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\(\%\) - 5\(\%\) for a \(\{5,4\}\) tiling under minimum-weight decoding with noiseless syndrome extraction [2]. For larger tilings, the lower bound on the distance decreases, suggesting the threshold will also decrease.
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 for 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: \(50\%\) threshold error rate using the 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 Sierpinski prism model 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 \(17.2\%\) for the surface-like logical operator together with \(3.3\%\) for the line-like logical operator of the 3D cubic code under RG decoding [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].
Abelian two-block group-algebra code Qubit Abelian 2BGA families with fixed check weight and polynomially growing distance have vanishing rate [20]. When such a family has a unique threshold, its optimal code capacity threshold under bit-flip noise is constrained by the Kramers-Wannier self-duality of zero-rate PSD codes [20].
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 [21]. 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 [22]. 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 [23; Table I].
Bivariate bicycle (BB) code Fully postselected failure rates of weight-six BB codes under bit-flip noise cross near \(p=1/(2+\sqrt{2})\approx 0.2929\) [20]. This value is the self-dual critical point of zero-rate PSD code families with a unique threshold [20]. Without postselection, the \(k=12\) weight-six family of Ref. [24] has a threshold of \(8.38(6)\%\) under BP-OSD decoding [20].
Chamon model code Depolarizing noise: \(4.92\%\) with repetition-based decoder [25].
Checkerboard model code Independent \(X,Z\) noise: \(\approx 10.7\%\), higher than 3D surface code and color code [26,27].
Clifford-deformed surface code (CDSC) Depolarizing noise: the threshold under ML decoding corresponds to the value of a critical point of a two-dimensional random-bond Ising model (RBIM) with two- and four-body terms on the Nishimori line [4,28,29]. 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 [30]. 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 \(3.3 \%\) [31] (see also [32]).
Compass code See [33; Sec. IV] for tables of code-capacity thresholds against spatially dependent and biased noise.
Concatenated GKP code \(0.599\) threshold displacement standard deviation for GKP-repetition code [34].\(0.59\) threshold displacement standard deviation for GKP-color code [35].A concatenated threshold with GKP codes on the lowest level exists for general Markovian noise [36].There is an upper bound on the threshold under local update recovery that is derived via quantum optimal transport [37].
Concatenated Steane code This family is one of the first to admit a concatenated threshold [38–44]; see the book [45].The bit-flip phase boundary nearly coincides with that of the toric code [20]. This is explained by the Kramers-Wannier self-duality of zero-rate PSD codes with a unique threshold [20]. Below threshold, exponentially many minimum-weight logical operators compete with the favorable distance scaling \(d = n^{\log_7 3}\) [20]. The result is a finite-size crossover in physical overhead relative to the toric code [20].
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 [46].
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 [47].
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 [48]. See also Ref. [49].\(3.0\%\) 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 [48].
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 [50].
GKP-surface code \(0.55\) (\(0.54\)) threshold displacement standard deviation for GKP-toric (GKP-surface) codes without using GKP analog information [52][51; Sec. IV.B]. Using the continuous GKP syndrome information raises the GKP-toric threshold to \(\sigma_0\approx 0.6\), corresponding to a qubit error rate of about \(14\%\) [51; Sec. IV.B].Analog QEC on GKP-surface codes with ideal syndrome measurements yields a threshold displacement standard deviation of about \(0.607\), close to the hashing bound for the Gaussian quantum channel [52].\(0.67\) threshold displacement standard deviation for GKP-XZZX-surface code [53].\(0.602\) threshold displacement standard deviation for GKP-surface codes with analog side information using MWPM closest point decoder [54].
Generalized bicycle (GB) code Depolarizing noise: \(15\%\) for a family of 6-limited \([[2^{m+1}-2,2m]]\) GB codes with BP-OSD decoder [55; Appx. C].
Generalized five-squares code For depolarizing noise, the original five-squares code has a threshold around \(1.5\%\) under the simple decoder and around \(2\%\) under the improved decoder [56].
