The Error Correction Zoo

Victor V. Albert; Philippe Faist

Here is a list of all quantum codes that admit thresholds other than code capacity thresholds.

Name	Threshold
2D hyperbolic surface code	1\(\%\) - 5\(\%\) for a \({5,4}\) tiling under minimum-weight decoding [1]. For larger tilings, the lower bound on the distance decreases, suggesting the threshold will also decrease.
3D Bacon-Shor code	The 3D Bacon-Shor code has two entanglement transitions [2].
3D subsystem color code	Phenomenological noise: \(0.31\%\) under clustering decoder [3].
3D surface code	Phenomenological noise model for the 3D toric code: \(2.90(2)\%\) under BP-OSD decoder [4], \(7.1\%\) under improved BP-OSD [5], \(7.3\%\) under RG [6], and \(2.6\%\) under flip decoder [7]. For 3D surface code: \(3.08(4)\%\) under flip decoder [4].
Asymmetric quantum code	A lower bound on concatenated thresholds with CSS codes under biased noise [8].
Bacon-Shor code	Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise [9].The Bacon-Shor code has a measurement threshold of zero [10].Applying a period-four measurement schedule to the original Bacon-Shor code yields a numerical threshold under circuit-level noise [11].
CSS-Plaquette code	The CSS-Plaquette code has one entanglement transition [2].
Concatenated Steane code	Between \(1.78\%\) and \(11.5\%\) with faulty photon detectors when combined with the dual-rail code at the first concatenation step in a variant of the KLM protocol [12,13].Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise [9]; see also [14].A measurement threshold of one [10].
Concatenated qubit code	The first methods to achieve a concatenated threshold against local stochastic noise use concatenated qubit stabilizer codes [15–22]; see the book [23].
Fibonacci string-net code	Between \(10^{-2}\%\) and \(5\cdot 10^{-2}\%\) for pair-creation and measurement noise [24].
Fusion-based quantum computing (FBQC) code	\(11.98\%\) against erasure in fusion measurements.\(1.07\%\) against Pauli error.In linear optical systems, can tolerate \(10.4\%\) probability of photon loss in each fusion.\(43.2\%\) against fusion failure.FBQC applied to the surface code yields thresholds for logical gates that is consistent with the code capacity threshold [25].
GKP CV-cluster-state code	A lower bound on the squeezing required to obtain a particular error rate can be formulated in terms of the displacement noise strength. This in turn determines how much squeezing is required in order to be below threshold for a particular concatenated code. A threshold of \(10^{-6}\) yields a required squeezing of 20.5 dB [26]. Anti-squeezing does not affect the threshold [27].
GKP-surface code	The threshold under displacement noise using ML decoding of GKP-toric codes corresponds to the value of a critical point of a 3D compact QED model in the presence of a quenched random gauge field [28]. The GKP-toric decoder yields a threshold displacement standard deviation of \(\sigma = 0.243\) [28], but this noise model did not properly take into account error propagation [29].\(11.2\)dB of squeezing under displacement noise using MWPM decoding for GKP-rotated-surface codes [29,30]. The error threshold under ML decoding of GKP-rotated-surface codes comes close to \(\sigma\approx 0.6065\), at which the best-known lower bound [31] on the capacity vanishes [32].
Guth-Lubotzky code	Phenomenological noise: data consisted with a threshold of \(4\%\) with BP-OSD or cellular automaton decoders [33].
Haah cubic code (CC)	The encoding rate depends on the code implemented, but code CC0 has been shown to have \(k \ge L\) (on a periodic finite cubic lattice of side length \(L\). In general we expect the number of logical qubits to scale as \(k = \Omega(L)\).
Haar-random qubit code	Haar-random qubit codes have a measurement threshold of one [10].
Heavy-hexagon code	\(0.45\%\) for \(X\) errors under a full circuit-level depolarizing noise model (obtained from Monte Carlo simulations).\(Z\)-errors have no threshold given the \(X\)-type Bacon-Shor stabilizers.
