Here is a list of non-qubit-stabilizer codes related to block quantum codes that detect or correct an error on at most two subsystems.
| Code | Description |
|---|---|
| Amplitude-damping CWS code | Self-complementary CWS code that is designed to detect and correct AD errors. |
| Binary dihedral PI code | Multi-qubit PI code designed to realize gates from the binary dihedral group transversally. Can also be interpreted as a single-spin code. The codespace projection is a projection onto an irrep of the binary dihedral group \( \mathsf{BD}_{2N} = \langle\omega I, X, P\rangle \) of order \(8N\), where \( \omega \) is a \( 2N \)th root of unity, and \( P = \text{diag} ( 1, \omega^2) \). |
| 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 rotor. |
| Group-representation code | Code whose projector is onto an irreducible representation of a subgroup \(G\) of a group of canonical or distinguished unitary operations, e.g., transversal gates in the case of block quantum codes, Gaussian operations in the case of bosonic codes, or \(SU(2)\) operations in the case of single-spin codes. |
| 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 [1] system [2]. |
| Numerically optimized four-qubit AD code | One of several four-qubit codes that can (approximately) correct a single AD error with higher fidelity than the \([[4,1,2]]\) subcodes of the \([[4,2,2]]\) code. |
| Perfect quantum code | A type of block quantum code whose parameters satisfy the quantum Hamming bound with equality. |
| Quantum plane-curve code | Quantum AG code constructed from plane-curve codes via the Galois-qudit Hermitian construction. Code parameters are \([[q^3,q^3+q^2-3q-2r,r+2q-q^2]]_q\), where \(r\) is an integer satisfying \(q^2 - 2 \leq r \leq q^2 + q - 3\), and where the underlying plane curve is \(y^q + y = x^{q-1}\). |
| Self-complementary qubit code | A qubit code which admits a basis of codewords of the form \(|c\rangle+|\overline{c}\rangle\), where \(c\) is a bitstring and \(\overline{c}\) is its negation a.k.a. complement. Their codewords generalize the two-qubit Bell states and three-qubit GHZ states and are often called (qubit) cat states or poor-man’s GHZ states. Such codes were originally pointed out to perform well against AD noise [3]. |
| Small-distance block code | A block code of length \(n\) that either detects or corrects errors on at most two coordinates, i.e., has distance \(d \leq 5\). |
| Small-distance block quantum code | A block quantum code on \(n\) subsystems that either detects or corrects errors on at most two subsystems, i.e., have distance \(\leq 5\). |
| Smolin-Smith-Wehner (SSW) code | A family of \(((n=4k+2l+3,M_{k,l},2))\) self-complementary CWS codes, where \(M_{k,l} \approx 2^{n-2}(1-\sqrt{2/(\pi(n-1))})\). For \(n \geq 11\), these codes have a logical subspace whose dimension is larger than that of the largest stabilizer code for the same \(n\) and \(d\). Ref. [4] augments a star-graph-based subfamily \(((4n+1,M_n,2))\) by one additional graph-state basis word, yielding \(((4n+1,M_n+1,2))\) codes with \(M_n=2^{4n-1}-\frac{1}{2}\binom{4n}{2n}\). In the CWS description of Ref. [5], the underlying stabilizer state is locally Clifford-equivalent to a GHZ state and its standard-form graph is a star graph. |
| Subset-Sum-Linear-Programming (SS-LP) code | Qubit block quantum code that encodes a logical qubit and that is constructed using the Subset-Sum-Linear-Programming (SS-LP) numerical construction. SS-LP codes are optimized to admit diagonal gates transversally and include \(((7,2,3))\) codes that realize the \(\mathsf{BD}_{16}\) and \(\mathsf{BD}_{32}\) groups transversally, yielding \(T\) and \(\sqrt{T}\) gates, respectively. Larger codes include an \(((8,2,3))\) code that transversally realizes \(\mathsf{BD}_{64}\). |
