Here is a list of quantum CSS codes for qubits and qudits.
| 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). |
| 2D subsystem color code | A subsystem version of the 2D color code. |
| 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 subsystem color code | A subsystem version of the 3D color code. |
| 3D subsystem surface code | Subsystem generalization of the surface code on a 3D cubic lattice with gauge-group generators of weight at most three. |
| 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. |
| Approximate secret-sharing code | A family of \( [[n,k,d]]_q \) CSS codes approximately correcting errors on up to \(\lfloor (n-1)/2 \rfloor\) qubits, i.e., with approximate distance approaching the no-cloning bound \(n/2\). Constructed using a non-degenerate CSS code, such as a polynomial quantum code, and a classical authentication scheme. The code can be viewed as an \(t\)-error tolerant secret sharing scheme. Since the code yields a small logical subspace using large registers that contain both classical and quantum information, it is not useful for practical error correction problems, but instead demonstrates the power of approximate quantum error correction. |
| Bacon-Shor code | Subsystem CSS code defined on an \(m_1 \times m_2\) lattice of qubits that generalizes the \([[9,1,3]]\) (subspace) Shor code. It is said to be symmetric when \(m_1=m_2\), and asymmetric otherwise. |
| Balanced product (BP) code | Family of CSS quantum codes based on products of two classical codes which share common symmetries. The balanced product can be understood as taking the usual tensor/hypergraph product and then factoring out the symmetries factored. This reduces the overall number of physical qubits \(n\), while, under certain circumstances, leaving the number of encoded qubits \(k\) and the code distance \(d\) invariant. This leads to a more favourable encoding rate \(k/n\) and normalized distance \(d/n\) compared to the tensor/hypergraph product. |
| Ball color code | A color code defined on a \(D\)-dimensional colex. This family includes 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. |
| Bravyi-Bacon-Shor (BBS) code | An \([[n,k,d]]\) CSS subsystem stabilizer code generalizing Bacon-Shor codes to a larger set of qubit geometries. Defined through a binary matrix \(A\) such that physical qubits live on sites \((i,j)\) whenever \(A_{i,j}=1\). The gauge group is generated by 2-qubit operators, including \(XX\) interations between any two qubits sharing a column in \(A\), and \(ZZ\) interations between two qubits sharing a row. The code parameters are: \(n=\sum_{i,j}A_{i,j}\), \(k=\text{rank}(A)\), and the distance is the minimum weight of any row or column. |
| 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 [2]. |
| 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. |
| Capped color code (CCC) | A non-geometrically local subsystem color code consisting of two layers of 2D color code stacked together and topped (or capped) by a single qubit. Gauge fixing yields two types of codes, capped color codes in H or T form. Layers of 2D color codes can also be stacked together in a recursive construction, yielding recursive capped color codes (RCCCs). |
| 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. |
| 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 [3] 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 [4]. |
| Color code | Member of a family of qubit CSS codes defined on particular \(D\)-dimensional graphs. |
| Compass code | Subspace or subsystem CSS code defined by gauge-fixing the Bacon-Shor code, i.e., the code whose gauge group consists of terms in the compass model Hamiltonian [5–7] on a square lattice. Families of random codes perform well against biased noise and spatially dependent (i.e., asymmetric) noise. |
| Concatenated Steane code | A member of the family of \([[7^m,1,3^m]]\) CSS codes, each of which is a recursive level-\(m\) concatenatenation of the Steane code. This family is one of the first to admit a concatenated threshold [8–12]. |
| 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 [13], later generalized [14; Thm. 4.2], can yield QLDPC codes [13; Thm. 1]. |
| Doubled color code | Family of \([[2t^3+8t^2+6t-1,1,2t+1]]\) subsystem color codes (with \(t\geq 1\)), constructed using a generalization of the doubling transformation [15], that admit a Clifford + \(T\) transversal gate set using gauge fixing. |
| Expander LP code | Family of \(G\)-lifted product codes constructed using two random classical Tanner codes defined on expander graphs [16]. 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. |
| Fibonacci fractal spin-liquid code | A fractal type-I fracton CSS code defined on a cubic lattice [17; Eq. (D23)]. |
| 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 [18,19] are not strictly QLDPC. |
| 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-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 [20]. The underlying classical codes form classical self-correcting memories [21–23]. |
| Freedman-Meyer-Luo code | Hyperbolic surface code constructed using cellulation of a Riemannian Manifold \(M\) exhibiting systolic freedom [24]. 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 [25]. 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 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 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 surface code | Extension of the surface code to 2D lattices of Galois qudits. |
