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 For correlated Pauli noise, bounds on code capacity thresholds can be obtained by mapping the effect of noise on the code to a statistical mechanical model. The first such threshold, based on the planar code, is \(0.017\%\) [6].\(10.9\%\) (\(10.31\%\)) with pure \(Z\)-dephasing noise for square tiling using tensor-network [7][8] (minimum-weight perfect matching [9]) decoder. \(18.9\%\) with depolarizing noise for square tiling [10].\(50\%\) with erasure errors for square tiling [11].\(3.3\%\) with phenomenological noise for square tiling [12].
Quantum low-density parity-check (QLDPC) code For correlated Pauli noise, bounds on code capacity thresholds for families of QLDPC codes can be obtained by mapping the effect of noise on the code to a statistical mechanical model [6][1][13].Bounds on code capacity thresholds for various noise models exist in terms of stabilizer generator weights [2].
Qubit stabilizer code For correlated Pauli noise, bounds on code capacity thresholds for any stabilizer codes can be obtained by mapping the effect of noise on the code to a statistical mechanical model [6][1][13][14].
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 [14].
XY surface code \(50\%\) at infinite \(Z\) bias with maximum-likelihood decoder [15].\(18.7\%\) for standard depolarising noise with maximum-likelihood decoder [15].
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 [16].\(4.15\%\) 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 [17].

References

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Pedro Parrado-Rodríguez, Manuel Rispler, and Markus Müller, “Rescaling decoder for 2D topological quantum color codes on 4.8.8 lattices”. 2112.09584
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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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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
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H. Bombin et al., “Strong Resilience of Topological Codes to Depolarization”, Physical Review X 2, (2012). DOI; 1202.1852
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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
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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
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Alexey A. Kovalev and Leonid P. Pryadko, “Spin glass reflection of the decoding transition for quantum error correcting codes”. 1311.7688
[14]
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
[15]
D. K. Tuckett et al., “Tailoring Surface Codes for Highly Biased Noise”, Physical Review X 9, (2019). DOI; 1812.08186
[16]
Oscar Higgott et al., “Fragile boundaries of tailored surface codes”. 2203.04948
[17]
Yue Wu et al., “Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays”. 2201.03540