Here is a list of all quantum codes that admit code capacity thresholds.
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].\(2.02 \times 10^{-5}\) concatenated threshold for the concatenated \([[9,1,3,3]]\) Bacon-Shor code [23].
Chamon model code Depolarizing noise: \(4.92\%\) with repetition-based decoder [24].
Checkerboard model code Independent \(X,Z\) noise: \(\approx 7.5\%\), higher than 3D surface code and color code [25].
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 [4,26,27]. 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 [28]. 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 \(3.3 \%\) [29] (see also [30]).
Compass code See [31; 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 [32].\(0.59\) threshold displacement standard deviation for GKP-color code [33].
Concatenated Steane code This family is one of the first to admit a concatenated threshold [34–39]; see the book [40].
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 [41].
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 [42]. See also Ref. [43].\(3.0\%\) 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 [42].
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 [44].
GKP-surface code \(0.55\) (\(0.54\)) threshold displacement standard deviation for GKP-toric (GKP-surface) codes with no analog side information [45] ([46]). Using rectangular lattices accounts for asymmetric noise and improves the GKP-surface threshold to \(0.58\) [45].\(0.67\) threshold displacement standard deviation for GKP-XZZX-surface code [47].\(0.602\) threshold displacement standard deviation for GKP-surface codes with analog side information using MWPM closest point decoder [48].
Generalized bicycle (GB) code Depolarizing noise: \(15\%\) for a family of 6-limited \([[2^{m+1}-2,2m]]\) GB codes with BP-OSD decoder [49; Appx. C].
Generalized five-squares code Original five-squares code has a threshold of at least \(2\%\) against depolarizing noise [50].
Heptagon holographic code \(~33\%\) under erasures using optimal erasure decoder for the finite-rate family, and \(50\%\) for the zero-rate family [51].Depolarizing noise: \(9.4\%\) using tensor-network decoder, and \(\approx 7\%\) using integer optimization decoder [52].\(18.985\%\) against depolarizing noise for zero-rate code under tensor-network decoder [53].
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 [54].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 [55]. 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 [56].
Honeycomb (6.6.6) color code Independent \(X,Z\) noise: \(p_X = 7.8\%\) under message-passing decoder [57], \(8.7\%\) under projection decoder [58], \(\geq 6\%\) under rescaling decoder [59], \(9.0\%\) under Möbius matching decoder [60], \(10.1\%\) under MaxSAT-based decoder [61], and \(8.2\%\) under concatenated MWPM decoder [62]. 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 [26,63], calculated to be \(10.9(2)\%\) in Ref. [63] and \(10.97(1)\%\) in Ref. [64].Depolarizing channel: \(12.6\%\) under the restriction decoder [65] and the projection decoder [58], and \(\approx 14.5\%\) under AMBP4 decoding [66; 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 [67]. 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 [68].\(40\%\) under erasure noise for constant-rate version of the code [68].
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 [69] 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 \(11.003\%\) [70].
NTRU-GKP code A lower bound on the threshould for displacement noise can be formulated in terms of code parameters [71; Appx. B].
Pastawski-Yoshida-Harlow-Preskill (HaPPY) code \(26\%\) for boundary erasure errors on the pentagon-hexagon HaPPY code under the greedy decoder [54].Lower bound of \(1/12 \approx 8.3\%\) for boundary erasure errors on the single-qubit HaPPY code under hierarchical recovery [54]. 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 [54].\(16.3\%\) for boundary Pauli errors on the single-qubit HaPPY code with 3 layers using integer optimization decoder [72].\(50\%\) against biased Pauli noise for single-qubit HaPPY code under tensor-network decoder [53].
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 [73].
Quantum Tanner code Independent \(X,Z\) noise: lower bound under potetial-based decoder [74; Corr. 15].
Quantum repetition code Independent \(X\) noise: \(50\%\) with RG decoder for quantum repetition code arranged on a 1D or 2D lattice [18].
Qubit CSS code Bounds on code capacity thresholds for various noise models exist in terms of stabilizer generator weights [5,73].
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 [75].
Six-qubit-tensor holographic code \(18.8\%\) under depolarizing noise using tensor-network decoder [76].
Square-octagon (4.8.8) color code Independent \(X,Z\) noise: \(p_X = 10.56(1)\%\) under IP decoder [77], \(8.87\%\) under matching decoder [78], \(7.60(2)\%\) under projection decoder [79], and \(8.7\%\) under two-copy surface-code decoder [80] (see [77; 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 [26,63], calculated to be \(10.9(2)\%\) in Ref. [63] and \(10.925(5)\%\) in Ref. [64].
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 [26,81], calculated to be around \(7\%\) in Ref. [82].
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 [72].
Toric code Independent \(X,Z\) noise: \(p_X = 10.31\%\) under MWPM decoding [83] (see also Ref. [84]), \(9.9\%\) under BP-OSD decoding [85], and \(8.9\%\) under GBP decoding [86]. The threshold under ML decoding corresponds to the value of a critical point of the two-dimensional random-bond Ising model (RBIM) on the Nishimori line [4,26], calculated to be \(10.94 \pm 0.02\%\) in Ref. [87], \(10.93(2)\%\) in Ref. [88], \(10.9187\%\) in Ref. [89], \(10.917(3)\%\) in Ref. [90], \(10.939(6)\%\) in Ref. [91], and estimated to be between \(10.9\%\) and \(11\%\) in Ref. [84]. Above values are for one type of noise only, and the ML threshold for combined \(X\) and \(Z\) noise is \(2p_X - p_X^2 \approx 20.6\%\).Depolarizing noise: between \(17\%\) and \(18.5\%\) under BSV tensor-network decoding [84], \(14\%\) under GBP decoding [86], \(16.5\%\) under recursive MWPM [92], between \(16\%\) and \(17.5\%\) under AMBP4 (depending on whether surface or toric code is considered) [93], and between \(15\%\) and \(16\%\) under RG [94], Markov-chain [95], or MWPM [96] decoding. The threshold under ML decoding corresponds to the value of a critical point of the disordered eight-vertex Ising model, calculated to be \(18.9(3)\%\) [97] (see also APS Physics viewpoint [98]).Erasure noise: \(50\%\) for square tiling [99,100]. There is an inverse relationship between coordination number of the syndrome graph, with the threshold corresponding to a percolation transition [101].Correlated noise: \(10.04(6)\%\) under mildly correlated bit-flip noise [7].The toric code has a measurement threshold of one [102].
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 [103].
Twisted XZZX toric code Depolarizing noise: \(17.5\%\) under AMBP4 decoding for the \([[(m^2+1)/2,1,m]]\) family [66; Fig. 10].Biased noise: between \(20\%\) and \(45\%\) at noise bias ranging from 1 to 10 under MWPM [104; 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 [26,63], calculated to be \(10.9\%\) [105].
X-cube Floquet code It is argued that this code has a threshold in Ref. [106].
X-cube model code Independent \(X,Z\) noise: \(\approx 7.5\%\), higher than 3D surface code and color code [25].
XY surface code \(50\%\) at infinite \(Z\) bias with maximum-likelihood decoder [107].\(18.7\%\) for standard depolarizing noise with maximum-likelihood decoder [107].
XYZ color code \(50\%\) threshold for noise infinitely biased towards \(X\) or \(Y\) or \(Z\) errors using cellular-automaton decoder [108].Independent \(X,Y\) noise: threshold value of the sum of both noise probabilities is between \(9\%\) and \(14\%\), depending on the noise bias [108].
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 [109] and \(17.5\%\) under AMBP4 [93].

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