Hierarchical code	Threshold exists for the locally decaying error model; see [34; Thm. 1.3]. However, the logical error rate below threshold falls super-polynomially (as opposed to exponentially) with the code distance. The code family possesses a threshold equal to that of surface codes given by tuning the inner code size for any fixed physical error rate.
Honeycomb (6.6.6) color code	The threshold under ML decoding with measurement errors corresponds to the value of a critical point of a three-dimensional disordered Ising model, estimated to be \(4.8(2)\%\) [35].Circuit-level noise: \(0.2\%\) using two flag qubits per stabilizer generator and the restriction decoder [36], and \(0.46\%\) under concatenated MWPM decoder [37].A measurement threshold of one [10].
Honeycomb Floquet code	\(0.2\%-0.3\%\) in a controlled-not circuit model with a correlated minimum-weight perfect-matching decoder [38].\(1.5\%<p<2.0\%\) in a circuit model with native weight-two measurements and a correlated minimum-weight perfect-matching decoder [38]. Here, \(p\) is the collective error rate of the two-body measurement gate, including both measurement and correlated data depolarization error processes.Against circuit-level noise: within \(0.2\% − 0.3\%\) for SD6 (standard depolarizing 6-step cycle), \(0.1\% − 0.15\%\) for SI1000 (superconducting-inspired 1000 ns cycle), and \(1.5\% − 2.0\%\) for EM3 (entangling-measurement 3-step cycle) [39,40].
Hyperbolic Floquet code	\(0.1\%\) under standard circuit-level depolarizing noise [41].\(0.1\%\) under phenomenological error model including depolarizing and measurement errors for the octagonal codes [42].
Hypergraph product (HGP) code	Circuit-level noise: \(0.1\%\) with all-to-all connected syndrome extraction circuits [43] and DiVincenzo-Aliferis syndrome extraction circuits [44] combined with non-local gates [45]. No threshold observed above physical noise rates at or above \(10^{-6}\) using 2D geometrically local syndrome extraction circuits.
Kitaev surface code	Circuit-level noise: \(1.8\%\) under correlated CNO\(T\)-gate errors and single-qubi\(T\)-gate depolarizing noise [46] with optimal decoder [47], and \(0.35\%\) under independent \(X,Z\) noise with optimal decoder [47]. Also, \(0.57\%\) for depolarizing noise on data and syndrome qubits as well initialization, gate, and measurement errors under MPWM decoding [48]. For this model, a logical qubit with a \(10^{-14}\) logical error rate requires between \(10^3\) to \(10^4\) physical qubits and a target gate fidelity above \(99.9\%\). Later work showed that arbitrarily large computations are possible for a physical error rate of approximately \(10^{-4}\) [49]. Thresholds of \(0.5-2.9\%\) have been observed for various noise models [47,50–55]. A threshold of \(0.41\%\) when concatenated with the \([[4,2,2]]\) code [56].Phenomenological noise: \(3.3\%\) for square tiling [57], and \(2.93(2)\%\) using several rounds of syndrome measurement [50].Fabrication errors [58].Quasistatic phase damping and readout noise: \(2.85\%\) [59].Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise [9].Thresholds for various measurement schedules, including that of the 3CX surface code, have been obtained [60].
Lift-connected surface (LCS) code	\(6.7\%\) and \(7.7\%\) under bit-flip noise and BP+OSD decoding for two families of LCS codes.
Loop toric code	Phenomenological noise model for the 4D loop toric code: \(4.35\%\) with RG decoder [6], and \(4.3\%\) under improved BP-OSD decoder [5].Gate-based depolarizing noise: \(0.31\%\) with RG decoder for 4D loop toric code [6].\(1.59\%\) for independent \(X,Z\) noise and faulty syndrome measurements using the Hastings decoder [61].
Modular-qudit code	There is a threshold against depolarizing noise for any modular-qudit gate that determines if the gate is non-Clifford [62].
