Here is a list of codes related to dynamically generated codes.

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

Brown-Fawzi random Clifford-circuit code | An \([[n,k]]\) stabilizer code whose encoder is a random Clifford circuit of depth \(O(\log^3 n)\). |

Clifford-deformed surface code (CDSC) | A generally non-CSS derivative of the surface code defined by applying a constant-depth Clifford circuit to the original (CSS) surface code. Unlike the surface code, CDSCs include codes whose thresholds and subthreshold performance are enhanced under noise biased towards dephasing. Examples of CDSCs include the XY code, XZZX code, and random CDSCs. |

Cluster-state code | A code based on a cluster state and often used in measurement-based quantum computation (MBQC) [1] (a.k.a. one-way quantum processing), which substitutes the temporal dimension necessary for decoding a conventional code with a spatial dimension. This is done by encoding the computation into the features of the cluster state''s graph. |

Crystalline-circuit qubit code | Code dynamically generated by unitary Clifford circuits defined on a lattice with some crystalline symmetry. A notable example is the circuit defined on a rotated square lattice with vertices corresponding to iSWAP gates and edges decorated by \(R_X[\pi/2]\), a single-qubit rotation by \(\pi/2\) around the \(X\)-axis. This circuit is invariant under space-time translations by a unit cell \((T, a)\) and all transformations of the square lattice point group \(D_4\). |

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. |

Dynamically-generated QECC | Block quantum code whose natural definition is in terms of a many-body scaling limit of a local dynamical process. Such processes, which are often non-deterministic, update the code structure and can include random unitary evolution or non-commuting projective measurements. |

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. |

Fusion-based quantum computing (FBQC) code | Code whose codewords are resource states used in an FBQC scheme. Related to a cluster state via Hadamard transformations. |

Haar-random qubit code | Haar-random codewords are generated in a process involving averaging over unitary operations distributed accoding to the Haar measure. Haar-random codes are used to prove statements about the capacity of a quantum channel to transmit quantum information [2], but encoding and decoding in such \(n\)-qubit codes quickly becomes impractical as \(n\to\infty\). |

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 [3] 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 [4]. The code has also been generalized to arbitrary non-chiral, Abelian topological order [5]. |

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. |

Local Haar-random circuit qubit code | An \(n\)-qubit code whose codewords are a pair of approximately locally indistinguishable states produced by starting with any two orthogonal \(n\)-qubit states and acting with a random unitary circuit of depth polynomial in \(n\). Two states are locally indistinguishable if they cannot be distinguished by local measurements. A single layer of the encoding circuit is composed of about \(n/2\) two-qubit nearest-neighbor gates run in parallel, with each gate drawn randomly from the Haar distribution on two-qubit unitaries. |

Monitored random-circuit code | Error-correcting code arising from a monitored random circuit. Such a circuit is described by a series of intermittant random local projective Pauli measurements with random unitary time-evolution operators. An important sub-family consists of Clifford monitored random circuits, where unitaries are sampled from the Clifford group [6]. When the rate of projective measurements is independently controlled by a probability parameter \(p\), there can exist two stable phases, one described by volume-law entanglement entropy and the other by area-law entanglement entropy. The phases and their transition can be understood from the perspective of quantum error correction, information scrambling, and channel capacities [7,8]. |

Quantum LDPC (QLDPC) code | Member of a family of \([[n,k,d]]\) modular-qudit or Galois-qudit stabilizer codes for which the number of sites participating in each stabilizer generator and the number of stabilizer generators that each site participates in are both bounded by a constant \(w\) as \(n\to\infty\); can be denoted by \([[n,k,d,w]]\). Sometimes, the two parameters are explicitly stated: each site of an an \((l,w)\)-regular QLDPC code is acted on by \(\leq l\) generators of weight \(\leq w\). QLDPC codes can correct many stochastic errors far beyond the distance, which may not scale as favorably. Together with more accurate, faster, and easier-to-parallelize measurements than those of general stabilizer codes, this property makes QLDPC codes interesting in practice. |

Random stabilizer code | An \(n\)-qubit, modular-qudit, or Galois-qudit stabilizer code whose construction is non-deterministic. Since stabilizer encoders are Clifford circuits, such codes can be thought of as arising from random Clifford circuits. |

Random-circuit code | Code whose encoding is naturally constructed by randomly sampling from a large set of (not necessarily unitary) quantum circuits. |

Spacetime circuit code | Qubit stabilizer code used to correct faults in Clifford circuits, i.e., circuits up made of Clifford gates and Pauli measurements. The code utilizes redundancy in the measurement outcomes of a circuit to correct circuit faults, which correspond to Pauli errors of the code. |

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. [9]. |

## References

- [1]
- R. Raussendorf, D. Browne, and H. Briegel, “The one-way quantum computer--a non-network model of quantum computation”, Journal of Modern Optics 49, 1299 (2002) arXiv:quant-ph/0108118 DOI
- [2]
- “Preface to the Second Edition”, Quantum Information Theory xi (2016) arXiv:1106.1445 DOI
- [3]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
- [4]
- M. Davydova, N. Tantivasadakarn, and S. Balasubramanian, “Floquet Codes without Parent Subsystem Codes”, PRX Quantum 4, (2023) arXiv:2210.02468 DOI
- [5]
- J. Sullivan, R. Wen, and A. C. Potter, “Floquet codes and phases in twist-defect networks”, (2023) arXiv:2303.17664
- [6]
- Y. Li, X. Chen, and M. P. A. Fisher, “Measurement-driven entanglement transition in hybrid quantum circuits”, Physical Review B 100, (2019) arXiv:1901.08092 DOI
- [7]
- S. Choi et al., “Quantum Error Correction in Scrambling Dynamics and Measurement-Induced Phase Transition”, Physical Review Letters 125, (2020) arXiv:1903.05124 DOI
- [8]
- M. J. Gullans and D. A. Huse, “Dynamical Purification Phase Transition Induced by Quantum Measurements”, Physical Review X 10, (2020) arXiv:1905.05195 DOI
- [9]
- A. Dua et al., “Engineering 3D Floquet Codes by Rewinding”, PRX Quantum 5, (2024) arXiv:2307.13668 DOI