Here is a list of codes related to qubit stabilizer codes that detect or correct an error on at most two qubits.
| Code | Description |
|---|---|
| Ball code | A distance-two color code defined on a colorable \(D\)-ball, equivalently on a \(D\)-colex with boundary [1; Appx. A]. In the morphing construction of Ref. [1], ball codes arise as the child codes associated with the morphed ball-like regions. This family includes hypercube codes (defined on balls constructed from hyperoctahedra) and 3D ball codes (defined on duals of certain Archimedean solids). |
| 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 [2]. Certain double covers of hyperbolic tilings also yield admissible tilings [3]. Other admissible hyperbolic tilings can be obtained via a fattening procedure [4]; see also a construction based on the more general quantum pin codes [5]. |
| Kitaev chain code | A Majorana stabilizer code obtained from the ground-state subspace of the Kitaev Majorana chain in its fermionic topological phase [6]. Its codespace is stabilized by nearest-neighbor Majorana bilinears, while two unpaired edge Majoranas furnish one logical fermionic mode. Under parity-preserving noise, it behaves as the Majorana analogue of the repetition code [7]. |
| Majorana box qubit | A family of Majorana stabilizer codes obtained by fixing the total fermion parity of \(n\) fermionic modes, equivalently \(2n\) Majorana zero modes, within the ground-state subspace of \(n\) Kitaev Majorana chain Hamiltonians. The resulting positive-parity subspace encodes \(n-1\) logical qubits and has Majorana distance \(2\). |
| 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 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 \(|\phi_1\rangle = |-\rangle\). |
| Reinforcement-learning quantum code | An approximate qubit code obtained from a numerical optimization involving a reinforcement learning agent. |
| Small-distance qubit stabilizer code | A qubit stabilizer code that either detects or corrects errors on at most two subsystems, i.e., has distance \(\leq 5\). |
| Tetron code | A \([[2,1,2]]_{f}\) Majorana box qubit encoding a logical qubit into four Majorana modes, equivalently into the fixed-total-parity sector of two physical fermionic modes. Four Majorana zero modes are the smallest aggregate that supports a qubit in a fixed fermion-parity sector [8]. This code can be concatenated with various qubit codes such as surface codes and color codes. Four-boundary Majorana surface-code patches are logical tetrons, i.e., higher-distance analogues of this physical tetron block [9]. |
| 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). |
| Trapezoid subsystem code | A member of a family of BBS codes with weight-two (two-body) gauge generators designed to suppress errors in adiabatic quantum computation. |
| \(((7,2))\) QETC | Seven-qubit QETC that transmutes all single-qubit Pauli errors to logical phase errors. See [10; Table 1] for its stabilizer generators. |
| \([[10,1,2]]\) Vasmer-Kubica code | A stabilizer code obtained by morphing the \([[15,1,3]]\) quantum Reed-Muller code on a subset whose child code is the \([[8,3,2]]\) smallest interesting color code [1]. It is the smallest known stabilizer code with a fault-tolerant logical \(T\) gate, implemented via physical \(T\), \(T^{\dagger}\), and \(CCZ\) gates [1]. |
| \([[10,1,3;1,3,4]]\) EAOA Hamming code | An EAOA qubit stabilizer code constructed from the dual of a \([10,6,3]\) code obtained by shortening the classical \([15,11,3]\) Hamming code at five positions. In the notation of the parent entry, the example of Ref. [11] is a \([[10,1,3;1,3,4]]\) code: it encodes one logical qubit and four classical strings (equivalently, two classical bits), while retaining one gauge qubit and using three ebits. |
| \([[11,1,5]]\) quantum dodecacode | Eleven-qubit pure stabilizer code that is the smallest qubit stabilizer code to correct two-qubit errors. It can be obtained from the dodecacode by puncturing [12; Table IV]. |
| \([[12,2,2]]\) CSS code | CSS code that admits a logical \(CS\) gate via application of physical \(T\) and \(T^{\dagger}\) gates. |
