Here is a list of all quantum codes that admit thresholds other than code capacity thresholds.
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].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 [5].The three dimensional version offers the possibility of being a self-correcting quantum memory [6].
Color code $$\geq 6.25\%$$ threshold for 2D color codes with error-free syndrome extraction, and $$0.1\%$$ with faulty syndrome extraction [7].$$0.46\%$$ for 3D codes with clustering decoder [8].$$1.9\%$$ 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 [8].$$0.31\%$$ noise threshold error rate for gauge code using clustering decoder [9].$$0.2\%$$ with depolarizing circuit-level noise using two flag-qubits per stabilizer generator and the restriction decoder [10].$$0.143\%$$ with depolarizing circuit-level noise using perfect-matching decoder [11].$$>0\%$$ threshold with sweep decoder [12].
Concatenated quantum code The first method to achieve a fault-tolerant computational threshold uses concatenated stabilizer codes [13][14][15][16].
Fibonacci string-net code Between $$10^{-2}\%$$ and $$5\cdot 10^{-2}\%$$ for pair-creation and measurement noise [17].
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 [18].
GKP-stabilizer code Thresholds against displacement noise cannot be obtained without ideal (i.e., non-normalizable) codewords [19].
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.
Higher-dimensional surface 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 [20].$$1.5\%<p<2.0\%$$ in a circuit model with native two-body measurements and a correlated minimum-weight perfect-matching decoder [20]. 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) [21][22].
Kitaev surface code $$1.8\%$$ for circuit-level depolarizing noise under optimal decoder [23]. $$0.57\%$$ for depolarizing noise on data and syndrome qubits as well initialization, gate, and measurement errors under MPWM decoding [24]. 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 $$10^{-4}$$ [25].$$0.35\%$$ for circuit-level independent $$X,Z$$ noise under optimal decoder [23].Phenomenological noise: $$3.3\%$$ for square tiling [26].Phenomenological noise model for the 2D toric code: $$2.93(2)\%$$ using several rounds of syndrome measurement [27].$$0.5-2.9\%$$ for various noise models [28] (see also Refs. [27][29]).
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 [30]
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 [31].There is no threshold for the pentagon HaPPY code as a constant number of errors (two) can make bulk recovery impossible.
Quantum Reed-Muller code Between $$10^{-3}$$ and $$10^{-6}$$ for depolarizing noise (assuming ideal decoders), see [32]
Quantum expander code Current estimate of $$2.7 \cdot 10^{-16}$$ in locally stochastic noise model [33].
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 [34].
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 [13][14][15][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 people can always increase the probability of success by increasing the number of physical bit $$n$$.
Subsystem surface code $$0.81\%$$ threshold for circuit-level depolarizing noise under a variant of MWPM and using gauge-fixing and specific measurement schedules [35], improving the $$0.67\%$$ threshold for standard measurement schedules [36].$$2.22\%$$ threshold for circuit-level infinitely biased noise under a variant of MWPM and using gauge-fixing and specific measurement schedules [35], improving the $$0.52\%$$ threshold with standarn measurement schedules.
Triorthogonal code Approximately $$\frac{1}{3k + 1}$$ [37].
Two-dimensional hyperbolic surface code 1$$\%$$ - 5$$\%$$ for a $${5,4}$$ tiling under minimum-weight decoding [38]. 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 [39].