Haah cubic code (CC) Cubic code 1 has a threshold of \(7.97(4)\%\) under BP-OSD decoding for code capacity bit-flip noise [20]. This estimate uses periodic lattices of odd side length on which the code encodes \(k=2\) logical qubits [20]. Under full postselection, the clean statistical mechanical model of cubic code 1 is the self-dual fractal Ising model [20]. This model has a unique transition at \(p=1/(2+\sqrt{2})\approx 0.2929\), the self-dual value shared by zero-rate PSD codes with a unique threshold [20].
Heptagon holographic code \(\approx 33\%\) under erasures using an optimal erasure decoder for the finite-rate family [57].Depolarizing noise: \(9.4\%\) using tensor-network decoder, and \(\approx 7\%\) using integer optimization decoder [58].\(18.985\%\) against depolarizing noise for zero-rate code under tensor-network decoder [59].
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 [60].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 [61]. 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 [62].
Honeycomb (6.6.6) color code Independent \(X,Z\) noise: \(p_X = 7.8\%\) under message-passing decoder [63], \(8.7\%\) under projection decoder [64], \(\geq 6\%\) under rescaling decoder [65], \(9.0\%\) under Möbius matching decoder [66], \(10.1\%\) under MaxSAT-based decoder [67], \(8.2\%\) under concatenated MWPM decoder [68], and close to the optimal value under the Frontier decoder [69]. 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 [28,70], calculated to be \(10.9(2)\%\) in Ref. [70] and \(10.97(1)\%\) in Ref. [71].Depolarizing channel: \(12.6\%\) under the restriction decoder [72] and the projection decoder [64], and \(\approx 14.5\%\) under AMBP4 decoding [73; 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 [74]. 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 [75].\(40\%\) under erasure noise for a constant-rate version of the code [75].
Lattice stabilizer code A threshold exists for the offline message-passing decoder [76].
Lift-connected surface (LCS) code \(6.7\%\) and \(7.7\%\) under bit-flip noise and BP+OSD decoding for two families of LCS codes.
Majorana-XYZ code Independent \(X,Z\) noise: \(1.4\% \pm 0.1\%\) for odd distance and \(1.8\% \pm 0.2\%\) for even distance under BP-OSD decoding of the stabilizer syndromes [77]. The fitted finite-size scaling exponents are \(\nu_{\mathrm{odd}}=1.5\pm0.5\) and \(\nu_{\mathrm{even}}=1.5\pm1.0\).
NTRU-GKP code A lower bound on the threshold for displacement noise can be formulated in terms of code parameters [78; Appx. B].
Pastawski-Yoshida-Harlow-Preskill (HaPPY) code \(26\%\) for boundary erasure errors on the pentagon-hexagon HaPPY code under the greedy decoder [60].Lower bound of \(1/12 \approx 8.3\%\) for boundary erasure errors on the single-qubit HaPPY code under hierarchical recovery [60]. 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 [60].\(16.3\%\) for boundary Pauli errors on the single-qubit HaPPY code with 3 layers using integer optimization decoder [79].\(50\%\) against biased Pauli noise for single-qubit HaPPY code under tensor-network decoder [59].
Permutationally self-dual (PSD) CSS code The clean statistical mechanical model of a PSD code under bit-flip noise is finite-size Kramers-Wannier self-dual up to a Hadamard mixing of logical sectors [20; Thm. 1]. For a PSD code family with a unique threshold, this model is self-dual in the thermodynamic limit iff the asymptotic rate vanishes [20; Corr. 1.1]. For such zero-rate families, the fully postselected critical point is pinned to the self-dual value \(p = 1/(2+\sqrt{2}) \approx 0.2929\) [20].For zero-rate PSD families with a unique threshold, the principal Boltzmann factor construction in the replica limit gives an approximate phase boundary above the Nishimori line under bit-flip noise [20]. Treating erasures as bond dilution extends the same approximation to mixed bit-flip and erasure noise [20].