Monitored random-circuit code	Above the critical measurement rate \( p_c\), the natural error correction properties of the circuit can no longer protect the information. This can be interpreted as the code threshold.These dynamically generated codes saturate the trade off between density of encoded information and the error rate threshold [63]
Oscillator-into-oscillator GKP code	Thresholds against displacement noise cannot be obtained without ideal (i.e., non-normalizable) codewords [64].
Pastawski-Yoshida-Harlow-Preskill (HaPPY) code	A single-qubit HaPPY code has a measurement threshold of one [10] (see also [65]).
Quantum LDPC (QLDPC) code	QLDPC codes with a constant encoding rate can reduce the overhead of fault-tolerant quantum computation to be constant [66].
Quantum expander code	Locally stochastic noise: \(2.7 \cdot 10^{-16}\) [67].
Quantum parity code (QPC)	All optical scheme using QPCs concatenated with either Steane or Golay codes [68].
Quantum repetition code	Phenomenological noise: \(11\%\) and \(17.2\%\) with RG decoder for quantum repetition code arranged on a 1D and 2D lattice, respectively [6].
Qubit BCH code	Semi-analytical estimates of concatenated thresholds [69]. Qubit BCH codes are difficult to study numerically [9].
Qubit code	Computational threshold: A fault-tolerant computational threshold is the maximum noise rate in a particular single-parameter noise model below which any logical computation of size \(M\) can be executed on a physical-qubit architecture to arbitrary accuracy and with an overhead of order \(O(M\text{polylog}M)\). The first methods to achieve a computational threshold use recursively concatenated stabilizer code families [15–22]; such a threshold is called a concatenated threshold. Initially proven under local stochastic noise, the concatenated threshold theorem also holds for various types of non-Markovian noise [20,21,70,71] and leakage errors [72]. This theorem can be rephrased in terms of Bernoulli site percolation [73]. The resulting concatenated code is highly degenerate, with all but an exponentially small fraction of generators having small weights. Circuit and measurement designs have to take care of the few stabilizer generators with large weights in order to be fault tolerant, but measurement duration may not pose a threat to scalability [44]. Concatenated methods require constant-space and polylogarithmic-time overhead, but concatenations using quantum Hamming codes improve this to quasi-polylogarithmic time [74,75], and concatenations of the Steane code and certain QLDPC codes further improve this to polylogarithmic time [76]. Subsequently, thresholds were determined for infinite families of lattice stabilizer codes, starting with the toric code [77]; such a threshold is colloquially called a topological threshold. Fault-tolerant computations with no notion of locality can be made local on a 2D or 3D geometry with minimal overhead [78]. There is a threshold against depolarizing noise for any single-qubit gate that determines if the gate enables universal quantum computation [62,79]. Measurement threshold: One can derive conditions quantifying how many random single-qubit measurements can be made without destroying the logical information [10]. The measurement threshold is the maximum total probability that a single qubit is measured in a random \(X\), \(Y\), or \(Z\) basis at which the logical information is still recoverable. The measurement threshold is at least as large as the erasure threshold [10; Thm. 4]. There is a dynamical phase transition between a bounded-error and an unbounded-error phase for a model of qubits weakly coupled to a refrigerator [80].
Qubit stabilizer code	Certain operations can be implemented in a fault-tolerant version [81,82] of holonomic quantum computation [83].
Raussendorf-Bravyi-Harrington (RBH) cluster-state code	Various thresholds for optical quantum computing scheme with RBH codes [84,85].\(0.75\%\) for preparation, gate, storage, and measurement errors [51].\(24.9\%\) under erasure noise [86].Concatenation of the RBH code with small codes such as the \([[2,1,1]]\) repetition code, \([[4,1,1,2]]\) subsystem code, or Steane code can improve thresholds [87].
Rotated surface code	Thresholds for various amounts of erasure, Pauli, correlated, and measurement noise are known [88,89].
SYK code	The coherent information of noise channels that either break or conserve fermion parity has been calculated for the SYK thermofield double state [90].
