Here is a list of all quantum codes that admit code capacity thresholds.

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Name Threshold
2D hyperbolic surface code Bounds on code capacity thresholds using ML decoding can be obtained by mapping the effect of noise on the code to a statistical mechanical model [1].\(1.3\%\) for a phenomenological noise model for the \(\{4,5\}\)-hyperbolic surface code [2].
2D lattice stabilizer code Noise thresholds can be formulated as anyon condensation transitions in a topological field theory [3], generalizing the mapping of the effect of noise on a code state to a statistical mechanical model [4–7]. Namely, the noise threshold for a noise channel \(\cal{E}\) acting on a 2D stabilizer state \(|\psi\rangle\) can be obtained from the properties of the resulting (mixed) state \(\mathcal{E}(|\psi\rangle\langle\psi|)\) [3,8–11].
2D subsystem color code The threshold under ML decoding under depolarizing noise corresponds to the value of a critical point of a disordered spin model, calculated to be \(5.5(2)\%\) in Ref. [12].Erasure noise: \(50\%\) noise threshold error rate under erasure noise using optimal erasure decoder [13], and \(9.7\%\) and \(44\%\) using gauge-fixing decoders [14,15].
3D lattice stabilizer code Applying Clifford deformations to various 3D stabilizer codes, including the 3D surface code, 3D color code, X-cube model code, and SFSL code, yields a \(50\%\) code capacity threshold under infinitely biased Pauli noise [16].
3D surface code Independent \(X,Z\) noise: \(12\%\) for bit-flip and \(3\%\) for phase-flip channels with MWPM decoder for 3D toric code [17], and \(17.2\%\) and \(3.3\%\) with RG decoder for 3D toric code [18].Erasure noise: \(24.8\%\) with generalization of linear-time ML erasure decoder [19] to 3D surface codes [17]. No threshold was observed for the 3D welded surface code [17].
Bacon-Shor code The number of check operators scales sublinearly with system size, so the Bacon-Shor codes alone do not exhibit a topological threshold in the \(m_1,m_2 \to \infty\) limit [20]. However, a threshold can be obtained from concatenated Bacon-Shor codes that are further restricted to planar geometries, whose recovery circuit is a subset of a circuit used by a larger bona-fide Bacon-Shor code [21]. This threshold differs from a concatenated threshold in that there are no long-range connectivity requirements.Lower bounds for the concatenated threshold of various small Bacon-Shor codes are tabulated in [22; Table I].
Chamon model code Depolarizing noise: \(4.92\%\) with repetition-based decoder [23].
Checkerboard model code Independent \(X,Z\) noise: \(\approx 7.5\%\), higher than 3D surface code and color code [24].
Clifford-deformed surface code (CDSC) Depolarizing noise: the threshold under ML decoding corresponds to the value of a critical point of the weight-two (two-body) two-dimensional random-bond Ising model (RBIM) on the Nishimori line [4,25,26]. Utilizing this statistical mechanical mapping yields a phase diagram for a 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.
Cluster-state code Independent \(X,Z\) noise: \(p_X = 2.9\%\) under MWPM decoding [27]. The threshold under ML decoding corresponds to the value of a critical point of the 3D random-plaquette \(\mathbb{Z}_2\) gauge theory (3D-RPGM) via the statistical mechanical mapping [4], calculated to be \(3.3 \%\) [28] (see also [29]).
Compass code See [30; Sec. IV] for tables of code capacity threholds against spatially dependent and biased noise.
Concatenated GKP code \(0.599\) threshold displacement standard deviation for GKP-repetition code [31].\(0.59\) threshold displacement standard deviation for GKP-color code [32].A concatenated threshold with GKP codes on the lowest level exists for general Markovian noise [33].
Concatenated Steane code This family is one of the first to admit a concatenated threshold [34–40]; see the book [41].
Conformal-field theory (CFT) code Threshold under dephasing depends on the structure of the conformal field theory, with the 1D critical Ising model admitting a finite threshold against certain dephasing noise [42].