Quantum Tanner code Independent \(X,Z\) noise: lower bound under potential-based decoder [80; Corr. 15].
Quantum repetition code Independent \(X\) noise: \(50\%\) with RG decoder for quantum repetition code arranged on a 1D or 2D lattice [18].A nonzero threshold exists under the single-rule local automaton decoders [81].
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 [47,82].
Qubit CSS code Bounds on code capacity thresholds for various noise models exist in terms of stabilizer generator weights [5,83].Consider a CSS code family with bounded check weights and asymptotic rate \(R\), in which \(\max(d_X,d_Z)\) strictly increases. Its decodable region is the set of error probabilities \(p\) and decoder temperatures \(T=1/K\) at which decoding succeeds with probability approaching one as the family grows. A temperature off the Nishimori line corresponds to a decoder that assumes an incorrect error probability. For \(X\) and \(Z\) errors of equal probability, the coupling \(K_{\max}=1/T_{\max}\) at the upper temperature boundary of this region satisfies \(K_{\max} - K_{\max}^{*} \geq R \ln 2\) [74; Thm. 3]. Here, the Kramers-Wannier dual coupling \(K^{*}\) is defined by \(\tanh K^{*} = e^{-2K}\). At zero rate, this bound is the self-dual point [74].
Qubit 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]. In particular, any family of qubit QLDPC codes with superlogarithmic distance achieves a threshold [5].Bounds on code capacity thresholds for various noise models exist in terms of stabilizer generator weights [83].
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 [84].
Rotated surface code Depolarizing noise: close to the optimal value of \(18.9\%\) under the Frontier decoder [69].
Six-qubit-tensor holographic code \(18.8\%\) under depolarizing noise using tensor-network decoder [85].
Square-octagon (4.8.8) color code Independent \(X,Z\) noise: \(p_X = 10.56(1)\%\) under IP decoder [86], \(8.87\%\) under matching decoder [87], \(7.60(2)\%\) under projection decoder [88], and \(8.7\%\) under two-copy surface-code decoder [89] (see [86; 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 [28,70], calculated to be \(10.9(2)\%\) in Ref. [70] (and in the Union Jack formulation in Ref. [90]) and \(10.925(5)\%\) in Ref. [71].
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 [28,91], calculated to be around \(7\%\) in Ref. [92].
Surface-code-fragment (SCF) holographic code \(7.1\%\) and \(8.2\%\) for even- and odd-radii reduced-rate codes, respectively, under depolarizing noise using the integer-optimization decoder [79].
Toric code Independent \(X,Z\) noise: \(p_X = 10.31\%\) under MWPM decoding [93] (see also Ref. [94]), \(9.9\%\) under BP-OSD decoding [95], and \(8.9\%\) under GBP decoding [96]. 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 [4,28], calculated to be \(10.94 \pm 0.02\%\) in Ref. [97], \(10.93(2)\%\) in Ref. [98], \(10.9187\%\) in Ref. [99], \(10.917(3)\%\) in Ref. [100], \(10.939(6)\%\) in Ref. [101], and estimated to be between \(10.9\%\) and \(11\%\) in Ref. [94]. The model for the case of the toric code has been thoroughly investigated [102,103]. The Bravyi-Suchara-Vargo (BSV) tensor network decoder [94] 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 [104]. ML decoding [4] is \(\#P\)-hard in general for the surface code [105]. Above values are for one type of noise only, and the ML threshold for combined \(X\) and \(Z\) noise is \(2p_X - p_X^2 \approx 20.6\%\) [96; Table 1]. Thresholds for various lattices have been obtained in Refs. [106,107]. Depolarizing noise: between \(17\%\) and \(18.5\%\) under BSV tensor-network decoding [94], \(14\%\) under GBP decoding [96], \(16.5\%\) under recursive MWPM [108], between \(16\%\) and \(17.5\%\) under AMBP4 (depending on whether surface or toric code is considered) [109], and between \(15\%\) and \(16\%\) under RG [110], Markov-chain [111], or MWPM [112] decoding. The threshold under ML decoding corresponds to the value of a critical point of the disordered eight-vertex Ising model, calculated to be \(18.9(3)\%\) [113] (see also APS Physics viewpoint [114]).Erasure noise: \(50\%\) for square tiling [115,116]. There is an inverse relationship between the coordination number of the syndrome graph and the threshold, with the latter corresponding to a percolation transition [117].AD noise: \(39\%\) [118].Correlated noise: the threshold under ML decoding corresponds to the value of a critical point of a particular random-bond Ising model (RBIM) [119,120]. 