| \([[12,2,4]]\) carbon code | Twelve-qubit CSS code based on Knill’s \(C_4/C_6\) scheme [13]. Using the concatenation convention of the Zoo, the carbon code can be viewed as a block concatenation with inner code \([[4,2,2]]\) and outer code \(C_6\): three inner \([[4,2,2]]\) blocks encode six intermediate qubits, which are then encoded into two logical qubits by the outer \([[6,2,2]]\) code. |
| \([[13,1,5]]\) quantum QR code | Thirteen-qubit cyclic Hermitian qubit code derived from a quaternary quadratic-residue code using the Hermitian construction [15][14; pg. 11]. The code admits a check matrix whose rows are cyclic permutations of the Pauli string \(XXZZIZIIIZIZZ\). |
| \([[13,1,5]]\) twisted toric code | Thirteen-qubit twisted toric 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 [16; 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 [17] stemming from the geometry of the polytope. A local-Clifford-equivalent clean realization has only \(X\)- and \(Z\)-type operators on its four-valent vertices, and its symplectic double is a \([[28,6,3]]\) genus-three code [18]. |
| \([[15, 7, 3]]\) 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 RM code | A \([[15,1,3]]\) quantum Reed-Muller code that is most easily thought of as a tetrahedral 3D color code. |
| \([[16,4,3]]\) dodecahedral code | A \([[16,4,3]]\) non-CSS qubit stabilizer code whose encoder-respecting form is the graph of vertices of a dodecahedron [19]. |
| \([[16,6,4]]\) Tesseract color code | A (hyperbolic 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 [20]. |
| \([[17,1,5]]\) 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 \(n=17\) [21]. It is also a normal self-dual CSS code whose transversal Hadamard acts logically, making it suitable as an inner code for fifth-order magic-state distillation [22]. |
| \([[2(m+1),m,2]]\) single-loss AD code | A member of a class of \([[2(m+1),m,2]]\) CSS codes for \(m\geq 1\) that generalizes the \([[4,1,2]]\) approximate amplitude-damping code of Ref. [23]. Its \(Z\)-type generators are \(m+1\) pairwise products, with each qubit participating in only one check; the single \(X\)-type generator is the all-\(X\) string. |
| \([[23, 1, 7]]\) Quantum Golay code | A \([[23, 1, 7]]\) self-dual CSS code with eleven stabilizer generators of each type, and with each generator being weight eight. |
| \([[2^D,D,2]]\) hypercube quantum code | Member of a family of codes defined by placing qubits on a \(D\)-dimensional hypercube, \(Z\)-stabilizers on all two-dimensional faces, and an \(X\)-stabilizer on all vertices. These codes realize gates at the \((D-1)\)-st level of the Clifford hierarchy. The measured physical bit string can be post-processed into both a logical output string and stabilizer checks, enabling end-of-circuit error detection directly from classical samples [24]. |
| \([[2^r+r, 2^r-r-2, 3]]\) Ring CPC code | A family of \([[2^r+r, 2^r-r-2, 3]]\) CPC codes for \(r \geq 3\) whose matrices are based on the shortened version of the \([2^r-1,2^r-r-1,3]\) Hamming code. See [25; Thm. 4] for their stabilizer generator matrix. |
| \([[2^r, 2^r-r-2, 3]]\) Gottesman code | A family of pure [12] non-CSS stabilizer codes of distance \(3\) that saturate the asymptotic quantum Hamming bound. |
| \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code | Member of a family of self-dual CSS 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 a color-code family constructed from a punctured first-order RM\((1,m=r)\) 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 [26,27]. Each code is a color code defined on a simplex in \(r-1\) dimensions [28,29], where qubits are placed on the vertices, edges, and faces as well as on the simplex itself. |
| \([[2^{m-1},2^{m-1}-m-1,4]]_{f}\) Hamming Majorana code | A member of the \([[2^{m-1},2^{m-1}-m-1,4]]_{f}\) family of Majorana stabilizer codes for \(m \geq 3\) constructed from a self-orthogonal first-order RM code (whose dual is the extended Hamming code). A shortened \([[2^{m-1}-1,2^{m-1}-m-2,3]]_{f}\) version can also be defined [30; Prop. 2.5.1]. The logical subspace of the \([[8,3,4]]_{f}\) Hamming Majorana code is a Cartan subspace of the \(E_8\) Lie algebra [31]. |
