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][3][4][5].
Calderbank-Shor-Steane (CSS) stabilizer code Bounds on code capacity thresholds for various noise models exist in terms of stabilizer generator weights [3][6].
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 [7]. 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 \(3.3 \%\) [8] (see also [9]).
Color code \(12.6\%\) threshold for triangular color code with the restriction decoder [10].\(12.6\%\) threshold for triangular color code with the projection decoder ([11]) [12]\(8.7\%\) threshold for phase errors for the hexagonal color code with the projection decoder [11]\(\geq 6\%\) threshold with rescaling-based decoder [13].
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 [14]. See also Ref. [15].\(3.0\%\) 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 [14].
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 [16].Holographic codes are argued to have a algebraic threshold, for which the error rate scales polynomially (as opposed to exponentially) in the thermodynamic limit [17]. Such a threshold is governed by the underlying conformal field theory describing the boundary.
Kitaev surface code Independent \(X,Z\) noise: \(p_X = 10.31\%\) under MWPM decoding [18] (see also Ref. [19]), \(9.9\%\) under BP-OSD decoding [20], and \(8.9\%\) under GBP decoding [21]. The threshold under ML decoding corresponds to the value of critical point of the two-dimensional random-bond Ising model on the Nishimori line [22][2] (see also [23]), calculated to be \(10.94 \pm 0.02\%\) in Ref. [24], \(10.93(2)\%\) in Ref. [25], and estimated to be between \(10.9\%\) and \(11\%\) in Ref. [19]. 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 [19], \(14\%\) under GBP decoding [21], \(16.5\%\) under recursive MWPM [26], and between \(15\%\) and \(16\%\) under RG [27], Markov-chain [28], or MWPM [29] decoding. The threshold under ML decoding corresponds to the value of critical point in the disordered eight-vertex Ising model, calculated to be \(18.9(3)\%\) [30] (see also APS Physics viewpoint [31]).Erasure noise: \(50\%\) for square tiling [32]. There is an inverse relationship between coordination number of the syndrome graph, with the threshold corresponding to a percolation transition [33].
Quantum low-density parity-check (QLDPC) code Bounds on code capacity thresholds using maximum-likelihood (ML) decoding can be obtained by mapping the effect of noise on the code to a statistical mechanical model [2][3][4].Bounds on code capacity thresholds for various noise models exist in terms of stabilizer generator weights [6].
Qubit stabilizer code Bounds on code capacity thresholds using maximum-likelihood (ML) decoding can be obtained by mapping the effect of noise on the code to a statistical mechanical model [2][3][4][5].
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 [22][34], calculated to be around \(7\%\) in Ref. [35].
XY surface code \(50\%\) at infinite \(Z\) bias with maximum-likelihood decoder [36].\(18.7\%\) for standard depolarising noise with maximum-likelihood decoder [36].
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 [37].\(4.15\%\) 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 [38].

