [39]; see also a construction based on the more general quantum pin codes [40]. |
| Hyperbolic surface code | An extension of the Kitaev surface code construction to hyperbolic manifolds. Given a cellulation of a hyperbolic manifold of arbitrary dimension, 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. |
| Hypersphere product code | Homological code based on products of hyperspheres. The hypersphere product code family has asymptotically diminishing soundness that scales as order \(O(1/\log (n)^2)\), locality of stabilizer generators scaling as order \(O(\log n/ \log\log n)\), and distance of order \(\Theta(\sqrt{n})\). |
| Kitaev chain code | An \([[n,1]]_{f}\) 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 [41]. |
| 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. |
| Layer code | Member of a family of qubit QLDPC 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 (or a more general qubit stabilizer 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. |
| 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. |
| 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 [42] yields a \(c^3\)-LTC [43]. Using two-sided expanders, which are only conjectured to exist, yields an asymptotically good QLDPC code family [44]. |
| 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 [45]. |
| 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 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. |
| Projective-plane surface code | A family of Kitaev surface codes on the non-orientable 2-dimensional compact manifold \(\mathbb{R}P^2\) (in contrast to a genus-\(g\) surface). Whereas genus-\(g\) surface codes require \(2g\) logical qubits, qubit codes on \(\mathbb{R}P^2\) are made from a single logical qubit. |
| Quantum 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. See Ref. [46] for explicit instances based on dihedral groups. This construction has been generalized to Schreier graphs [47]. |
| Quantum convolutional code | 1D translationally invariant qubit stabilizer code 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 expander code | CSS code constructed from a hypergraph product of bipartite expander graphs [48] 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 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 [49] and stabilizer constructions [50]. In either case, the stabilizer generator matrix is constructed by “spatially” coupling sub-matrix blocks in chain-like fashion (or, more generally, in grid-like fashion) to yield a band matrix. The sub-matrix blocks have to satisfy certain conditions amongst themselves so that the resulting band matrix is a stabilizer generator matrix. Matrices corresponding to translationally invariant chains are called time-variant, and otherwise are called time-invariant. |
| Quantum turbo code | A quantum version of the turbo code, obtained from an interleaved serial quantum concatenation [51; Def. 30] of quantum convolutional codes. |
| 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 QLDPC code | Member of a family of \([[n,k,d]]\) qubit 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 \([[n,k,d,w]]\). Sometimes, the two parameters are explicitly stated: each site of an an \((l,w)\)-regular qubit QLDPC code is acted on by \(\leq l\) generators of weight \(\leq w\). Qubit 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. |
| 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. |
| Sierpinski prism model 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]. |
| Spacetime circuit code | Qubit stabilizer code used to correct faults in Clifford circuits, i.e., circuits made up of Clifford gates and Pauli measurements. The code utilizes redundancy in the measurement outcomes of a circuit to correct circuit faults, which correspond to Pauli errors of the code. |
| Square homological product code | Homological product code whose underlying quantum-code boundary operators are square matrices (see Qubit CSS-to-homology correspondence). |
| Square-lattice cluster-state code | A code based on the cluster state on a square lattice that was used in the first proposal for MBQC [52,53]. |
| 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 [39]. |
| Stellated color code | A non-CSS color code on a lattice patch with a single twist defect at the center of the patch. |
| Tensor-product HDX code | A code constructed in a similar way as the HDX code, but utilizing iterated homological 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. |
| 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 [54]. |
| Tetron code | A \([[2,1,2]]_{f}\) Majorana box qubit, encoding two fixed-fermion states into the 4D 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 [55] and other qubit Hamiltonians on certain graphs [56,57]. It has been used throughout condensed matter physics under the name of the Majorana representation [58,59] or parton construction [60], allowing for a mean-field treatment of many models that are otherwise not amenable. Majorana stabilizer groups can be converted into ordinary qubit stabilizer groups via the parton mapping, while their corresponding states are converted via the Gutzwiller projection [61]. |
| 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 [62] and that can be used as a resource state for fault-tolerant MBQC [63]. The anyons at the boundary of the lattice are described by the 3F anyon theory. |
| Tillich-Zémor code | A family of \([[n^2 + m^2, (n - \text{rank}([C \mid M]))^2 + (m - \text{rank}([C \mid M]^\top))^2, d]]\) quantum LDPC codes constructed via the hypergraph product of two classical \((n, m, r)\)-structured LDPC seed codes |
| Toric code | Version of the Kitaev surface code on a square lattice with periodic boundary conditions, encoding two logical qubits. Being the first manifestation of the surface code, “toric code” is often an alternative name for the general construction. Twisted toric code [64; 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). |
| Triangular surface code | A surface code with weight-four stabilizer generators defined on a honeycomb tiling that are examples of twist-defect surface code with a single twist defect at the center of the patch. The codes use about \(25\%\) fewer physical per logical qubit for a given distance compared to the surface code. |
| Truncated trihexagonal (4.6.12) color code | 2D color code defined on a patch of the 4.6.12 (truncated trihexagonal or square-hexagon-dodecagon) tiling. |