| 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 [26] 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 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 \(t\) [27; Def. V.1]. Such codes admit gates at the \(\nu\)th level of the Clifford hierarchy. Such codes can also be level-lifted [27; 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. |
| 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. |
| Heavy-hexagon code | Subsystem stabilizer code on the heavy-hexagonal lattice that combines Bacon-Shor and surface-code stabilizers. Encodes one logical qubit into \(n=(5d^2-2d-1)/2\) physical qubits with distance \(d\). The heavy-hexagonal lattice allows for low degree (at most 3) connectivity between all the data and ancilla qubits, which is suitable for fixed-frequency transom qubits subject to frequency collision errors. The code can be split into a surface and a Bacon-Shor code, with the idling qubits of one code serving as the physical qubits of the other [28]. |
| 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. |
| 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 [29,30]. 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 [31]. Certain double covers of hyperbolic tilings also yield admissible tilings [32]. Other admissible hyperbolic tilings can be obtained via a fattening procedure [33]; see also a construction based on the more general quantum pin codes [34]. |
| 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 [35]. |
| 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 [36] system [37]. |
| 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. |
| 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. |
| 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 [38,39]. |
| 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 [40] yields a \(c^3\)-LTC [41]. Using two-sided expanders, which are only conjectured to exist, yields an asymptotically good QLDPC code family [42]. |
| Modular-qudit CSS code | An \(((n,K,d))_q\) modular-qudit stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type Pauli strings. Codes can be defined from two classical codes and/or chain complexes over the ring \(\mathbb{Z}_q\) via an extension of qubit CSS-to-homology correspondence to modular qudits. The homology group of the logical operators has a torsion component because the chain complexes are defined over a ring, which yields codes whose logical dimension is not a power of \(q\). |
| Modular-qudit GKP code | Modular-qudit analogue of the GKP code. Encodes a qudit into a larger qudit and protects against Pauli shifts up to some maximum value. |
| Modular-qudit 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 [43] or constructing a star-bipartition; see [44; 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 subsystem color code | An extension of subsystem color codes to modular qudits. Codes are defined analogous to qubit subsystem color codes, but a directionality is required in order to make the modular-qudit stabilizer commute [44; Sec. VII]. |
| Modular-qudit surface code | Extension of the surface code to prime-dimensional [45,46] and more general modular qudits [47]. Stabilizer generators are few-body \(X\)-type and \(Z\)-type Pauli strings associated to the stars and plaquettes, respectively, of a tessellation of a two-dimensional surface. Since qudits have more than one \(X\) and \(Z\)-type operator, various sets of stabilizer generators can be defined. Ground-state degeneracy and the associated phase depends on the qudit dimension and the stabilizer generators. |
| 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 [48]. |
| 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 non-zero entries in self-orhogonal rows to \(X\) operators, and \(Z\) operators for each row in the orthogonal complement [49,50]. |
| 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 AG code | A Galois-qudit CSS code constructed using two linear AG codes. |
| 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. |
| Quantum Goppa code | A Galois-qudit CSS code constructed using two Goppa codes. |
| 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 \(r\) [51; 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 [52]. |
| 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 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 expander code | CSS code constructed from a hypergraph product of bipartite expander graphs [16] 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 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 [53; Thm. 40] and \(d \geq \sqrt{n}\) when \(n\equiv 1\) modulo 4 [53; 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 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. |
| 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 CSS code | An \([[n,k,d]]\) stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type Pauli strings. Codes can be defined from two classical codes and/or chain complexes over \(\mathbb{Z}_2\) per the qubit CSS-to-homology correspondence below. Strong CSS codes are codes for which there exists a set of \(X\) and \(Z\) stabilizer generators of equal weight. |
| 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 [17; Eq. (D22)]. The code admits an excitation-moving operator shaped like a Sierpinski triangle [17; 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 [54,55]. Sampling a description of this code can be done with an efficient randomized algorithm with \(2^{-\Omega(n)}\) failure probability. |
| Skew-cyclic CSS code | A member of a family of Galois-qudit CSS codes constructed from skew-cyclic classical codes over rings [56; Thm. 5.8]. See related study [57] that uses cyclic codes over rings. |
| 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 [33]. |
