Name | Threshold |
---|---|

Calderbank-Shor-Steane (CSS) stabilizer code | Bounds on code capacity thresholds for various noise models exist in terms of stabilizer generator weights [1][2]. |

Clifford-deformed surface code (CDSC) | A class of random CDSCs, parametrized by the probabilities \(\Pi_{XZ},~ \Pi_{YZ}\) of \(X\leftrightarrow Z\) and \(Y\leftrightarrow Z\) Pauli permutations, respectively, has \(50\%\) code capacity threshold at infinite \(Z\) bias.Certain translation-invariant CDSCs such as the XY code and the XZZX code also have \(50\%\) code capacity threshold at infinite \(Z\) bias.XZZX code and the \((0.5,\Pi_{YZ})\) random CDSCs have a \(50\%\) code capacity threshold for noise infinitely biased towards either Pauli-\(X\), \(Y\), or \(Z\) errors. |

Color code | \(\geq 6\%\) threshold with rescaling-based decoder [3]. |

Holographic code | The ideal holographic code (perfect representation of AdS/CFT) should be able to protect a central bulk operator against erasures of half of the physical qubits on the boundary, in line with AdS-Rindler reconstruction [4].Holographic codes are argued to have a algebraic threshold, for which the error rate scales polynomially (as opposed to exponentially) in the thermodynamic limit [5]. Such a threshold is governed by the underlying conformal field theory describing the boundary. |

Kitaev surface code | Independent \(X,Z\) noise: \(10.31\%\) under MWPM decoding [6] (see also Ref. [7]). The threshold under ML decoding corresponds to the value of critical point of the two-dimensional random-bond Ising model on the Nishimori line [8][9], calculated to be \(10.94 \pm 0.02\%\) in Ref. [10], \(10.93(2)\%\) in Ref. [11], and estimated to be between \(10.9\%\) and \(11\%\) in Ref. [7].Depolarizing noise: between \(17\%\) and \(18.5\%\) under tensor-network decoding [7], and between \(15\%\) and \(16\%\) under RG [12], Markov-chain [13], or MWPM [14] decoding. The threshold under ML decoding corresponds to the value of critical point in the disordered eight-vertex Ising model, calculated to be \(18.9(3)\%\) [15] (see also APS Physics viewpoint [16]).Erasure noise: \(50\%\) for square tiling [17]. There is an inverse relationship between coordination number of the syndrome graph, with the threshold corresponding to a percolation transition [18].Phenomenological noise: \(3.3\%\) for square tiling [19]. |

Quantum low-density parity-check (QLDPC) code | Bounds on code capacity thresholds using maximum-likelihood (ML) decoding can be obtained by mapping the effect of noise on the code to a statistical mechanical model [9][1][20].Bounds on code capacity thresholds for various noise models exist in terms of stabilizer generator weights [2]. |

Qubit stabilizer code | Bounds on code capacity thresholds using maximum-likelihood (ML) decoding can be obtained by mapping the effect of noise on the code to a statistical mechanical model [9][1][20][21]. |

Subsystem qubit stabilizer code | For correlated Pauli noise, bounds can be obtained by mapping the effect of noise on the code to a statistical mechanical model [21]. |

Subsystem surface code | Independent \(X,Z\) noise: the threshold under ML decoding corresponds to the value of critical point of the two-dimensional hexagonal-lattice random-bond Ising model on the Nishimori line [8][22], calculated to be around \(7\%\) in Ref. [23]. |

XY surface code | \(50\%\) at infinite \(Z\) bias with maximum-likelihood decoder [24].\(18.7\%\) for standard depolarising noise with maximum-likelihood decoder [24]. |

XYZ\(^2\) hexagonal stabilizer code | \(50\%\) for pure \(Z\), \(Y\), or \(Z\) noise under maximum-likelihood decoding.Threshold matches that of the \(XZZX\) code for various bias levels of \(X\), \(Y\), or \(Z\) biased noise under maximum-likelihood decoding.\(\sim 18\%\) for depolarizing noise under maximum-likelihood decoding. |

XZZX surface code | \(50\%\) threshold for noise infinitely biased towards \(X\) or \(Y\) or \(Z\) errors using a maximum-likelihood decoder.For large but finite \(X\)- or \(Z\)-biased noise, the code's thresholds exceed the zero-rate hashing bound. The difference of the threshold from the hashing bound exceeds \(2.9\%\) at a \(Z\) or \(X\) bias of 300.\(18.7\%\) for standard depolarising noise with maximum-likelihood decoder.\(0.817\%\) and \(0.940\%\) with minimum-weight perfect matching and belief-matching decoder, respectively, for biased circuit-level noise [25].\(4.15\%\) when \(98\%\) of depolarizing errors are coverted into erasure errors with union-find decoder on a planar code, vs. \(0.937\%\) for pure depolarizing noise. In Rydberg atomic devices, erasure conversion during gates is promising because the dominant source of noise is spontaneous decay into detectable energy levels outside of the computational subspace [26]. |

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