Here is a list of all quantum codes that admit code capacity thresholds.
Name Threshold
Abelian topological code Noise thresholds can be formulated as anyon condensation transitions in a topological field theory [1], generalizing the mapping of the effect of noise on a code state to a statistical mechanical model [2–5].
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.
Cluster-state code Independent \(X,Z\) noise: \(p_X = 2.9\%\) under MWPM decoding [6]. The threshold under ML decoding corresponds to the value of critical point of the 3D random-plaquette \(\mathbb{Z}_2\) gauge theory (3D-RPGM) via the statistical mechanical mapping [2], calculated to be \(3.3 \%\) [7] (see also [8]).
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 [9]. See also Ref. [10].\(3.0\%\) 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 [9].
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 [11].Holographic codes are argued to have a algebraic threshold, for which the error rate scales polynomially (as opposed to exponentially) in the thermodynamic limit [12]. Such a threshold is governed by the underlying conformal field theory describing the boundary.
Hypergraph product (HGP) code The threshold under ML decoding corresponds to the value of critical point of a two-dimensional random-bond Ising model on the Nishimori line [2,13,14] (see also [15]).
Kitaev surface code Independent \(X,Z\) noise: \(p_X = 10.31\%\) under MWPM decoding [16] (see also Ref. [17]), \(9.9\%\) under BP-OSD decoding [18], and \(8.9\%\) under GBP decoding [19]. The threshold under ML decoding corresponds to the value of critical point of the two-dimensional random-bond Ising model on the Nishimori line [2,13] (see also [15]), calculated to be \(10.94 \pm 0.02\%\) in Ref. [20], \(10.93(2)\%\) in Ref. [21], and estimated to be between \(10.9\%\) and \(11\%\) in Ref. [17]. Above values are for one type of noise only, and 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 tensor-network decoding [17], \(14\%\) under GBP decoding [19], \(16.5\%\) under recursive MWPM [22], and between \(15\%\) and \(16\%\) under RG [23], Markov-chain [24], or MWPM [25] decoding. The threshold under ML decoding corresponds to the value of critical point in the disordered eight-vertex Ising model, calculated to be \(18.9(3)\%\) [26] (see also APS Physics viewpoint [27]).Erasure noise: \(50\%\) for square tiling [28]. There is an inverse relationship between coordination number of the syndrome graph, with the threshold corresponding to a percolation transition [29].
Quantum low-density parity-check (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 [2–4].Bounds on code capacity thresholds for various noise models exist in terms of stabilizer generator weights [30].
Qubit CSS code Bounds on code capacity thresholds for various noise models exist in terms of stabilizer generator weights [3,30].
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 [2–5].
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. [31].\(0.31\%\) noise threshold error rate using clustering decoder [32].\(9.7\%\) noise threshold error rate under erasure noise using erasure decoder [33]. This can be improved to \(44\%\) using gauge fixing [34].\(50\%\) noise threshold error rate under erasure noise using optimal erasure decoder [34].
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 [5].
Subsystem surface code Independent \(X,Z\) noise: the threshold under ML decoding corresponds to the value of critical point of the two-dimensional hexagonal-lattice random-bond Ising model on the Nishimori line [13,35], calculated to be around \(7\%\) in Ref. [36].
Triangular color code \(12.6\%\) threshold for triangular color code with the restriction decoder [37] and the projection decoder [38,39].\(8.7\%\) threshold for phase errors for the honeycomb triangular color code with the projection decoder [38].\(\geq 6\%\) threshold with rescaling-based decoder on the 4-8-8 triangular color code [40].\(44\%\) threshold under erasure noise for the 4-8-8 triangular color code [41] (see also [33]).
X-cube model code Independent \(X,Z\) noise: \(\sim 7.5\%\), higher than 3D surface code and color code [42].
XY surface code \(50\%\) at infinite \(Z\) bias with maximum-likelihood decoder [43].\(18.7\%\) for standard depolarising noise with maximum-likelihood decoder [43].
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 [44].\(4.15\%\) 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 [45].

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