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

3D surface code | Phenomenological noise model for the 3D toric code: \(2.90(2)\%\) under BP-OSD decoder [1], \(7.1\%\) under improved BP-OSD [2], and \(2.6\%\) under flip decoder [3]. For 3D surface code: \(3.08(4)\%\) under flip decoder [1]. |

Bacon-Shor code | The number of check operators scales sublinearly with system size, so the Bacon-Shor codes alone do not exhibit a threshold [4]. However, a threshold can be obtained from concatenated Bacon-Shor codes restricted to planar geometries, whose recovery circuit is a subset of a circuit used by a larger bona-fide Bacon-Shor code [5].A lower bound of \(1.94 \times 10^{-4}\) for the accuracy threshold was proved for Bacon-Shor code with 5 levels of concatenation, using Steane method of FTEC [6].The three dimensional version offers the possibility of being a self-correcting quantum memory [7].The Bacon-Shor code has a measurement threshold of zero [8]. |

Bivariate bicycle code | \(0.8\%\) pseudothreshold for circuit-level noise under BP-OSD decoder [9] (cf. [10]). |

Concatenated quantum code | The first methods to achieve a fault-tolerant computational threshold use concatenated stabilizer codes [11–17]. Such methods require constant-space and polylogarithmic time overhead, but concatentions using quantum Hamming codes improve this to quasi-polylogarithmic time [18]. |

Fibonacci string-net code | Between \(10^{-2}\%\) and \(5\cdot 10^{-2}\%\) for pair-creation and measurement noise [19]. |

Fusion-based quantum computing (FBQC) code | \(11.98\%\) against erasure in fusion measurements.\(1.07\%\) against Pauli error.In linear optical systems, can tolerate \(10.4\%\) probability of photon loss in each fusion.\(43.2\%\) against fusion failure.FBQC applied to the surface code yields thresholds for logical gates that is consistent with the code capacity threshold [20]. |

GKP-stabilizer code | Thresholds against displacement noise cannot be obtained without ideal (i.e., non-normalizable) codewords [21]. |

Haah cubic code | The encoding rate depends on the code implemented, but code 0 has been shown to have \(k \ge L\) (on a periodic finite cubic lattice of side length \(L\). In general we expect the number of logical bits to scale as \(k \sim L\). |

Haar-random qubit code | Haar-random qubit codes have a measurement threshold of one [8]. |

Heavy-hexagon code | \(0.45\%\) for \(X\) errors under a full circuit-level depolarizing noise model (obtained from Monte Carlo simulations).\(Z\)-errors have no threshold given the \(X\)-type Bacon-Shor stabilizers. |

Hierarchical code | Threshold exists for the locally decaying error model; see [22; Thm. 1.3]. However, the logical error rate below threshold falls super-polynomially (as opposed to exponentially) with the code distance. The code family possesses a threshold equal to that of surface codes given by tuning the inner code size for any fixed physical error rate. |

Homological code | Phenomenological noise model for the 4D toric code: \(4.3\%\) under improved BP-OSD decoder [2]. |

Honeycomb Floquet code | \(0.2\%-0.3\%\) in a controlled-not circuit model with a correlated minimum-weight perfect-matching decoder [23].\(1.5\%<p<2.0\%\) in a circuit model with native two-body measurements and a correlated minimum-weight perfect-matching decoder [23]. Here, \(p\) is the collective error rate of the two-body measurement gate, including both measurement and correlated data depolarization error processes.Against circuit-level noise: within \(0.2\% − 0.3\%\) for SD6 (standard depolarizing 6-step cycle), \(0.1\% − 0.15\%\) for SI1000 (superconducting-inspired 1000 ns cycle), and \(1.5\% − 2.0\%\) for EM3 (entangling-measurement 3-step cycle) [24,25]. |

Hyperbolic Floquet code | \(0.1\%\) under standard circuit-level depolarising noise [26].\(0.1\%\) under phenomenological error model including depolarizing and measurement errors for the octagonal codes [27]. |