Dihedral \(G=D_m\) quantum-double code Behavior under \(X\)-type noise (namely, diffusion of certain anyons) for the \(G=D_4\) case is related to the phase diagram of a disordered net model [43].
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 [44]. See also Ref. [45].\(3.0\%\) 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 [44].
Finite-dimensional quantum error-correcting code Coherent information of the state under the action of a noise channel can be used to estimate the optimal threshold [46].
GKP-surface code \(0.55\) (\(0.54\)) threshold displacement standard deviation for GKP-toric (GKP-surface) codes with no analog side information [47] ([48]). Using rectangular lattices accounts for asymmetric noise and improves the GKP-surface threshold to \(0.58\) [47].\(0.67\) threshold displacement standard deviation for GKP-XZZX-surface code [49].\(0.602\) threshold displacement standard deviation for GKP-surface codes with analog side information using MWPM closest point decoder [50].
Generalized bicycle (GB) code Depolarizing noise: \(15\%\) for a family of 6-limited \([[2^{m+1}-2,2m]]\) GB codes with BP-OSD decoder [51; Appx. C].
Generalized five-squares code Original five-squares code has a threshold of at least \(2\%\) against depolarizing noise [52].
Heptagon holographic code \(~33\%\) under erasures using optimal erasure decoder for the finite-rate family, and \(50\%\) for the zero-rate family [53].Depolarizing noise: \(9.4\%\) using tensor-network decoder, and \(\approx 7\%\) using integer optimization decoder [54].\(18.985\%\) against depolarizing noise for zero-rate code under tensor-network decoder [55].
Holographic tensor-network code The ideal holographic tensor-network 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 [56].Holographic tensor-network codes are argued to have a algebraic threshold, for which the error rate scales polynomially (as opposed to exponentially) in the thermodynamic limit [57]. Such a threshold is governed by the underlying conformal field theory describing the boundary.
Homological code \(>0\%\) threshold with sweep decoder for lattice surface codes in various dimensions [58].
Honeycomb (6.6.6) color code Independent \(X,Z\) noise: \(p_X = 7.8\%\) under message-passing decoder [59], \(8.7\%\) under projection decoder [60], \(\geq 6\%\) under rescaling decoder [61], \(9.0\%\) under Möbius matching decoder [62], \(10.1\%\) under MaxSAT-based decoder [63], and \(8.2\%\) under concatenated MWPM decoder [64]. The threshold under ML decoding corresponds to the value of a critical point of the two-dimensional three-body random-bond Ising model (RBIM) on the Nishimori line [25,65], calculated to be \(10.9(2)\%\) in Ref. [65] and \(10.97(1)\%\) in Ref. [66].Depolarizing channel: \(12.6\%\) under the restriction decoder [67] and the projection decoder [60], and \(\approx 14.5\%\) under AMBP4 decoding [68; Fig. 12].
Hypergraph product (HGP) code Some thresholds were determined in Ref. [5].Bounds on code capacity thresholds using ML decoding can be obtained by mapping the effect of noise on the code to a statistical mechanical model [69]. For example, a threshold of \(7\%\) was obtained under independent \(X\) and \(Z\) noise for codes obtained from random \((3,4)\)-regular Gallager codes.
Hyperinvariant tensor-network (HTN) code \(19.1\%\) under depolarizing noise and \(50\%\) under erasure noise for a \(\{5,4\}\) tiling [70].\(40\%\) under erasure noise for constant-rate version of the code [70].
Lift-connected surface (LCS) code \(2.9\%\) and \(3.2\%\) under phenomenological noise and BP+OSD decoding for two families of LCS codes.
Loop toric code Independent \(X,Z\) noise: \(2.117\%\) with Hastings decoder [71] and \(7.3\%\) with RG decoder for 4D surface code [18]. It is conjectured via a statistical-mechanical mapping that the optimal ML decoder yields a threshold of \(11.003\%\) [72].