Single-shot code	Residual errors do not become unwieldy after some system-size-independent number cycles of faulty syndrome measurements, and a perfect decoder would be able to recover the information if the final residual error is correctable. Consider acting on a state \(\rho\) with a noise channel \(\mathcal N\) with noise rate \(p\), followed by \(t\) rounds of faulty syndrome measurements \(\mathcal R\) with noise rate \(\eta\) and one perfect recovery (which can be substituted with destructive physical-qubit measurements in practice). The failure probability of a single-shot code should decrease exponentially with the distance of the code, \begin{align} p_{\text{fail}}&=1-F\left(\mathcal{R}[\mathcal{R}_{\eta}\mathcal{N}_{p}]^{t}(\rho),\rho\right)\tag{(1)}\\&=t\left(p/p_{\star}\right)^{d}~, \tag{(2)}\end{align} where \(F\) is a state fidelity, and where \(p_{\star}\) is called the sustainable threshold [4]. For any \(p\) below this threshold, some maximum measurement noise \(\eta_{\star}>0\) can be tolerated after sufficiently large \(t\). The final ideal decoding step \(\mathcal{R}\) cannot be done non-destructively in practice due to noisy syndrome measurements, but information can still be recovered by measuring all logical qubits in the computational basis and correcting the outcomes. If the code is single-shot, then such a procedure will output the correct logical information.
Square-octagon (4.8.8) color code	Phenomenological noise: \(3.05(4)\%\) under IP decoder [91; Table I] and \(2.08(1)\%\) under projection decoder [92].Circuit-level noise: \(0.082(3)\%\) under IP decoder, \(0.143(1)\%\) under projection decoder [92], \(0.143\%\) under matching decoder [93], and an analytic lower bound of \(\approx 0.1\%\) [94] (see [91; Table I]).
Subsystem surface code	\(0.81\%\) threshold for circuit-level depolarizing noise under a variant of MWPM and using gauge-fixing and specific measurement schedules [95], improving the \(0.67\%\) threshold for standard measurement schedules [96].\(2.22\%\) threshold for circuit-level infinitely biased noise under a variant of MWPM and using gauge-fixing and specific measurement schedules [95], improving the \(0.52\%\) threshold with standarn measurement schedules.
Tetrahedral color code	\(0.46\%\) with clustering decoder [97].\(1.9\%\) for 1D string-like logical operators and \(27.6\%\) for 2D sheet-like operators for 3D codes with noise models using optimal decoding and perfect measurements [97].
Toric code	The threshold under ML decoding with measurement errors corresponds to the value of a critical point of a three-dimensional random plaquette model [50,77].\(0.133\%\) for independent \(X,Z\) noise and faulty syndrome measurements using a local automaton decoder [61].Toric-code thresholds for post-selected QEC can be studied with statistical mechanical models [98].
Triangular surface code	\(3.2\%\) bit-flip error-correction threshold for noisy syndrome measurements and \(2.6\%\) for bit-phase flip noise. The decoder used is a decoding graph as describe above [99].In general, the triangular surface code has a threshold of similar magnitude to the toric code for uncorrelated \(X\) and \(Z\) errors. For correlated errors, the triangle code has a lower threshold of a factor of about \(36\) [99].
Twisted XZZX toric code	Phenomenological noise: between \(3\%\) and \(10\%\) at noise bias ranging from 1 to 4 under MWPM [100; Fig. 5].
XY surface code	\(6.32(3)\%\) for infinite \(Z\) bias, and thresholds of \(\approx 5\%\) for \(Z\) bias around \(\eta = 100\) using a variant of the minimum-weight perfect matching decoder [101].
XYZ ruby Floquet code	Circuit-level noise: \(\approx 0.18\%\) using BP-OSD decoder [102].