| \([[2m,2m-2,2]]\) 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 [32; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [12]. |
| \([[3, 1, 3;2]]\) EA code | Distance-three EA stabilizer code encoding one logical qubit and using two ebits. It is the smallest example of an EA code correcting an arbitrary single-qubit error. |
| \([[30,8,3]]\) 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. It 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 [33][34; Sec. VI.C]. |
| \([[4,1,1,2]]\) Four-qubit subsystem code | Error-detecting four-qubit subsystem stabilizer code encoding one logical qubit and one gauge qubit. |
| \([[4,1,2]]\) Leung-Nielsen-Chuang-Yamamoto (LNCY) code | A four-qubit CSS stabilizer code that is the only qubit CSS code with such parameters [35; ID 6]. |
| \([[4,1,2]]\) twist-defect code | A four-qubit non-CSS stabilizer code that can be interpreted as the smallest triangular color code with \(x\)-, \(y\)-, and \(z\)-type Pauli boundaries [36; Fig. 7], and equivalently as a small twist-defect surface code on a tetrahedron inscribed in a sphere [18]. It is the only non-CSS qubit stabilizer code with parameters \([[4,1,2]]\) [35; ID 8]. The code admits weight-three stabilizer generators \(\{IXXX,YIYY,ZZZI\}\) and weight-two logical Pauli \(X,Y,Z\) operators. |
| \([[4,2,2]]\) Four-qubit code | A four-qubit hyperbolic self-dual CSS stabilizer code that is the smallest two-logical-qubit stabilizer code to detect a single-qubit error. It is unique for its parameters [37; Thm. 8][35; ID 9]. |
| \([[49,1,5]]\) triorthogonal code | Triorthogonal and quantum divisible code which is the smallest distance-five stabilizer code to admit a transversal \(T\) gate [38][33; Appx. B]. It is one example of a level-three generalized divisible code obtainable from the doubling transformation [34; Sec. VI.D]. Its \(X\)-type stabilizers form a triply even linear binary code in the symplectic representation. |
| \([[5,1,2]]\) rotated surface code | Rotated surface code on one rung of a ladder, with one qubit on the rung, and four qubits surrounding it. |
| \([[5,1,3]]\) Five-qubit perfect code | Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error. |
| \([[54,6,5]]\) five-covered icosahedral code | A \([[54,6,5]]\) qubit stabilizer code whose encoder-respecting form is the graph of a five-cover of the icosahedron [19]. The covering-space construction avoids the weight-three logical operators that occur for the bare icosahedral graph [19]. |
| \([[6,1,2]]\) semi-self-dual CSS code | A six-qubit CSS stabilizer code with generators \(ZIZIIZ\), \(IZZIZI\), \(IIIZZZ\), \(XIXXXI\), and \(IXXXIX\) [35; ID 59]. It is a semi-self-dual CSS code, i.e., a CSS code whose \(X\)-type stabilizers are contained in the \(Z\)-type stabilizers [39]. |
| \([[6,1,3]]\) Six-qubit stabilizer code | A degenerate, non-trivial \([[6,1,3]]\) stabilizer code. It is one of two six-qubit distance-three codes that are unique up to equivalence [12], with the other code being decomposable and an extension of the five-qubit code [40]. The code admits fault-tolerant syndrome extraction using only one ancilla per stabilizer generator measurement. |
| \([[6,1,3]]_{f}\) Vijay-Fu Majorana code | A Majorana stabilizer code encoding a logical fermion into six physical fermions. This code is the shortest code correcting single fermion-parity flips [41]. |
| \([[6,2,2]]\) \(C_6\) code | Error-detecting normal self-dual CSS code on three qubit pairs that encodes a logical qubit pair and detects any error acting on one pair [42]. In Knill’s \(C_4/C_6\) architecture, this code is used at the second and higher concatenation levels. A choice of check operators used in that construction is \(XIIXXX\), \(XXXIIX\), \(ZIIZZZ\), and \(ZZZIIZ\), with logical operators \(X_L = IXXIII\), \(Z_L = IIZZIZ\), \(X_S = XIXXII\), and \(Z_S = IIIZZI\) [42][35; ID 126]. |
| \([[6,4,2]]\) 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 [12; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [43]. |