References

[1]
Y. Bao et al., “Mixed-state topological order and the errorfield double formulation of decoherence-induced transitions”, (2023) arXiv:2301.05687
[2]
E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
[3]
A. A. Kovalev and L. P. Pryadko, “Fault tolerance of quantum low-density parity check codes with sublinear distance scaling”, Physical Review A 87, (2013) arXiv:1208.2317 DOI
[4]
A. A. Kovalev and L. P. Pryadko, “Spin glass reflection of the decoding transition for quantum error correcting codes”, (2014) arXiv:1311.7688
[5]
C. T. Chubb and S. T. Flammia, “Statistical mechanical models for quantum codes with correlated noise”, Annales de l’Institut Henri Poincaré D 8, 269 (2021) arXiv:1809.10704 DOI
[6]
I. Dumer, A. A. Kovalev, and L. P. Pryadko, “Thresholds for Correcting Errors, Erasures, and Faulty Syndrome Measurements in Degenerate Quantum Codes”, Physical Review Letters 115, (2015) arXiv:1412.6172 DOI
[7]
R. Raussendorf, J. Harrington, and K. Goyal, “A fault-tolerant one-way quantum computer”, Annals of Physics 321, 2242 (2006) arXiv:quant-ph/0510135 DOI
[8]
T. Ohno et al., “Phase structure of the random-plaquette gauge model: accuracy threshold for a toric quantum memory”, Nuclear Physics B 697, 462 (2004) arXiv:quant-ph/0401101 DOI
[9]
K. Takeda, T. Sasamoto, and H. Nishimori, “Exact location of the multicritical point for finite-dimensional spin glasses: a conjecture”, Journal of Physics A: Mathematical and General 38, 3751 (2005) arXiv:cond-mat/0501372 DOI
[10]
C. Chamberland et al., “Triangular color codes on trivalent graphs with flag qubits”, New Journal of Physics 22, 023019 (2020) arXiv:1911.00355 DOI
[11]
N. Delfosse, “Decoding color codes by projection onto surface codes”, Physical Review A 89, (2014) arXiv:1308.6207 DOI
[12]
N. Maskara, A. Kubica, and T. Jochym-O’Connor, “Advantages of versatile neural-network decoding for topological codes”, Physical Review A 99, (2019) arXiv:1802.08680 DOI
[13]
P. Parrado-Rodríguez, M. Rispler, and M. Müller, “Rescaling decoder for two-dimensional topological quantum color codes on 4.8.8 lattices”, Physical Review A 106, (2022) arXiv:2112.09584 DOI
[14]
A. Schotte et al., “Quantum error correction thresholds for the universal Fibonacci Turaev-Viro code”, (2021) arXiv:2012.04610
[15]
S. Burton, C. G. Brell, and S. T. Flammia, “Classical simulation of quantum error correction in a Fibonacci anyon code”, Physical Review A 95, (2017) arXiv:1506.03815 DOI
[16]
F. Pastawski et al., “Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence”, Journal of High Energy Physics 2015, (2015) arXiv:1503.06237 DOI
[17]
N. Bao, C. Cao, and G. Zhu, “Deconfinement and error thresholds in holography”, Physical Review D 106, (2022) arXiv:2202.04710 DOI
[18]
C. Wang, J. Harrington, and J. Preskill, “Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory”, Annals of Physics 303, 31 (2003) arXiv:quant-ph/0207088 DOI
[19]
S. Bravyi, M. Suchara, and A. Vargo, “Efficient algorithms for maximum likelihood decoding in the surface code”, Physical Review A 90, (2014) arXiv:1405.4883 DOI
[20]
J. Roffe et al., “Decoding across the quantum low-density parity-check code landscape”, Physical Review Research 2, (2020) arXiv:2005.07016 DOI
[21]
J. Old and M. Rispler, “Generalized Belief Propagation Algorithms for Decoding of Surface Codes”, (2022) arXiv:2212.03214
[22]
H. Nishimori, “Geometry-Induced Phase Transition in the ±JIsing Model”, Journal of the Physical Society of Japan 55, 3305 (1986) DOI
[23]
R. Fan et al., “Diagnostics of mixed-state topological order and breakdown of quantum memory”, (2023) arXiv:2301.05689
[24]
A. Honecker, M. Picco, and P. Pujol, “Universality Class of the Nishimori Point in the 2D±JRandom-Bond Ising Model”, Physical Review Letters 87, (2001) arXiv:cond-mat/0010143 DOI
[25]
F. Merz and J. T. Chalker, “Two-dimensional random-bond Ising model, free fermions, and the network model”, Physical Review B 65, (2002) arXiv:cond-mat/0106023 DOI
[26]
A. deMarti iOlius et al., “Performance enhancement of surface codes via recursive MWPM decoding”, (2023) arXiv:2212.11632
[27]
G. Duclos-Cianci and D. Poulin, “Fast Decoders for Topological Quantum Codes”, Physical Review Letters 104, (2010) arXiv:0911.0581 DOI
[28]
A. Hutter, J. R. Wootton, and D. Loss, “Efficient Markov chain Monte Carlo algorithm for the surface code”, Physical Review A 89, (2014) arXiv:1302.2669 DOI
[29]
D. S. Wang et al., “Threshold error rates for the toric and surface codes”, (2009) arXiv:0905.0531
[30]
H. Bombin et al., “Strong Resilience of Topological Codes to Depolarization”, Physical Review X 2, (2012) arXiv:1202.1852 DOI
[31]
D. Gottesman, “Keeping One Step Ahead of Errors”, Physics 5, (2012) DOI
[32]
T. M. Stace, S. D. Barrett, and A. C. Doherty, “Thresholds for Topological Codes in the Presence of Loss”, Physical Review Letters 102, (2009) arXiv:0904.3556 DOI
[33]
N. Nickerson and H. Bombín, “Measurement based fault tolerance beyond foliation”, (2018) arXiv:1810.09621
[34]
S. Bravyi et al., “Subsystem surface codes with three-qubit check operators”, (2013) arXiv:1207.1443
[35]
S. L. A. de Queiroz, “Multicritical point of Ising spin glasses on triangular and honeycomb lattices”, Physical Review B 73, (2006) arXiv:cond-mat/0510816 DOI
[36]
D. K. Tuckett et al., “Tailoring Surface Codes for Highly Biased Noise”, Physical Review X 9, (2019) arXiv:1812.08186 DOI
[37]
O. Higgott et al., “Fragile boundaries of tailored surface codes and improved decoding of circuit-level noise”, (2022) arXiv:2203.04948
[38]
Y. Wu et al., “Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays”, Nature Communications 13, (2022) arXiv:2201.03540 DOI
  • Home
  • Code graph
  • Code lists
  • Concepts glossary
  • Search

Classical Domain

  • Binary Kingdom
  • Galois-field Kingdom
  • Matrix Kingdom
  • Analog Kingdom
  • Spherical Kingdom
  • Ring Kingdom
  • Group Kingdom

Quantum Domain

  • Qubit Kingdom
  • Modular-qudit Kingdom
  • Galois-qudit Kingdom
  • Bosonic Kingdom
  • Spin Kingdom
  • Group Kingdom
  • Category Kingdom

Classical-quantum Domain

  • Binary c-q Kingdom
  • Bosonic/analog c-q Kingdom

  • Add new code
  • Additional resources
  • Team
  • About
≡
Error correction zoo by Victor V. Albert, Philippe Faist, and many contributors. This work is licensed under a CC-BY-SA License. See how to contribute.