XZZX surface code	\(\approx 4.5\%\) using minimum-weight perfect matching decoder for depolarizing noise (bias \(\eta=0.5\)); \(\approx 10\%\) for infinite \(Z\) bias.\(4.15\%\) when \(98\%\) of depolarizing errors are converted into erasure errors with union-find decoder on a planar code, vs. \(0.937\%\) for pure depolarizing noise. In Rydberg atomic devices, the dominant source of noise is spontaneous decay into detectable energy levels outside of the computational subspace. Since that decay occurs in a Rydberg level that is accessible from only of the hyperfine states used for storage, the resulting channel is biased erasure [103].\(0.817\%\) and \(0.940\%\) with minimum-weight perfect matching and belief-matching decoder, respectively, for biased circuit-level noise [104].
\([[15,1,3]]\) quantum Reed-Muller code	Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise [9].
\([[23, 1, 7]]\) Quantum Golay code	\(1.32\times 10^{-3}\)-per gate error rate for depolarizing noise upon recursive concatenation [105], improving previous lower bounds [9,69,106]. The first numerical study [69] found that the Golay code achieved the highest threshold among a dozen well-known codes at the time [9].
\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code	Concatenated threshold requiring constant-space and quasi-polylogarithmic time overhead [74].

References

[1]: 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
[2]: B. Placke and S. A. Parameswaran, “Slow measurement-only dynamics of entanglement in Pauli subsystem codes”, (2024) arXiv:2405.14927
[3]: B. J. Brown, N. H. Nickerson, and D. E. Browne, “Fault-tolerant error correction with the gauge color code”, Nature Communications 7, (2016) arXiv:1503.08217 DOI
[4]: A. O. Quintavalle, M. Vasmer, J. Roffe, and E. T. Campbell, “Single-Shot Error Correction of Three-Dimensional Homological Product Codes”, PRX Quantum 2, (2021) arXiv:2009.11790 DOI
[5]: O. Higgott and N. P. Breuckmann, “Improved Single-Shot Decoding of Higher-Dimensional Hypergraph-Product Codes”, PRX Quantum 4, (2023) arXiv:2206.03122 DOI
[6]: K. Duivenvoorden, N. P. Breuckmann, and B. M. Terhal, “Renormalization Group Decoder for a Four-Dimensional Toric Code”, IEEE Transactions on Information Theory 65, 2545 (2019) arXiv:1708.09286 DOI
[7]: T. R. Scruby and K. Nemoto, “Local Probabilistic Decoding of a Quantum Code”, Quantum 7, 1093 (2023) arXiv:2212.06985 DOI
[8]: P. Aliferis and J. Preskill, “Fault-tolerant quantum computation against biased noise”, Physical Review A 78, (2008) arXiv:0710.1301 DOI
[9]: A. W. Cross, D. P. DiVincenzo, and B. M. Terhal, “A comparative code study for quantum fault-tolerance”, (2009) arXiv:0711.1556
[10]: D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
[11]: M. S. Alam, J. Chen, and T. R. Scruby, “Bacon-Shor Board Games”, (2025) arXiv:2504.02749
[12]: M. Silva, “Erasure Thresholds for Efficient Linear Optics Quantum Computation”, (2004) arXiv:quant-ph/0405112
[13]: M. Silva, M. Rötteler, and C. Zalka, “Thresholds for linear optics quantum computing with photon loss at the detectors”, Physical Review A 72, (2005) arXiv:quant-ph/0502101 DOI
[14]: B. W. Reichardt, “Improved ancilla preparation scheme increases fault-tolerant threshold”, (2004) arXiv:quant-ph/0406025
[15]: E. Knill, R. Laflamme, and W. H. Zurek, “Resilient quantum computation: error models and thresholds”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 365 (1998) arXiv:quant-ph/9702058 DOI