| \([[6k+2,3k,2]]\) Campbell-Howard code | Family of \([[6k+2,3k,2]]\) qubit stabilizer codes with quasi-transversal \(CCZ^{\otimes k}\) gates that are relevant to magic-state distillation. In the synthillation framework, these distance-two codes realize batches of logical \(CCZ\) gates using physical \(T\) gates followed by a Clifford correction. |
| \([[6r,2r,2]]\) Ganti-Onunkwo-Young code | A member of the family of \([[6r,2r,2]]\) CSS codes designed to suppress errors in adiabatic quantum computation. All but two of its stabilizer generators are weight-two (two-body), and the remaining two are weight-\(4r\). |
| \([[7, 1:1, 3]]\) hybrid stabilizer code | A distance-three seven-qubit hybrid stabilizer code storing one qubit and one classical bit. Admits a stabilizer generator set with a weight-two generator, which delineates the underlying classical code [44; Eq. (3)]. |
| \([[7,1,3]]\) Steane code | A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [40][35; ID 226]. The code is constructed using the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors. |
| \([[7,1,3]]\) bare code | A \([[7,1,3]]\) code that admits fault-tolerant syndrome extraction using only one ancilla per stabilizer generator measurement. |
| \([[7,1,3]]\) twist-defect surface code | A \([[7,1,3]]\) code (different from the Steane code) that is a small example of a twist-defect surface code. |
| \([[8, 2:1, 3]]\) hybrid stabilizer code | A code obtained from the \([[8,3,3]]\) Gottesman code by using one of its logical qubits as a classical bit. One can also use two logical qubits as classical bits, obtaining an \([[8,1:2,3]]\) hybrid stabilizer code. |
| \([[8, 3, 3]]\) Eight-qubit Gottesman code | Eight-qubit non-degenerate code that can be obtained from a modified CSS construction using the \([8,4,4]\) extended Hamming code and a \([8,7,2]\) even-weight code [45]. The modification introduces signs between the codewords. |
| \([[8,1,2]]\) Shen-Wang-Cao code | A stabilizer code that admits a logical \(T\) gate via application of physical \(T\) gates and a \(CZ\)-like gate. |
| \([[8,2,2]]\) 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. In encoded IQP sampling, the final measurement outcomes determine both the logical sample and stabilizer checks, enabling end-of-circuit error detection or postselected decoding directly from the classical samples [24]. |
| \([[8,3,2]]\) 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 [17] stemming from the geometry of the polytope. |
| \([[9,1,3]]\) Shor code | Nine-qubit CSS code that is the first quantum error-correcting code. Among indecomposable \([[9,1,3]]\) CSS codes, the Shor code has the largest automorphism group [35]. |
| \([[9,1,3]]\) Surface-17 code | A \([[9,1,3]]\) rotated surface code named for the sum of its 9 data qubits and 8 syndrome qubits. It is one of the four inequivalent CSS gauge fixings of the nine-qubit Bacon-Shor code [35]. It uses the smallest number of qubits to perform fault-tolerant error correction on a surface code with parallel syndrome extraction. |
| \([[9,1,4,3]]\) Nine-qubit Bacon-Shor code | Error-correcting nine-qubit subsystem stabilizer code encoding one logical qubit and four gauge qubits. There are exactly four inequivalent CSS gauge fixings of the code, including the Shor code and the surface-17 code [35]. |
| \([[9,3,3]]\) Quadric code | Nine-qubit pure Hermitian qubit code constructed from the almost MDS \([9,3,6]_4\) Hermitian self-orthogonal code. It is the only pure Hermitian code with its parameters [35; ID 170235] and is the highest-distance qubit stabilizer code for its \(n\) and \(k\). |
| \([[k+4,k,2]]\) 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 stabilizer generators are \(X_1X_2X_3X_4\), \(Z_1Z_2Z_3Z_4\), \(X_1X_2X_5X_6...X_{k+4}\), and \(Z_1Z_2Z_5Z_6...Z_{k+4}\). |
| \([[n,n-2k,4]]\) Quantum cap code | A distance-four pure Hermitian qubit code constructed from a Hermitian self-orthogonal \([n,k]_4\) code associated with an \(n\)-cap in \(PG(k-1,4)\). |
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