NTRU-GKP code A lower bound on the threshould for displacement noise can be formulated in terms of code parameters [73; Appx. B].
Pastawski-Yoshida-Harlow-Preskill (HaPPY) code \(26\%\) for boundary erasure errors on the pentagon-hexagon HaPPY code under the greedy decoder [56].Lower bound of \(1/12 \approx 8.3\%\) for boundary erasure errors on the single-qubit HaPPY code under hierarchical recovery [56]. Numerical evidence indicates the threshold may be closer to \(50\%\).There is no threshold for the pentagon HaPPY code as a constant number of errors (four) can make bulk recovery impossible [56].\(16.3\%\) for boundary Pauli errors on the single-qubit HaPPY code with 3 layers using integer optimization decoder [74].\(50\%\) against biased Pauli noise for single-qubit HaPPY code under tensor-network decoder [55].
Quantum LDPC (QLDPC) code Bounds on code capacity thresholds using ML decoding can be obtained by mapping the effect of noise on the code to a statistical mechanical model [4–6].Bounds on code capacity thresholds for various noise models exist in terms of stabilizer generator weights [75].
Quantum Tanner code Independent \(X,Z\) noise: lower bound under potetial-based decoder [76; Corr. 15].
Quantum repetition code Independent \(X\) noise: \(50\%\) with RG decoder for quantum repetition code arranged on a 1D or 2D lattice [18].
Quantum-double code Behavior under particular \(X\)-type noise (namely, diffusion of an anyon that squares to the trivial anyon) is related to the phase diagram of a disordered \(D_4\) rotor model [43,77].
Qubit CSS code Bounds on code capacity thresholds for various noise models exist in terms of stabilizer generator weights [5,75].
Qubit stabilizer code Bounds on code capacity thresholds using ML decoding can be obtained by mapping the effect of noise on the code to a statistical mechanical model [4–7]. The AQEC relative entropy is related to the resulting threshold [78].
Six-qubit-tensor holographic code \(18.8\%\) under depolarizing noise using tensor-network decoder [79].
Square-octagon (4.8.8) color code Independent \(X,Z\) noise: \(p_X = 10.56(1)\%\) under IP decoder [80], \(8.87\%\) under matching decoder [81], \(7.60(2)\%\) under projection decoder [82], and \(8.7\%\) under two-copy surface-code decoder [83] (see [80; Table I]). The threshold under ML decoding corresponds to the value of a critical point of a two-dimensional three-body random-bond Ising model (RBIM) on the Nishimori line [25,65], calculated to be \(10.9(2)\%\) in Ref. [65] and \(10.925(5)\%\) in Ref. [66].
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 [7].
Subsystem surface code Independent \(X,Z\) noise: the threshold under ML decoding corresponds to the value of a critical point of the two-dimensional hexagonal-lattice random-bond Ising model (RBIM) on the Nishimori line [25,84], calculated to be around \(7\%\) in Ref. [85].
Surface-code-fragment (SCF) holographic code \(7.1\%\) and \(8.2\%\) for even and odd raddi reduced-rate codes, respectively, under depolarizing using the integer optimization decoder [74].
