Here is a list of Floquet codes and other related dynamical codes.

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

Dynamical automorphism (DA) code | Dynamically-generated stabilizer-based code whose (not necessarily periodic) sequence of few-body measurements implements state initialization, logical gates and error detection. |

Floquet color code | Floquet code on a trivalent 2D lattice whose parent topological phase is the \(\mathbb{Z}_2\times\mathbb{Z}_2\) 2D color-code phase and whose measurements cycle logical quantum information between the nine \(\mathbb{Z}_2\) surface-code condensed phases of the parent phase. The code's ISG is the stabilizer group of one of the nine surface codes. |

Fracton Floquet code | Floquet code whose qubits are placed on vertices of a truncated cubic honeycomb. Its weight-two check operators are placed on edges of each truncated cube, while weight-three check operators are placed on each triangle. Its ISG can be that of the X-cube model code or the checkerboard model code. On a three-torus of size \(L_x \times L_y \times L_z\), the code consists of \(n= 48L_xL_yL_z\) physical qubits and encodes \(k= 2(L_x+L_y+L_z)-6\) logical qubits. |

Hastings-Haah Floquet code | DA code whose sequence of check-operator measurements is periodic. The first instance of a dynamical code. |

Honeycomb Floquet code | Floquet code based on the Kitaev honeycomb model [1] whose logical qubits are generated through a particular sequence of measurements. A CSS version of the code has been proposed which loosens the restriction of which sequences to use [2]. The code has also been generalized to arbitrary non-chiral, Abelian topological order [3]. |

Hyperbolic Floquet code | Floquet code whose check-operators correspond to edges of a hyperbolic lattice of degree 3. |

Ladder Floquet code | Floquet code defined on a ladder qubit geometry, with one qubit per vertex. The check operators consist of \(ZZ\) checks on each rung and alternating \(XX\) and \(YY\) check on the legs. |

X-cube Floquet code | Floquet code whose qubits are placed on vertices of a truncated cubic lattice. Its weight-two check operators are placed on various edges. Its ISG can be that of the X-cube model code or that of several decoupled surface codes. |

XYZ ruby Floquet code | Floquet code whose qubits are placed on vertices of a ruby lattice. Its weight-two check operators are placed on various edges. One third of the time during its measurement schedule, its ISG is that of the 6.6.6 color code concatenated with a three-qubit repetition code. Together, all ISGs generate the gauge group of the 3F subsystem code. A Floquet code with the same underlying subsystem code but with a different measurement schedule was developed in Ref. [4]. |

## References

- [1]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
- [2]
- M. Davydova, N. Tantivasadakarn, and S. Balasubramanian, “Floquet Codes without Parent Subsystem Codes”, PRX Quantum 4, (2023) arXiv:2210.02468 DOI
- [3]
- J. Sullivan, R. Wen, and A. C. Potter, “Floquet codes and phases in twist-defect networks”, (2023) arXiv:2303.17664
- [4]
- A. Dua et al., “Engineering 3D Floquet Codes by Rewinding”, PRX Quantum 5, (2024) arXiv:2307.13668 DOI