[16]: J. Preskill, “Reliable quantum computers”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 385 (1998) arXiv:quant-ph/9705031 DOI
[17]: D. Gottesman, “Fault-tolerant quantum computation with local gates”, Journal of Modern Optics 47, 333 (2000) arXiv:quant-ph/9903099 DOI
[18]: D. Aharonov and M. Ben-Or, “Fault-Tolerant Quantum Computation With Constant Error Rate”, (1999) arXiv:quant-ph/9906129
[19]: K. M. Svore, B. M. Terhal, and D. P. DiVincenzo, “Local fault-tolerant quantum computation”, Physical Review A 72, (2005) arXiv:quant-ph/0410047 DOI
[20]: P. Aliferis, D. Gottesman, and J. Preskill, “Quantum accuracy threshold for concatenated distance-3 codes”, (2005) arXiv:quant-ph/0504218
[21]: P. Aliferis, “Level Reduction and the Quantum Threshold Theorem”, (2011) arXiv:quant-ph/0703230
[22]: K. M. Svore, D. P. DiVincenzo, and B. M. Terhal, “Noise Threshold for a Fault-Tolerant Two-Dimensional Lattice Architecture”, (2006) arXiv:quant-ph/0604090
[23]: D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
[24]: A. Schotte, L. Burgelman, and G. Zhu, “Fault-tolerant error correction for a universal non-Abelian topological quantum computer at finite temperature”, (2022) arXiv:2301.00054
[25]: H. Bombín, C. Dawson, R. V. Mishmash, N. Nickerson, F. Pastawski, and S. Roberts, “Logical Blocks for Fault-Tolerant Topological Quantum Computation”, PRX Quantum 4, (2023) arXiv:2112.12160 DOI
[26]: N. C. Menicucci, “Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States”, Physical Review Letters 112, (2014) arXiv:1310.7596 DOI
[27]: B. W. Walshe, L. J. Mensen, B. Q. Baragiola, and N. C. Menicucci, “Robust fault tolerance for continuous-variable cluster states with excess antisqueezing”, Physical Review A 100, (2019) arXiv:1903.02162 DOI
[28]: C. Vuillot, H. Asasi, Y. Wang, L. P. Pryadko, and B. M. Terhal, “Quantum error correction with the toric Gottesman-Kitaev-Preskill code”, Physical Review A 99, (2019) arXiv:1810.00047 DOI
[29]: K. Noh and C. Chamberland, “Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code”, Physical Review A 101, (2020) arXiv:1908.03579 DOI
[30]: K. Noh, C. Chamberland, and F. G. S. L. Brandão, “Low-Overhead Fault-Tolerant Quantum Error Correction with the Surface-GKP Code”, PRX Quantum 3, (2022) arXiv:2103.06994 DOI
[31]: A. S. Holevo and R. F. Werner, “Evaluating capacities of Bosonic Gaussian channels”, (1999) arXiv:quant-ph/9912067
[32]: M. Lin and K. Noh, “Exploring the quantum capacity of a Gaussian random displacement channel using Gottesman-Kitaev-Preskill codes and maximum likelihood decoding”, (2024) arXiv:2411.04277
[33]: N. P. Breuckmann and V. Londe, “Single-Shot Decoding of Linear Rate LDPC Quantum Codes With High Performance”, IEEE Transactions on Information Theory 68, 272 (2022) arXiv:2001.03568 DOI
[34]: C. A. Pattison, A. Krishna, and J. Preskill, “Hierarchical memories: Simulating quantum LDPC codes with local gates”, (2025) arXiv:2303.04798
[35]: R. S. Andrist, H. G. Katzgraber, H. Bombin, and M. A. Martin-Delgado, “Tricolored lattice gauge theory with randomness: fault tolerance in topological color codes”, New Journal of Physics 13, 083006 (2011) arXiv:1005.0777 DOI
[36]: C. Chamberland, A. Kubica, T. J. Yoder, and G. Zhu, “Triangular color codes on trivalent graphs with flag qubits”, New Journal of Physics 22, 023019 (2020) arXiv:1911.00355 DOI
[37]: S.-H. Lee, A. Li, and S. D. Bartlett, “Color code decoder with improved scaling for correcting circuit-level noise”, Quantum 9, 1609 (2025) arXiv:2404.07482 DOI