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 [121].The phase boundary of the RBIM nearly coincides with those of the concatenated Steane code and the recursively concatenated surface-17 code [20]. The three families therefore have similar optimal bit-flip thresholds [20]. This coincidence arises from a common Kramers-Wannier self-duality of the associated statistical mechanical models [20]. Zero-rate PSD codes with a unique threshold share this self-duality [20].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 [122,123]. Another statistical mechanical mapping has been studied for \(X\)-type noise channels interpolating between coherent and incoherent noise [124].Threshold of \(1.5\%\) under real-time geometrically local decoder based on introducing an ancillary buffer and confining spacetime interactions between anyons [125].
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 compute the total weight of any path [126].
Twisted XZZX toric code Depolarizing noise: \(17.5\%\) under AMBP4 decoding for the \([[(m^2+1)/2,1,m]]\) family [73; Fig. 10].Biased noise: between \(20\%\) and \(45\%\) at noise bias ranging from 1 to 10 under MWPM [127; Fig. 5].
X-cube Floquet code It is argued that this code has a threshold in Ref. [128].
X-cube model code Independent \(X,Z\) noise: minimum threshold \(\approx 7.5\%\), higher than those reported for the 3D surface code and color code [26].
XY surface code \(50\%\) at infinite \(Z\) bias with maximum-likelihood decoder [129].\(18.8(2)\%\) for standard depolarizing noise with maximum-likelihood decoder [129].
XYZ color code \(50\%\) threshold for noise infinitely biased towards \(X\) or \(Y\) or \(Z\) errors using cellular-automaton decoder [130].Independent \(X,Y\) noise: threshold value of the sum of both noise probabilities is between \(9\%\) and \(14\%\), depending on the noise bias [130].
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.\(18.3\%\) under biased noise [131].
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 [132] and \(17.5\%\) under AMBP4 [109].
\((2,2)\) Loop toric code Independent \(X,Z\) noise: \(2.117\%\) with Hastings decoder [133] and \(7.3\%\) with RG decoder for the open-boundary 4D tesseract code [18]. It is conjectured via a statistical-mechanical mapping that the optimal ML decoder yields a threshold of \(11.003\%\) [134].
\([[144,12,12]]\) gross code Bit-flip noise: pseudo-threshold of \(\approx 5\%\) for the block logical error rate under BP-OSD decoding [135].
\([[8, 3, 3]]\) Eight-qubit Gottesman code \(4\%\) pseudo-threshold under depolarizing noise with a weight-one lookup-table decoder [136; Sec. III.A].
\([[9,1,3]]\) Surface-17 code Recursively concatenating the surface-17 code yields a zero-rate family. For a suitable concatenation map, every level is finite-size Kramers-Wannier self-dual under the statistical mechanical mapping of bit-flip decoding [20; Lemma 1]. If the family has a unique threshold, this self-duality constrains its optimal bit-flip threshold [20]. The bit-flip phase boundary of the family nearly coincides with that of the toric code [20].
\([[9,1,4,3]]\) Nine-qubit Bacon-Shor code \(2.02 \times 10^{-5}\) concatenated threshold for the recursively concatenated code [137].
List (property_set_excluding_descendants): All codes with Code Capacity Threshold that are not descendants of Error-correcting code (ECC).

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