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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