Here is a list of generalized homological product and related codes.

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

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 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) [1; Sec. III.E]. A particular family with \(G=\mathbb{Z}_{\ell}\) yields codes with constant rate and nearly constant distance. |

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). |

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. |

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 [2; Eq. (D38)]. |

Color code | Member of a family of qubit CSS codes defined on particular \(D\)-dimensional graphs. |

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 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 [3], later generalized [4; Thm. 4.2], can yield QLDPC codes [3; Thm. 1]. |

Expander LP code | Family of \(G\)-lifted product codes constructed using two random classical Tanner codes defined on expander graphs [5]. For certain parameters, this construction yields the first asymptotically good QLDPC codes. Classical codes resulting from this construction are one of the first two families of \(c^3\)-LTCs. |

Fiber-bundle code | 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. |

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 [6]. The underlying classical codes form classical self-correcting memories [7–9]. |

Freedman-Meyer-Luo code | Hyperbolic surface code constructed using cellulation of a Riemannian Manifold \(M\) exhibiting systolic freedom [10]. 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 [11]. Improved codes are obtained by studying a weak family of Riemann metrics on closed 4-dimensional manifolds \(S^2\otimes S^2\) with the \(Z_2\)-homology. |

Galois-qudit 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 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 surface code | Extension of the surface code to 2D lattices of Galois qudits. |

Generalized bicycle (GB) code | A quasi-cyclic Galois-qudit CSS code constructed using a generalized version of the bicycle ansatz [12] 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 [1; 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. |

Golden code | Variant of the Guth-Lubotzky hyperbolic surface code that uses regular tessellations for 4-dimensional hyperbolic space. |

Guth-Lubotzky code | Hyperbolic surface code based on cellulations of certain four-dimensional manifolds. The manifolds are shown to have good homology and systolic properties for the purposes of code construction, with corresponding codes exhibiting linear rate. |

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 \(\Theta(\sqrt{n})\). |

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 [13,14]. Combining the quantum code obtained from a Ramanujan complex and a good classical LDPC code, which can be thought of as coming from a 1-dimensional chain complex, yields a new quantum code that is defined on a 2-dimensional chain complex. This 2-dimensional chain complex is obtained by the co-complex of the product of the 2 co-complexes. The length, dimension and distance of the new quantum code depend on the input codes. |

Homological code | CSS-type extenstion of the Kitaev surface code to arbitrary manifolds. The version on a Euclidean manifold of some fixed dimension is called the \(D\)-dimensional "surface" or \(D\)-dimensional toric code. |

Homological product code | CSS code formulated using the tensor product of two chain complexes (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 | Triangular color code defined on a patch of the 6.6.6 (honeycomb) tiling. |

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 [15]. Certain double covers of hyperbolic tilings also yield admissible tilings [16]. Other admissible hyperbolic tilings can be obtained via a fattening procedure [17]; see also a construction based on the more general quantum pin codes [18]. |

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 [19]. |

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 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 [20] system [21]. |

Kitaev surface code | A family of Abelian topological CSS stabilizer codes whose generators are few-body \(X\)-type and \(Z\)-type Pauli strings associated to the stars and plaquettes, respectively, of a cellulation of a two-dimensional surface (with a qubit located at each edge of the cellulation). Codewords correspond to ground states of the surface code Hamiltonian, and error operators create or annihilate pairs of anyonic charges or vortices. |

La-cross code | Code constructed using the 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. |

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. |

Long-range enhanced surface code (LRESC) | Code constructed using the 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 [22,23]. |

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 [24] yields a \(c^3\)-LTC [25]. Using two-sided expanders, which are only conjectured to exist, yields an asymptotically good QLDPC code family [26]. |

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 [27] or constructing a star-bipartition; see [28; 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 surface code | Extension of the surface code to prime-dimensional [29,30] and more general modular qudits [31]. 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. |

Projective-plane surface code | A family of Kitaev surface codes on the non-orientable 2-dimensional compact manifold \(\mathbb{R}P^2\) (in contrast to a genus-\(g\) surface). Whereas genus-\(g\) surface codes require \(2g\) logical qubits, qubit codes on \(\mathbb{R}P^2\) are made from a single logical qubit. |

Quantum Reed-Muller code | A CSS code formed from a classical Reed-Muller (RM) code or its punctured/shortened versions. Such codes often admit transversal logical gates in the Clifford hierarchy. |

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 [32]. |

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 expander code | CSS code constructed from a hypergraph product of bipartite expander graphs [5] 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 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 rainbow code | A CSS code whose qubits are associated with vertices of a simplex graph with \(m+1\) colors. |

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. |

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. |

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

Square-octagon (4.8.8) color code | Triangular 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 [17]. |

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. |

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. |

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 [33]. |

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. |

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 [34; Fig. 8] refers to the construction on a torus with twisted (a.k.a. shifted) boundary conditions. |

Truncated trihexagonal (4.6.12) color code | Triangular color code defined on a patch of the 4.6.12 (truncated trihexagonal or square-hexagon-dodecagon) tiling. |

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 [35]. |

Union-Jack color code | Triangular color code defined on a patch of the Tetrakis square tiling (a.k.a. the Union Jack lattice). |

XYZ product code | A non-CSS QLDPC code constructed from a particular hypergraph product of three classical 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 [36]. |

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 [20] system after a choice of grounding [21]. |

\((1,3)\) 4D toric code | A generalization of the Kitaev surface code defined on a 4D lattice. The code is called a \((1,3)\) toric code because it admits 1D \(Z\)-type and 3D \(X\)-type logical operators. |

\(D\)-dimensional twisted toric code | Extenstion of the Kitaev toric code to higher-dimensional lattices with 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 [37]. |

\([[144,12,12]]\) gross code | A BB QLDPC 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 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. |

\([[16,6,4]]\) Tesseract color code | A 4D color code defined on a tesseract, with stabilizer generators of both types supported on each cube. |

\([[2^D,D,2]]\) 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 [38]. Various other concatenations give families with increasing distance (see cousins). |

\([[2^r-1, 2^r-2r-1, 3]]\) 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 \(2^{r-1}\). |

\([[2^r-1,1,3]]\) 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 [39,40]. Each code is a color code defined on a simplex in \(r-1\) dimensions [27,41], 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 \(r \geq 2\). |

\([[2m,2m-2,2]]\) 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 [42; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [43]. |

\([[30,8,3]]\) 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. |

\([[6,4,2]]\) 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 [43; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [44]. |

\([[7,1,3]]\) Steane code | A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [45]. The code is constructed using the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors. |

\([[8,2,2]]\) hyperbolic color code | An \([[8,2,2]]\) hyperbolic color code defined on the projective plane. |

\([[8,3,2]]\) CSS 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. |

\([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code | Modular-qudit CSS code that generalizes the \([[9,1,3]]\) Shor code using properties of the multiplicative group \(\mathbb{Z}_q\). |

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