Toric code Independent \(X,Z\) noise: \(p_X = 10.31\%\) under MWPM decoding [86] (see also Ref. [87]), \(9.9\%\) under BP-OSD decoding [88], and \(8.9\%\) under GBP decoding [89]. The threshold under ML decoding corresponds to the value of a critical point of a two-dimensional random-bond Ising model (RBIM) on the Nishimori line [4,25], calculated to be \(10.94 \pm 0.02\%\) in Ref. [90], \(10.93(2)\%\) in Ref. [91], \(10.9187\%\) in Ref. [92], \(10.917(3)\%\) in Ref. [93], \(10.939(6)\%\) in Ref. [94], and estimated to be between \(10.9\%\) and \(11\%\) in Ref. [87]. The Bravyi-Suchara-Vargo (BSV) tensor network decoder [87] exactly solves the ML decoding problem under independent \(X,Z\) noise for the surface code and has complexity of order \(O(n^2)\); the decoder provides an efficient tensor-network contraction for the partition function resulting from the statistical mechanical mapping, which is known to be solvable for an Ising model on a planar graph [95]. ML decoding [4] is \(\#P\)-hard in general for the surface code [96]. Above values are for one type of noise only, and the ML threshold for combined \(X\) and \(Z\) noise is \(2p_X - p_X^2 \approx 20.6\%\) [89; Table 1]. Thresholds for various lattices have been obtained in Refs. [97,98]. Depolarizing noise: between \(17\%\) and \(18.5\%\) under BSV tensor-network decoding [87], \(14\%\) under GBP decoding [89], \(16.5\%\) under recursive MWPM [99], between \(16\%\) and \(17.5\%\) under AMBP4 (depending on whether surface or toric code is considered) [100], and between \(15\%\) and \(16\%\) under RG [101], Markov-chain [102], or MWPM [103] decoding. The threshold under ML decoding corresponds to the value of a critical point of the disordered eight-vertex Ising model, calculated to be \(18.9(3)\%\) [104] (see also APS Physics viewpoint [105]).Erasure noise: \(50\%\) for square tiling [106,107]. There is an inverse relationship between coordination number of the syndrome graph, with the threshold corresponding to a percolation transition [108].AD noise: \(39\%\) [109].Correlated noise: the threshold under ML decoding corresponds to the value of a critical point of a particular random-bond Ising model (RBIM) [110,111]. A threshold of \(10.04(6)\%\) under mildly correlated bit-flip noise is obtained in Ref. [7].The toric code has a measurement threshold of one [112].Coherent noise: the threshold under ML decoding corresponds to the value of a critical point of a particular random-bond Ising model (RBIM) called the complex-coupled Ashkin-Teller model [113,114]. Another statistical mechanical mapping has been studied for \(X\)-type noise channels interpolating between coherent and incoherent noise [115].
Triangular surface code \(10\%\) under either bit-flip or bit-phase noise for ideal syndrome measurements. The decoder used is a decoding graph with the same weight assigned to each edge, and Dijkstra's algorithm is used to computre the total weight of any path [116].
Twisted XZZX toric code Depolarizing noise: \(17.5\%\) under AMBP4 decoding for the \([[(m^2+1)/2,1,m]]\) family [68; Fig. 10].Biased noise: between \(20\%\) and \(45\%\) at noise bias ranging from 1 to 10 under MWPM [117; Fig. 5].
Union-Jack color code Independent \(X,Z\) noise: The threshold under ML decoding corresponds to the value of a critical point of a two-dimensional three-body random-bond Ising model (RBIM) on the Nishimori line [25,65], calculated to be \(10.9\%\) [118].
X-cube Floquet code It is argued that this code has a threshold in Ref. [119].
X-cube model code Independent \(X,Z\) noise: \(\approx 7.5\%\), higher than 3D surface code and color code [24].
XY surface code \(50\%\) at infinite \(Z\) bias with maximum-likelihood decoder [120].\(18.7\%\) for standard depolarizing noise with maximum-likelihood decoder [120].
XYZ color code \(50\%\) threshold for noise infinitely biased towards \(X\) or \(Y\) or \(Z\) errors using cellular-automaton decoder [121].Independent \(X,Y\) noise: threshold value of the sum of both noise probabilities is between \(9\%\) and \(14\%\), depending on the noise bias [121].
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.\(\approx 18\%\) for depolarizing noise under maximum-likelihood decoding.
XZZX surface code 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.\(50\%\) threshold for noise infinitely biased towards \(X\) or \(Y\) or \(Z\) errors using a maximum-likelihood decoder.Depolarizing noise: \(18.7(1)\%\) under tensor-network decoder [122] and \(17.5\%\) under AMBP4 [100].
\([[9,1,4,3]]\) Nine-qubit Bacon-Shor code \(2.02 \times 10^{-5}\) concatenated threshold for the recursively concatenated code [123].

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