| Twisted \(1\)-group code | Block group-representation code realizing particular irreps of particular groups such that a distance of two is automatically guaranteed. Groups which admit irreps with this property are called twisted (unitary) \(1\)-groups and include the binary icosahedral group \(2I\), the \(\Sigma(360\phi)\) subgroup of \(SU(3)\), the family \(\{PSp(2b, 3), b \geq 1\}\), and the alternating groups \(A_{5,6}\). Groups whose irreps are images of the appropriate irreps of twisted \(1\)-groups also yield such properties, e.g., the binary tetrahedral group \(2T\) or qutrit Pauli group \(\Sigma(72\phi)\). |
| 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 [1] system after a choice of grounding [2]. |
| \(((10,24,3))\) qubit code | Ten-qubit nonadditive CWS code saturating the linear-programming bound for one-error-correcting codes on ten qubits [4]. It was constructed using the coding-clique graph-theoretic framework introduced in [4], which unifies additive and nonadditive code constructions. |
| \(((2m+1,3 \times 2^{2m-3},2))\) Rains code | Member of a family of pure odd-length distance-two CWS codes with parameters \(((2m+1,3 \times 2^{2m-3},2))\) for all \(m \geq 2\), constructed recursively from the \(((5,6,2))\) code [8,9][6; Lem. 5 and Thm. 4][7; Exam. 8]. |
| \(((3,2,2))_3\) Three-qutrit single-deletion code | Three-qutrit PI code that is the smallest qutrit PI code to correct one deletion error. |
| \(((3,6,2))_{\mathbb{Z}_6}\) Euler code | Three-qudit error-detecting code with logical dimension \(K=6\) that is obtained from a particular AME state that serves as a solution of a quantum analogue of the classical problem of 36 officers of Euler. The code is obtained from a \(((4,1,3))_{\mathbb{Z}_6}\) code. |
| \(((4,2,2))\) Four-qubit single-deletion code | Four-qubit PI code that is the smallest qubit code to correct one deletion error. |
| \(((5,3,2))_3\) qutrit code | Smallest qutrit block code realizing the \(\Sigma(360\phi)=3.A_6\) subgroup of \(SU(3)\) transversally. The next smallest code is \(((7,3,2))_3\). |
| \(((5,6,2))\) qubit code | Five-qubit cyclic CWS code detecting a single-qubit error. This code has a logical subspace whose dimension is larger than that of the \([[5,2,2]]\) code, the best five-qubit stabilizer code with the same distance [4]. |
| \(((6,2,3))\) transversal-\(\mathbb{Z}_{10}\) code | Six-qubit code that realizes gates from the group \(\mathbb{Z}_{10}\) transversally. This is the smallest known distance-three code supporting a transversal gate outside of the Clifford group. See Ref. [10] for the codewords. |
| \(((7,2,3))\) Pollatsek-Ruskai code | Seven-qubit PI code that realizes gates from the binary icosahedral group transversally. Can also be interpreted as a spin-\(7/2\) single-spin code. The codespace projection is a projection onto an irrep of the binary icosahedral group \(2I\). See Ref. [10] for other non-PI codes realizing \(2I\) gates transversally. |
| \(((8,2,3))\) Plenio-Vedral-Knight CE code | An eight-qubit single error-correcting code that is the first CE code. Each logical state is a superposition of computational basis states with four excitations. |
| \(((9,12,3))\) qubit code | Nine-qubit cyclic CWS code correcting a single-qubit error. This code has a logical subspace whose dimension is larger than that of the \([[9,3,3]]\) code, the best nine-qubit stabilizer code with the same distance [11]. |
