Here is a list of all quantum codes that admit code capacity thresholds.
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], calculated to be \(10.94 \pm 0.02\%\) in Ref. [9], \(10.93(2)\%\) in Ref. [10], 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 [11], Markov-chain [12], or MWPM [13] decoding. The threshold under ML decoding corresponds to the value of critical point in the disordered eight-vertex Ising model, calculated to be \(18.9(3)\%\) [14] (see also APS Physics viewpoint [15]).Erasure noise: \(50\%\) for square tiling [16].Phenomenological noise: \(3.3\%\) for square tiling [17].
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 [8][1][18].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 [8][1][18][19].
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 [19].
XY surface code \(50\%\) at infinite \(Z\) bias with maximum-likelihood decoder [20].\(18.7\%\) for standard depolarising noise with maximum-likelihood decoder [20].
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 [21].\(4.15\%\) 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 [22].

References

[1]
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
[2]
I. Dumer, A. A. Kovalev, and L. P. Pryadko, “Thresholds for Correcting Errors, Erasures, and Faulty Syndrome Measurements in Degenerate Quantum Codes”, Physical Review Letters 115, (2015). DOI; 1412.6172
[3]
Pedro Parrado-Rodríguez, Manuel Rispler, and Markus Müller, “Rescaling decoder for 2D topological quantum color codes on 4.8.8 lattices”. 2112.09584
[4]
F. Pastawski et al., “Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence”, Journal of High Energy Physics 2015, (2015). DOI; 1503.06237
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Ning Bao, Charles Cao, and Guanyu Zhu, “Deconfinement and Error Thresholds in Holography”. 2202.04710
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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
[7]
S. Bravyi, M. Suchara, and A. Vargo, “Efficient algorithms for maximum likelihood decoding in the surface code”, Physical Review A 90, (2014). DOI; 1405.4883
[8]
E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002). DOI; quant-ph/0110143
[9]
O. Derzhko, J. Richter, and O. Zaburannyi, “Spin-Peierls instability in the spin-1/2 transverse XX chain with Dzyaloshinskii-Moriya interaction”. cond-mat/0001014
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F. Merz and J. T. Chalker, “Two-dimensional random-bond Ising model, free fermions, and the network model”, Physical Review B 65, (2002). DOI; cond-mat/0106023
[11]
G. Duclos-Cianci and D. Poulin, “Fast Decoders for Topological Quantum Codes”, Physical Review Letters 104, (2010). DOI; 0911.0581
[12]
A. Hutter, J. R. Wootton, and D. Loss, “Efficient Markov chain Monte Carlo algorithm for the surface code”, Physical Review A 89, (2014). DOI; 1302.2669
[13]
D. S. Wang et al., “Threshold error rates for the toric and surface codes”. 0905.0531
[14]
H. Bombin et al., “Strong Resilience of Topological Codes to Depolarization”, Physical Review X 2, (2012). DOI; 1202.1852
[15]
D. Gottesman, “Keeping One Step Ahead of Errors”, Physics 5, (2012). DOI
[16]
T. M. Stace, S. D. Barrett, and A. C. Doherty, “Thresholds for Topological Codes in the Presence of Loss”, Physical Review Letters 102, (2009). DOI; 0904.3556
[17]
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
[18]
Alexey A. Kovalev and Leonid P. Pryadko, “Spin glass reflection of the decoding transition for quantum error correcting codes”. 1311.7688
[19]
C. T. Chubb and S. T. Flammia, “Statistical mechanical models for quantum codes with correlated noise”, Annales de l’Institut Henri Poincaré D 8, 269 (2021). DOI; 1809.10704
[20]
D. K. Tuckett et al., “Tailoring Surface Codes for Highly Biased Noise”, Physical Review X 9, (2019). DOI; 1812.08186
[21]
Oscar Higgott et al., “Fragile boundaries of tailored surface codes”. 2203.04948
[22]
Yue Wu et al., “Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays”. 2201.03540