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

Bacon-Shor code | 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 [1].The three dimensional version offers the possibility of being a self-correcting quantum memory [2]. |

Binary 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 people can always increase the probability of success by increasing the number of physical bit \(n\). |

Color code | \(\geq 6.25\%\) threshold for 2D color codes with error-free syndrome extraction, and \(0.1\%\) with faulty syndrome extraction [3].\(0.46\%\) for 3D codes with clustering decoder [4].\(1.9\%\) 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 [4].\(0.31\%\) noise threshold error rate for gauge code using clustering decoder [5].\(0.143\%\) with depolarising circuit-level noise using perfect-matching decoder [6].\(>0\%\) threshold with sweep decoder [7]. |

Concatenated quantum code | The first method to achieve a fault-tolerant computational threshold uses concatenated stabilizer codes [8][9][10][11]. |

Fibonacci string-net code | \(4.7\%\) for depolarizing noise, \(7.3\%\) for dephasing noise, and \(3.8\%\) for bit-flip noise with clustering decoder, assuming perfect measurements and gates [12].\(3.0\%\) for depolarizing noise, \(6.0\%\) for dephasing noise, and \(2.5\%\) for bit-flip noise with fusion-aware iterative MWPM decoder, assuming perfect measurements and gates [12]. |

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. |

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 \(k \sim L\). |

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. |

Honeycomb code | \(0.2\%-0.3\%\) in a controlled-not circuit model with a correlated minimum-weight perfect-matching decoder [13].\(1.5\%<p<2.0\%\) in a circuit model with native two-body measurements and a correlated minimum-weight perfect-matching decoder [13]. 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) [14]. |

Kitaev surface code | \(0.5-1.1\%\) for various error models [15][16]. |

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 [17] |

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 [18].There is no threshold for the pentagon HaPPY code as a constant number of errors (two) can make bulk recovery impossible. |

Polar code | Achieves Shannon capacity of the binary-input memoryless channel under successive cancellation decoder [19]. |

Quantum Reed-Muller code | Between \(10^{-3}\) and \(10^{-6}\) for depolarizing noise (assuming ideal decoders), see [20] |

Quantum expander code | Current estimate of \(2.7 \cdot 10^{-16}\) in locally stochastic noise model [21]. |

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 [22]. |

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 [8][9][10][11]. 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. |

Triorthogonal code | Approximately \(\frac{1}{3k + 1}\) [23]. |

Two-dimensional hyperbolic surface code | 1\(\%\) - 5\(\%\) for a \({5,4}\) tiling under minimum-weight decoding [24]. 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 [25]. |

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. |

## References

- [1]
- P. Aliferis and A. W. Cross, “Subsystem Fault Tolerance with the Bacon-Shor Code”, Physical Review Letters 98, (2007). DOI; quant-ph/0610063
- [2]
- D. Bacon, “Operator quantum error-correcting subsystems for self-correcting quantum memories”, Physical Review A 73, (2006). DOI; quant-ph/0506023
- [3]
- D. S. Wang et al., “Graphical algorithms and threshold error rates for the 2d colour code”. 0907.1708
- [4]
- A. Kubica et al., “Three-Dimensional Color Code Thresholds via Statistical-Mechanical Mapping”, Physical Review Letters 120, (2018). DOI; 1708.07131
- [5]
- B. J. Brown, N. H. Nickerson, and D. E. Browne, “Fault-tolerant error correction with the gauge color code”, Nature Communications 7, (2016). DOI; 1503.08217
- [6]
- Andrew J. Landahl and Ciaran Ryan-Anderson, “Quantum computing by color-code lattice surgery”. 1407.5103
- [7]
- A. M. Kubica, The Abcs of the Color Code: A Study of Topological Quantum Codes as Toy Models for Fault-tolerant Quantum Computation and Quantum Phases of Matter, California Institute of Technology, 2018. DOI
- [8]
- E. Knill, R. Laflamme, and W. H. Zurek, “Resilient quantum computation: error models and thresholds”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 365 (1998). DOI; quant-ph/9702058
- [9]
- Dorit Aharonov and Michael Ben-Or, “Fault-Tolerant Quantum Computation With Constant Error Rate”. quant-ph/9906129
- [10]
- J. Preskill, “Reliable quantum computers”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 385 (1998). DOI; quant-ph/9705031
- [11]
- Panos Aliferis, Daniel Gottesman, and John Preskill, “Quantum accuracy threshold for concatenated distance-3 codes”. quant-ph/0504218
- [12]
- Alexis Schotte et al., “Quantum error correction thresholds for the universal Fibonacci Turaev-Viro code”. 2012.04610
- [13]
- C. Gidney et al., “A Fault-Tolerant Honeycomb Memory”, Quantum 5, 605 (2021). DOI; 2108.10457
- [14]
- Craig Gidney and Michael Newman, “Benchmarking the Planar Honeycomb Code”. 2202.11845
- [15]
- A. M. Stephens, “Fault-tolerant thresholds for quantum error correction with the surface code”, Physical Review A 89, (2014). DOI; 1311.5003
- [16]
- M. Ohzeki, “Locations of multicritical points for spin glasses on regular lattices”, Physical Review E 79, (2009). DOI; 0811.0464
- [17]
- M. J. Gullans and D. A. Huse, “Dynamical Purification Phase Transition Induced by Quantum Measurements”, Physical Review X 10, (2020). DOI; 1905.05195
- [18]
- R. J. Harris et al., “Decoding holographic codes with an integer optimization decoder”, Physical Review A 102, (2020). DOI; 2008.10206
- [19]
- E. Arikan, “Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels”, IEEE Transactions on Information Theory 55, 3051 (2009). DOI
- [20]
- L. Luo et al., “Fault-tolerance thresholds for code conversion schemes with quantum Reed–Muller codes”, Quantum Science and Technology 5, 045022 (2020). DOI
- [21]
- Omar Fawzi, Antoine Grospellier, and Anthony Leverrier, “Efficient decoding of random errors for quantum expander codes”. 1711.08351
- [22]
- Daniel Gottesman, “Fault-Tolerant Quantum Computation with Constant Overhead”. 1310.2984
- [23]
- S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012). DOI; 1209.2426
- [24]
- A. A. Kovalev and L. P. Pryadko, “Fault tolerance of quantum low-density parity check codes with sublinear distance scaling”, Physical Review A 87, (2013). DOI; 1208.2317
- [25]
- D. K. Tuckett et al., “Fault-Tolerant Thresholds for the Surface Code in Excess of 5% Under Biased Noise”, Physical Review Letters 124, (2020). DOI; 1907.02554