| Subsystem CSS code | Subsystem stabilizer code which admits a set of gauge-group generators which consist of either all-\(Z\) or all-\(X\) Pauli strings. This ensures that the code's stabilizer group is also CSS. |
| Subsystem Galois-qudit CSS code | Galois-qudit subsystem stabilizer code which admits a set of gauge-group generators which consist of either all-\(Z\) or all-\(X\) Galois-qudit Pauli strings. |
| Subsystem color code | A subsystem version of the color code. One way to obtain it is by expanding the vertices of a two-colex embedded in a surface of genus \(g\). Vertex expansion consists of spl every vertex into a triangle and splitting every edge into a pair of edges. |
| Subsystem homological product code | A CSS subsystem code constructed from a product of two (subspace) CSS codes. The case for qubits is formulated below, but these codes have also been extended to Galois qudits [58]. |
| Subsystem hyperbolic surface code | Subsystem generalization of the surface code on a 2D hyperbolic tesselation with gauge-group generators of weight at most three. An \(\{r,s\}\) hyperbolic tesselation with \(E\) edges yields a \([[3E/2,(1/2-2/r)E+2,(1-2/r)E,d]]\) subsystem code. |
| Subsystem hypergraph product (SHP) code | A CSS subsystem version of the generalized Shor code that has the same parameters as the subspace version, but requires fewer stabilizer measurements, resulting in a simpler error recovery routine. The code can also be thought of as a subsystem version of an HGP code because two such codes reduce to an HGP code upon gauge fixing [59; Sec. III]. The code can be obtained from a generalized Shor code by removing certain stabilizers that do no affect the code distance. |
| Subsystem lifted-product (SLP) code | Member of a family of subsystem CSS codes constructed from a subsystem hypergraph product of a lifted binary linear code. |
| Subsystem modular-qudit CSS code | Modular-qudit subsystem stabilizer code which admits a set of gauge-group generators which consist of either all-\(Z\) or all-\(X\) modular-qudit Pauli strings. This ensures that the code's stabilizer group is also CSS. |
| Subsystem rotated surface code | Subsystem version of the rotated surface code. |
| Subsystem surface code | Subsystem version of the surface code defined on a square lattice with qubits placed at every vertex and center of everry edge. |
| 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. |
| 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). |
| 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 [60]. |
| 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. |
| 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 [61; Fig. 8] refers to the construction on a torus with twisted (a.k.a. shifted) boundary conditions. |
| Triorthogonal code | Qubit CSS code whose \(X\)-type logicals and stabilizer generators form a triorthogonal matrix (defined below) in the symplectic representation. |
| 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 CSS code | Galois-qudit CSS code whose stabilizer generator matrices \(H_X=(A,B)\) and \(H_Z=(B^T,-A^T)\), are constructed from a pair of square commuting matrices \(A\) and \(B\). |
| Two-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 [62]. |
| Type-II fractal spin-liquid code | A type-II fracton prime-qudit CSS code defined on a cubic lattice [17; Eqs. (D9-D10)]. |
| Union-Jack color code | Triangular 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 the membrane-coupled [63], twice-foliated [64], and several generalized [65] X-cube models. |
| 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 [36] system after a choice of grounding [37]. |
| \((1,3)\) 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 [66]. |
| \([[10,1,2]]\) CSS code | Smallest stabilizer code to implement a logical \(T\) gate via application of physical \(T\), \(T^{\dagger}\), and \(CCZ\) gates. |
| \([[11,1,5]]_3\) 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. |
| \([[12,2,4]]\) carbon code | Self-dual twelve-qubit CSS code. |
| \([[144,12,12]]\) 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 [67]. 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 [68]. 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 \(2^{r-1}\). |
| \([[2^r-1, 2^r-2r-1, 3]]_p\) 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 [69]. |
| \([[2^r-1,1,3]]\) 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 [70,71]. Each code is a color code defined on a simplex in \(r-1\) dimensions [43,72], 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 [73; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [74]. |
| \([[30,8,3]]\) 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. |
| \([[4,1,1,2]]\) Four-qubit subsystem code | Error-detecting four-qubit subsystem stabilizer code encoding one logical qubit and one gauge qubit. |
| \([[4,2,2]]\) Four-qubit code | Four-qubit CSS stabilizer code is the smallest qubit stabilizer code to detect a single-qubit error. |
| \([[49,1,5]]\) triorthogonal code | Triorthogonal and quantum divisible code which is the smallest distance-five stabilizer code to admit a transversal \(T\) gate [15,75,76]. 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. |
| \([[6,2,2]]\) \(C_6\) 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 [74; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [77]. |
| \([[7,1,3]]\) Steane code | A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [78]. 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 \(\mathbb{Z}_q\). |
| \([[9m-k,k,2]]_3\) 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 \(Z_1Z_2Z_5Z_6...Z_{k+4}\).' |
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