Hypergraph product (HGP) code | Circuit-level noise: \(0.1\%\) with all-to-all connected syndrome extraction circuits [28]. No threshold observed above physical noise rates at or above \(10^{-6}\) using 2D geometrically local syndrome extraction circuits. |

Kitaev surface code | \(1.8\%\) for circuit-level depolarizing noise under optimal decoder [29]. \(0.57\%\) for depolarizing noise on data and syndrome qubits as well initialization, gate, and measurement errors under MPWM decoding [30]. For this model, a logical qubit with a \(10^{-14}\) logical error rate requires between \(10^3\) to \(10^4\) physical qubits and a target gate fidelity above \(99.9\%\). Later work showed that arbitrarily large computations are possible for a physical error rate of approximately \(10^{-4}\) [31].\(0.35\%\) for circuit-level independent \(X,Z\) noise under optimal decoder [29].Phenomenological noise: \(3.3\%\) for square tiling [32].Phenomenological noise model for the 2D toric code: \(2.93(2)\%\) using several rounds of syndrome measurement [33].\(0.5-2.9\%\) for various noise circuit-level noise models [10,34] (see also Refs. [33,35]).Quasistatic phase damping and readout noise: \(2.85\%\) [36].The toric code has a measurement threshold of one [8]. |

Lift-connected surface (LCS) code | \(6.7\%\) and \(7.7\%\) under bit-flip noise and BP+OSD decoding for two families of LCS codes. |

Monitored random-circuit code | Above the critical measurement rate \( p_c\), the natural error correction properties of the circuit can no longer protect the information. This can be interpreted as the code threshold.These dynamically generated codes saturate the trade off between density of encoded information and the error rate threshold [37] |

Pastawski-Yoshida-Harlow-Preskill (HaPPY) code | \(26\%\) for boundary erasure errors on the the pentagon/hexagon HaPPY code, which has alternating layers of pentagons and hexagons in the tiling.\(\sim 50\%\) for boundary erasure errors on the single-qubit HaPPY code, which has a central pentagon encoding one bulk operator and hexagons tiling all other layers.\(16.3\%\) for boundary Pauli errors on the single-qubit HaPPY code with 3 layers [38].There is no threshold for the pentagon HaPPY code as a constant number of errors (two) can make bulk recovery impossible.The single-qubit HaPPY code has a measurement threshold of one [8] (see also [39]). |

Quantum Reed-Muller code | Between \(10^{-3}\) and \(10^{-6}\) for depolarizing noise (assuming ideal decoders), see [40] |

Quantum expander code | Locally stochastic noise: \(2.7 \cdot 10^{-16}\) [41]. |

Quantum low-density parity-check (QLDPC) code | QLDPC codes with a constant encoding rate can reduce the overhead of fault-tolerant quantum computation to be constant [42]. |

Qubit code | Computational threshold: A fault-tolerant computational threshold is the maximum noise rate in a noise model below which any logical computation of size \(M\) can be executed on a physical-qubit architecture to arbitrary accuracy and with an overhead of order \(O(M\text{polylog}M)\). The first methods to achieve a computational threshold use concatenated stabilizer codes [11–17]. Such methods require constant-space and polylogarithmic-time overhead, but concatentions using quantum Hamming codes improve this to quasi-polylogarithmic time [18]. Fault-tolerant computations with no notion of locality can be made local on a 2D or 3D geometry with minimial overhead [43]. Measurement threshold: One can derive conditions quantifying how many random single-qubit measurements can be made without destroying the logical information [8]. The measurement threshold is the maximum total probability that a single qubit is measured in a random \(X\), \(Y\), or \(Z\) basis at which the logical information is still recoverable. The measurement threshold is at least as large as the erasure threshold [8; Thm. 4]. |

Qubit stabilizer code | Computational thresholds against stochastic local noise can be achieved through repeated use of concatenatenation, and can rely on the same small code in every level [11,12,14,16]. The resulting code is highly degenerate, with all but an exponentially small fraction of generators having small weights. Circuit and measurement designs have to take case of the few stabilizer generators with large weights in order to be fault tolerant. |

