Here is a list of cyclic quantum codes.

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

Binary dihedral PI code | Multi-qubit 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 \( P = \text{diag} ( 1, \omega^2) \). |

Clifford spin code | A single-spin code designed to realize a discrete group of gates using \(SU(2)\) rotations. Codewords are subspaces of a spin's Hilbert space that house irreducible representations (irreps) of a discrete subgroup of \(SU(2)\). |

Combinatorial PI code | A member of a family of PI quantum codes whose correction properties are derived from solving a family of combinatorial identities. The code encodes one logical qubit in superpositions of Dicke states whose coefficients are square roots of ratios of binomial coefficients. |

Cyclic quantum code | A block quantum code such that cyclic permutations of the subsystems leave the codespace invariant. In other words, the automorphism group of the code contains the cyclic group \(\mathbb{Z}_n\). |

Five-qubit perfect code | Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error. |

Five-rotor code | Extension of the five-qubit stabilizer code to the integer alphabet, i.e., the angular momentum states of a planar rotor. The code is \(U(1)\)-covariant and ideal codewords are not normalizable. |

Four-qubit single-deletion code | Four-qubit PI code that is the smallest qubit code to correct one deletion error. |

Frobenius code | A cyclic prime-qudit stabilizer code whose length \(n\) divides \(p^t + 1\) for some positive integer \(t\). |

GNU PI code | PI code whose codewords can be expressed as superpositions of Dicke states with coefficients are square-roots of the binomial distribution. |

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. |

Ouyang-Chao constant-excitation PI code | A constant-excitation PI Fock-state code whose construction is based on integer partitions. |

PI qubit code | Block quantum code defined on two-dimensional subsystems such that any permutation of the subsystems leaves any codeword invariant. |

Permutation-invariant (PI) code | Block quantum code such that any permutation of the subsystems leaves any codeword invariant. In other words, the automorphism group of the code contains the symmetric group \(S_n\). |

Quantum repetition code | Encodes \(1\) qubit into \(n\) qubits according to \(|0\rangle\to|\phi_0\rangle^{\otimes n}\) and \(|1\rangle\to|\phi_1\rangle^{\otimes n}\). Also known as a bit-flip code when \(|\phi_i\rangle = |i\rangle\), and a phase-flip code when \(|\phi_0\rangle = |+\rangle\) and \(|\phi_1\rangle = |-\rangle\). |

Qudit GNU PI code | Extension of the GNU PI codes to those encoding logical qudits into physical qubits. Codewords can be expressed as superpositions of Dicke states with coefficients are square-roots of polynomial coefficients, with the case of binomial coefficients reducing to the GNU PI codes. |

Twisted XZZX toric code | A cyclic code that can be thought of as the XZZX toric code with shifted (a.k.a twisted) boundary conditions. Admits a set of stabilizer generators that are cyclic shifts of a particular weight-four \(XZZX\) Pauli string. For example, a seven-qubit \([[7,1,3]]\) variant has stabilizers generated by cyclic shifts of \(XZIZXII\) [1]. Codes encode either one or two logical qubits, depending on qubit geometry, and perform well against biased noise [2]. |

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 \(\Sigma(72\phi)\). |

Very small logical qubit (VSLQ) code | The two logical codewords are \(|\pm\rangle \propto (|0\rangle\pm|2\rangle)(|0\rangle\pm|2\rangle)\), where the total Hilbert space is the tensor product of two transmon qudits (whose ground states \(|0\rangle\) and second excited states \(|2\rangle\) are used in the codewords). Since the code is intended to protect against losses, the qutrits can equivalently be thought of as oscillator Fock-state subspaces. |

Wasilewski-Banaszek code | Three-oscillator constant-excitation Fock-state code encoding a single logical qubit. |

\(((5,3,2))_3\) qutrit code | Smallest qutrit block code realizing the \(\Sigma(360\phi)\) subgroup of \(SU(3)\) transversally. The next smallest code is \(((7,3,2))_3\). |

\(((5,6,2))\) qubit code | Six-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 [3]. |

\(((7,2,3))\) 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 \(2I\). |

\(((9,12,3))\) 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 [4]. |

\(((9,2,3))\) Ruskai code | Nine-qubit PI code that protects against single-qubit errors as well as two-qubit errors arising from exchange processes. |

\(((n,1,2))\) Bravyi-Lee-Li-Yoshida PI code | PI distance-two code on \(n\geq4\) qubits whose degree of entanglement vanishes asymptotically with \(n\) [5; Appx. D]. |

\([[13,1,5]]\) cyclic code | Thirteen-qubit twisted surface code for which there is a set of stabilizer generators consisting of cyclic permutations of the \(XZZX\)-type Pauli string \(XIZZIXIIIIIII\). The code can be thought of as a small twisted XZZX code [6; Ex. 11 and Fig. 3] or can be derived from a quaternary QR code using the Hermitian construction [7]; see [8; pg. 11] for details. |

\([[5,1,3]]_q\) 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 \(GF(q)\) over \(GF(p)\). |

\([[5,1,3]]_{\mathbb{R}}\) 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 \(\mathbb{Z}_q\) [9]; see also [10; Thm. 13]. It has four stabilizer generators consisting of \(X Z Z^\dagger X^\dagger I\) and its cyclic permutations. A concise expression for a set of codewords can be found in [11; Sec. VI.B]. |

## References

- [1]
- A. Robertson et al., “Tailored Codes for Small Quantum Memories”, Physical Review Applied 8, (2017) arXiv:1703.08179 DOI
- [2]
- Q. Xu et al., “Tailored XZZX codes for biased noise”, (2022) arXiv:2203.16486
- [3]
- S. Yu, Q. Chen, and C. H. Oh, “Graphical Quantum Error-Correcting Codes”, (2007) arXiv:0709.1780
- [4]
- A. R. Calderbank et al., “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
- [5]
- S. Bravyi et al., “How much entanglement is needed for quantum error correction?”, (2024) arXiv:2405.01332
- [6]
- A. A. Kovalev, I. Dumer, and L. P. Pryadko, “Design of additive quantum codes via the code-word-stabilized framework”, Physical Review A 84, (2011) arXiv:1108.5490 DOI
- [7]
- E. Rains, private communication, April 1997.
- [8]
- F. Vatan, V. P. Roychowdhury, and M. P. Anantram, “Spatially Correlated Qubit Errors and Burst-Correcting Quantum Codes”, (1997) arXiv:quant-ph/9704019
- [9]
- H. F. Chau, “Five quantum register error correction code for higher spin systems”, Physical Review A 56, R1 (1997) arXiv:quant-ph/9702033 DOI
- [10]
- E. M. Rains, “Nonbinary quantum codes”, (1997) arXiv:quant-ph/9703048
- [11]
- P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI