Here is a list of quantum dynamically generated codes.

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

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 | Code consisting of cluster states [1], which are stabilizer states defined on a graph. There is one stabilizer generator \(S_v\) per graph vertex \(v\) of the form \begin{align} S_v = X_{v} \prod_{w\in N(v)} Z_w~, \tag*{(1)}\end{align} where the neighborhood \(N(v)\) is the set of vertices which share an edge with \(v\). |

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 | Also called the CSS Floquet code. Floquet code on a trivalent 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 three \(\mathbb{Z}_2\) surface-code condensed phases of the parent phase. The code's ISG is the stabilizer group of one of the three surface codes. |

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. |

Honeycomb Floquet code | Floquet code based on the Kitaev honeycomb model [3] whose logical qubits are generated through a particular sequence of measurements. |

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. |

Low-depth random Clifford-circuit qubit code | An \([[n,k]]\) qubit stabilizer code whose encoder is an \(n\)-qubit unitary transformation that takes a \(k\)-qubit state as input (with \(k\leq n\), and the remaining \(n-k\) qubits initialized to \(|0\rangle^{\otimes n-k}\) ) to give a corresponding state in the codespace as the output. An \(n\)-qubit quantum circuit with random two-qubit Clifford gates can act as an encoder into a code with distance \(d\) with high probability, with a size (i.e. number of gates in the circuit) at most \(O(n^2 log n)\)). Noting that two gates acting on disjoint qubits could in fact be executed simultaneously, this is equivalent to the depth (number of time steps in the circuit) being at most \(O(log^3 n)\). |

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 [4]. 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 [5,6]. |

Quantum low-density parity-check (QLDPC) code | Also called a sparse quantum 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 as \(n\to\infty\). A geometrically local stabilizer code is a QLDPC code where the sites involved in any syndrome bit are contained in a fixed volume that does not scale with \(n\). As opposed to general stabilizer codes, syndrome extraction of the constant-weight check operators of a QLDPC codes can be done using a constant-depth circuit. |

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. |

## References

- [1]
- H. J. Briegel and R. Raussendorf, “Persistent Entanglement in Arrays of Interacting Particles”, Physical Review Letters 86, 910 (2001) arXiv:quant-ph/0004051 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]
- 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
- [5]
- S. Choi et al., “Quantum Error Correction in Scrambling Dynamics and Measurement-Induced Phase Transition”, Physical Review Letters 125, (2020) arXiv:1903.05124 DOI
- [6]
- M. J. Gullans and D. A. Huse, “Dynamical Purification Phase Transition Induced by Quantum Measurements”, Physical Review X 10, (2020) arXiv:1905.05195 DOI