Repetition code | Suppose each bit has probability \(p\) of being received correctly, independent for each bit. The probability that a repetition code is received correctly is \(\sum_{k=0}^{(n-1)/2}\frac{n!}{k!(n-k)!}p^{n-k}(1-p)^{k}\). If \(\frac{1}{2}\leq p\), then one can always increase the probability of success by increasing the number of physical bits \(n\); see section 2.2.1 Ref. [44] for a pedagogical explanation. |

Single-shot code | Residual errors do not become unwieldy after some system-size-independent number cycles of faulty syndrome measurements, and a perfect decoder would be able to recover the information if the final residual error is correctable. Consider acting on a state \(\rho\) with a noise channel \(\mathcal N\) with noise rate \(p\), followed by \(t\) rounds of faulty syndrome measurements \(\mathcal R\) with noise rate \(\eta\) and one perfect recovery (which can be substituted with destructive physical-qubit measurements in practice). The failure probability of a single-shot code should decrease exponentially with the distance of the code, \begin{align} fp_{\text{fail}} =1-F\left(\mathcal{R}[\mathcal{R}_{\eta}\mathcal{N}_{p}]^{t}(\rho),\rho\right) =t\left(p/p_{\star}\right)^{d}~, \tag*{(1)}\end{align} where \(F\) is a state fidelity, and where \(p_{\star}\) is called the sustainable threshold [1]. For any \(p\) below this threshold, some maximum measurement noise \(\eta_{\star}>0\) can be tolerated after sufficiently large \(t\). The final ideal decoding step \(\mathcal{R}\) cannot be done non-destructively in practice due to noisy syndrome measurements, but information can still be recovered by measuring all logical qubits in the computational basis and correcting the outcomes. If the code is single-shot, then such a procedure will output the correct logical information. |

Subsystem surface code | \(0.81\%\) threshold for circuit-level depolarizing noise under a variant of MWPM and using gauge-fixing and specific measurement schedules [45], improving the \(0.67\%\) threshold for standard measurement schedules [46].\(2.22\%\) threshold for circuit-level infinitely biased noise under a variant of MWPM and using gauge-fixing and specific measurement schedules [45], improving the \(0.52\%\) threshold with standarn measurement schedules. |

Three-dimensional color code | \(0.46\%\) for 3D codes with clustering decoder [47].\(1.9\%\) for 1D string-like logical operators and \(27.6\%\) for 2D sheet-like operators for 3D codes with noise models using optimal decoding and perfect measurements [47]. |

Triangular color code | The honeycomb color code appears to have a measurement threshold of one [8]. |

Two-dimensional color code | \(\geq 6.25\%\) threshold for 2D color codes with error-free syndrome extraction, and \(0.1\%\) with faulty syndrome extraction [48].\(0.2\%\) with depolarizing circuit-level noise using two flag-qubits per stabilizer generator and the restriction decoder [49].\(0.143\%\) with depolarizing circuit-level noise using perfect-matching decoder [50].\(>0\%\) threshold with sweep decoder [51]. |

Two-dimensional hyperbolic surface code | 1\(\%\) - 5\(\%\) for a \({5,4}\) tiling under minimum-weight decoding [52]. For larger tilings, the lower bound on the distance decreases, suggesting the threshold will also decrease. |

XY surface code | \(6.32(3)\%\) for infinite \(Z\) bias, and thresholds of \(\sim 5\%\) for \(Z\) bias around \(\eta = 100\) using a variant of the minimum-weight perfect matching decoder [53]. |

XZZX surface code | \(\sim 4.5\%\) using minimum-weight perfect matching decoder for depolarizing noise (bias \(\eta=0.5\)); \(\sim 10\%\) for infinite \(Z\) bias. |

\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code | Concatenated thresholds requiring constant-space and quasi-polylogarithmic time overhead [18]. |

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