| Twist-defect 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 [65] 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 \(XZIZXII\) [66]. Codes encode either one or two logical qubits, depending on qubit geometry, and perform well against biased noise [67]. See Ref. [65] for a table of some of these for small instances, where they are called genus-one genon codes. |
| Two-foliated fracton code | A type-I fracton code obtained by gauging [12–21] a topological phase with subsystem symmetry. |
| Union-Jack color code | 2D color code defined on a patch of the Tetrakis square tiling (a.k.a. the Union Jack lattice). |
| X-cube model code | A foliated type-I fracton code supporting a subextensive number of logical qubits. Variants include several generalized X-cube models [68]. A non-stabilizer commuting-projector code constructed by stacking layers of the double-semion string-net model, called the semionic X-cube model [69], is equivalent to the X-cube model [70] (see also Refs. [71,72]). |
| XY surface code | Non-CSS derivative of the surface code whose generators are \(XXXX\) and \(YYYY\), obtained by mapping \(Z \to Y\) in the surface code. |
| XYZ 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 \(ZZZXXX\) [73]. |
| 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 [74]. |
| XYZ\(^2\) hexagonal stabilizer code | An instance of the matching code based on the Kitaev honeycomb model. It is described on a honeycomb tiling with \(XYZXYZ\) stabilizers on each hexagonal plaquette. Each vertical pair of qubits has an \(XX\), \(YY\), or \(ZZ\) link stabilizer depending on the orientation of the plaquette stabilizers. |
| XZZX surface code | Non-CSS variant of the rotated surface code whose generators are \(XZZX\) Pauli strings associated, clock-wise, to the vertices of each face of a two-dimensional lattice (with a qubit located at each vertex of the tessellation). |
| \((1,3)\) 4D toric code | A generalization of the Kitaev surface code defined on a 4D lattice. The code is called a \((1,3)\) toric code because it admits 1D \(Z\)-type and 3D \(X\)-type logical operators. In the hypercubic lattice version, qubits are placed on edges, each \(Z\)-type stabilizer generator is supported on cubes on the boundary of a hypercube, and \(X\)-type stabilizers are placed on the edges neighboring every vertex [75]. |
| \((2,2)\) Loop toric code | A generalization of the Kitaev surface code defined on a 4D lattice. The code is called a \((2,2)\) toric code because it admits 2D membrane \(Z\)-type and \(X\)-type logical operators. Both types of operators create 1D (i.e., loop) excitations at their edges. The code serves as a self-correcting quantum memory [76,77]. |
| \((5,1,2)\)-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 sublinear distance scaling of \(n^{1-\epsilon}\) for any \(\epsilon>0\) and logarithmic-weight stabilizer generators [78,79]. At finite \(n\), twisting boundary conditions can reduce qubit overhead for a fixed distance [79]. |
| \([[108,8,10]]\) BB6 code | A bivariate bicycle (BB) code with parameters \([[108,8,10]]\) and weight-six stabilizer generators [80]. |
| \([[13,1,5]]\) 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 [81; 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 [82] 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 [83]. The name stems from the fact that a gross is a dozen dozen. |
| \([[15,1,3]]\) quantum RM code | A \([[15,1,3]]\) quantum Reed-Muller code that is most easily thought of as a tetrahedral 3D color code. |
| \([[16,6,4]]\) 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 [84]. |
| \([[17,1,5]]\) 4.8.8 color code | Seventeen-qubit doubly even 2D color code that admits a transversal implementation of the logical Clifford group. The smallest distance-five CSS code has \(n=17\) [85]. |
| \([[288,12,18]]\) double-gross code | A bivariate bicycle (BB) code with parameters \([[288,12,18]]\) and weight-six stabilizer generators [80]. |
| \([[2^D,D,2]]\) 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 [86]. Various other concatenations give families with increasing distance (see cousins). |
| \([[2^r-1,1,3]]\) 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 [87,88]. Each code is a color code defined on a simplex in \(r-1\) dimensions [89,90], where qubits are placed on the vertices, edges, and faces as well as on the simplex itself. |
| \([[2m,2m-2,2]]\) 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 [91; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [92]. |
| \([[30,8,3]]\) Bring code | A \([[30,8,3]]\) hyperbolic surface code on a quotient of the \(\{5,5\}\) hyperbolic tiling called Bring’s curve. Its qubits and stabilizer generators lie on the vertices of the small stellated dodecahedron. Admits a set of weight-five stabilizer generators. |
| \([[4,2,2]]\) Four-qubit code | Four-qubit CSS stabilizer code is the smallest qubit stabilizer code to detect a single-qubit error. |
| \([[5,1,2]]\) rotated surface code | Rotated surface code on one rung of a ladder, with one qubit on the rung, and four qubits surrounding it. |
| \([[5,1,3]]\) Five-qubit perfect code | Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error. |
| \([[6,2,2]]\) \(C_6\) code | Error-detecting 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. |
| \([[6,4,2]]\) 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 [92; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [93]. |
| \([[7,1,3]]\) Steane code | A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [94]. The code is constructed using the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors. |
| \([[7,1,3]]\) twist-defect surface code | A \([[7,1,3]]\) code (different from the Steane code) that is a small example of a twist-defect surface code. |
| \([[72,12,6]]\) BB6 code | A bivariate bicycle (BB) code with parameters \([[72,12,6]]\) and weight-six stabilizer generators [80]. |
| \([[8,2,2]]\) 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. |
| \([[8,3,2]]\) Surface code on a cube | An \([[8,3,2]]\) twist-defect surface code whose qubits lie on the vertices of a cube. It is obtained by three-coloring the faces of a cube and placing \(X\), \(Y\), and \(Z\) stabilizer generators on each pair of faces of the same color. Its non-CSS nature is due to twist defects [82] stemming from the geometry of the polytope. |
| \([[9,1,3]]\) 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. |
| \([[90,8,10]]\) BB6 code | A bivariate bicycle (BB) code with parameters \([[90,8,10]]\) and weight-six stabilizer generators [80]. |
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