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]
A. O. Quintavalle et al., “Single-Shot Error Correction of Three-Dimensional Homological Product Codes”, PRX Quantum 2, (2021). DOI; 2009.11790
[2]
Oscar Higgott and Nikolas P. Breuckmann, “Improved single-shot decoding of higher dimensional hypergraph product codes”. 2206.03122
[3]
T. R. Scruby and K. Nemoto, “Local Probabilistic Decoding of a Quantum Code”. 2212.06985
[4]
N. C. Brown, M. Newman, and K. R. Brown, “Handling leakage with subsystem codes”, New Journal of Physics 21, 073055 (2019). DOI; 1903.03937
[5]
P. Aliferis and A. W. Cross, “Subsystem Fault Tolerance with the Bacon-Shor Code”, Physical Review Letters 98, (2007). DOI; quant-ph/0610063
[6]
D. Bacon, “Operator quantum error-correcting subsystems for self-correcting quantum memories”, Physical Review A 73, (2006). DOI; quant-ph/0506023
[7]
D. S. Wang et al., “Graphical algorithms and threshold error rates for the 2d colour code”. 0907.1708
[8]
A. Kubica et al., “Three-Dimensional Color Code Thresholds via Statistical-Mechanical Mapping”, Physical Review Letters 120, (2018). DOI; 1708.07131
[9]
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
[10]
C. Chamberland et al., “Triangular color codes on trivalent graphs with flag qubits”, New Journal of Physics 22, 023019 (2020). DOI; 1911.00355
[11]
Andrew J. Landahl and Ciaran Ryan-Anderson, “Quantum computing by color-code lattice surgery”. 1407.5103
[12]
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
[13]
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
[14]
Dorit Aharonov and Michael Ben-Or, “Fault-Tolerant Quantum Computation With Constant Error Rate”. quant-ph/9906129
[15]
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
[16]
Panos Aliferis, Daniel Gottesman, and John Preskill, “Quantum accuracy threshold for concatenated distance-3 codes”. quant-ph/0504218
[17]
Alexis Schotte, Lander Burgelman, and Guanyu Zhu, “Fault-tolerant error correction for a universal non-Abelian topological quantum computer at finite temperature”. 2301.00054
[18]
Hector Bombin et al., “Logical blocks for fault-tolerant topological quantum computation”. 2112.12160
[19]
L. Hanggli and R. Konig, “Oscillator-to-Oscillator Codes Do Not Have a Threshold”, IEEE Transactions on Information Theory 68, 1068 (2022). DOI; 2102.05545
[20]
C. Gidney et al., “A Fault-Tolerant Honeycomb Memory”, Quantum 5, 605 (2021). DOI; 2108.10457
[21]
C. Gidney, M. Newman, and M. McEwen, “Benchmarking the Planar Honeycomb Code”, Quantum 6, 813 (2022). DOI; 2202.11845
[22]
Adam Paetznick et al., “Performance of planar Floquet codes with Majorana-based qubits”. 2202.11829
[23]
Bettina Heim, Krysta M. Svore, and Matthew B. Hastings, “Optimal Circuit-Level Decoding for Surface Codes”. 1609.06373
[24]
A. G. Fowler et al., “Surface codes: Towards practical large-scale quantum computation”, Physical Review A 86, (2012). DOI; 1208.0928
[25]
A. G. Fowler, “Proof of Finite Surface Code Threshold for Matching”, Physical Review Letters 109, (2012). DOI; 1206.0800
[26]
T. Ohno et al., “Phase structure of the random-plaquette gauge model: accuracy threshold for a toric quantum memory”, Nuclear Physics B 697, 462 (2004). DOI; quant-ph/0401101
[27]
C. Wang, J. Harrington, and J. Preskill, “Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory”, Annals of Physics 303, 31 (2003). DOI; quant-ph/0207088
[28]
M. Ohzeki, “Locations of multicritical points for spin glasses on regular lattices”, Physical Review E 79, (2009). DOI; 0811.0464
[29]
A. M. Stephens, “Fault-tolerant thresholds for quantum error correction with the surface code”, Physical Review A 89, (2014). DOI; 1311.5003
[30]
M. J. Gullans and D. A. Huse, “Dynamical Purification Phase Transition Induced by Quantum Measurements”, Physical Review X 10, (2020). DOI; 1905.05195
[31]
R. J. Harris et al., “Decoding holographic codes with an integer optimization decoder”, Physical Review A 102, (2020). DOI; 2008.10206
[32]
L. Luo et al., “Fault-tolerance thresholds for code conversion schemes with quantum Reed–Muller codes”, Quantum Science and Technology 5, 045022 (2020). DOI
[33]
O. Fawzi, A. Grospellier, and A. Leverrier, “Efficient decoding of random errors for quantum expander codes”, Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (2018). DOI; 1711.08351
[34]
Daniel Gottesman, “Fault-Tolerant Quantum Computation with Constant Overhead”. 1310.2984
[35]
O. Higgott and N. P. Breuckmann, “Subsystem Codes with High Thresholds by Gauge Fixing and Reduced Qubit Overhead”, Physical Review X 11, (2021). DOI; 2010.09626
[36]
Sergey Bravyi et al., “Subsystem surface codes with three-qubit check operators”. 1207.1443
[37]
S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012). DOI; 1209.2426
[38]
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
[39]
D. K. Tuckett et al., “Fault-Tolerant Thresholds for the Surface Code in Excess of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>5</mml:mn><mml:mo>%</mml:mo></mml:math> Under Biased Noise”, Physical Review Letters 124, (2020). DOI; 1907.02554