| \(((9,2,3))\) Ruskai code | Nine-qubit PI code that protects against single-qubit errors as well as two-qubit errors arising from exchange processes. |
| \(((n,1+n(q-1),2))_q\) union stabilizer code | Member of a family of \(((n,1+n(q-1),2))_q\) Galois-qudit union stabilizer codes for odd \(n\). |
| \(((n,2,2))\) Bravyi-Lee-Li-Yoshida PI code | PI distance-two code on \(n\geq4\) qubits whose degree of entanglement vanishes asymptotically with \(n\) [12; Appx. D] (cf. [13]). |
| \([[10,1,4]]_{G}\) tenfold code | A \([[10,1,4]]_{G}\) group code for finite Abelian \(G\). The code is defined using a graph that is closely related to the \([[5,1,3]]\) code. |
| \([[11,1,5]]_3\) qutrit Golay code | An \([[11,1,5]]_3\) code 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. |
| \([[2^r-1, 2^r-2r-1, 3]]_p\) 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 [14]. |
| \([[3,1,2]]_3\) Three-qutrit code | A \([[3,1,2]]_3\) prime-qudit CSS code that is the smallest qutrit stabilizer code to detect a single-qutrit error. It has 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. |
| \([[3,1,2]]_{\mathbb{Z}}\) 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 rotor. |
| \([[4,2,2]]_{G}\) four group-qudit code | \([[4,2,2]]_{G}\) group quantum code that is an extension of the four-qubit code to group-valued qudits. |
| \([[5,1,3]]_q\) Galois-qudit code | True stabilizer code that generalizes the five-qubit perfect code to Galois qudits of prime-power dimension \(q=p^m\). It has \(4(m-1)\) stabilizer generators expressed as \(X_{\gamma} Z_{\gamma} Z_{-\gamma} X_{-\gamma} I\) and its cyclic permutations, with \(\gamma\) iterating over basis elements of \(\mathbb{F}_q\) over \(\mathbb{F}_p\). |
| \([[5,1,3]]_{\mathbb{R}}\) Braunstein five-mode code | An analog stabilizer version of the five-qubit perfect code, encoding one mode into five and correcting arbitrary errors on any one mode. |
| \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code | Modular-qudit stabilizer code that generalizes the five-qubit perfect code using properties of the multiplicative group \(\mathbb{Z}_q\) [15]; see also [16; Thm. 13]. It has four stabilizer generators consisting of \(X Z Z^\dagger X^\dagger I\) and its cyclic permutations. |
| \([[5,1,3]]_{\mathbb{Z}}\) Five-rotor code | Extension of the five-qubit stabilizer code to the integer alphabet, i.e., the angular momentum states of a rotor. The code is \(U(1)\)-covariant and ideal codewords are not normalizable. |
| \([[6,2,3]]_{q}\) code | Six-qudit MDS error-detecting code defined for Galois-qudit dimension \(q=3\) [17], \(q=2^2\) [18], and \(q \geq 5\) [17][19; Exam. 33]. This code cannot exist for qubits (\(q=2\)). |
| \([[7,3,3]]_{q}\) code | Seven-qudit MDS error-detecting code defined for Galois-qudit dimension \(q=3\) [17] and \(q \geq 7\) [17][19; Exam. 33]. This code cannot exist for qubits (\(q=2\)). |
| \([[9,1,3]]_{\mathbb{R}}\) Lloyd-Slotine code | An analog stabilizer version of Shor’s nine-qubit code, encoding one mode into nine and correcting arbitrary errors on any one mode. |
| \([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code | Modular-qudit CSS code that generalizes the \([[9,1,3]]\) Shor code to \(q\)-level systems. |
| \([[9,1,5]]_3\) quantum Glynn code | Nine-qutrit pure Hermitian code that is the smallest qutrit stabilizer code to correct two-qutrit errors. |
| \([[9m-k,k,2]]_3\) triorthogonal code | Member of the \([[9m-k,k,2]]_3\) family of triorthogonal qutrit codes (for \(k\leq 3m-2\)) that admit a transversal diagonal gate in the third level of the qudit Clifford hierarchy and that are relevant for magic-state distillation. |
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