[38]: C. Gidney, M. Newman, A. Fowler, and M. Broughton, “A Fault-Tolerant Honeycomb Memory”, Quantum 5, 605 (2021) arXiv:2108.10457 DOI
[39]: C. Gidney, M. Newman, and M. McEwen, “Benchmarking the Planar Honeycomb Code”, Quantum 6, 813 (2022) arXiv:2202.11845 DOI
[40]: A. Paetznick, C. Knapp, N. Delfosse, B. Bauer, J. Haah, M. B. Hastings, and M. P. da Silva, “Performance of Planar Floquet Codes with Majorana-Based Qubits”, PRX Quantum 4, (2023) arXiv:2202.11829 DOI
[41]: O. Higgott and N. P. Breuckmann, “Constructions and Performance of Hyperbolic and Semi-Hyperbolic Floquet Codes”, PRX Quantum 5, (2024) arXiv:2308.03750 DOI
[42]: A. Fahimniya, H. Dehghani, K. Bharti, S. Mathew, A. J. Kollár, A. V. Gorshkov, and M. J. Gullans, “Fault-tolerant hyperbolic Floquet quantum error correcting codes”, (2024) arXiv:2309.10033
[43]: N. Delfosse, M. E. Beverland, and M. A. Tremblay, “Bounds on stabilizer measurement circuits and obstructions to local implementations of quantum LDPC codes”, (2021) arXiv:2109.14599
[44]: D. P. DiVincenzo and P. Aliferis, “Effective Fault-Tolerant Quantum Computation with Slow Measurements”, Physical Review Letters 98, (2007) arXiv:quant-ph/0607047 DOI
[45]: O. Chandra, G. Muraleedharan, and G. K. Brennen, “Non-local resources for error correction in quantum LDPC codes”, (2025) arXiv:2409.05818
[46]: D. S. Wang, A. G. Fowler, A. M. Stephens, and L. C. L. Hollenberg, “Threshold error rates for the toric and surface codes”, (2009) arXiv:0905.0531
[47]: B. Heim, K. M. Svore, and M. B. Hastings, “Optimal Circuit-Level Decoding for Surface Codes”, (2016) arXiv:1609.06373
[48]: A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: Towards practical large-scale quantum computation”, Physical Review A 86, (2012) arXiv:1208.0928 DOI
[49]: A. G. Fowler, “Proof of Finite Surface Code Threshold for Matching”, Physical Review Letters 109, (2012) arXiv:1206.0800 DOI
[50]: 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
[51]: R. Raussendorf and J. Harrington, “Fault-Tolerant Quantum Computation with High Threshold in Two Dimensions”, Physical Review Letters 98, (2007) arXiv:quant-ph/0610082 DOI
[52]: A. G. Fowler, A. M. Stephens, and P. Groszkowski, “High-threshold universal quantum computation on the surface code”, Physical Review A 80, (2009) arXiv:0803.0272 DOI
[53]: M. Ohzeki, “Locations of multicritical points for spin glasses on regular lattices”, Physical Review E 79, (2009) arXiv:0811.0464 DOI
[54]: D. S. Wang, A. G. Fowler, and L. C. L. Hollenberg, “Surface code quantum computing with error rates over 1%”, Physical Review A 83, (2011) arXiv:1009.3686 DOI
[55]: A. M. Stephens, “Fault-tolerant thresholds for quantum error correction with the surface code”, Physical Review A 89, (2014) arXiv:1311.5003 DOI
[56]: B. Criger and B. Terhal, “Noise thresholds for the [4,2,2]-concatenated toric code”, Quantum Information and Computation 16, 1261 (2016) arXiv:1604.04062 DOI
[57]: T. Ohno, G. Arakawa, I. Ichinose, and T. Matsui, “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
[58]: J. M. Auger, H. Anwar, M. Gimeno-Segovia, T. M. Stace, and D. E. Browne, “Fault-tolerance thresholds for the surface code with fabrication errors”, Physical Review A 96, (2017) arXiv:1706.04912 DOI
[59]: D. Pataki, Á. Márton, J. K. Asbóth, and A. Pályi, “Coherent errors in stabilizer codes caused by quasistatic phase damping”, Physical Review A 110, (2024) arXiv:2401.04530 DOI
[60]: I. Hesner, B. Hetényi, and J. R. Wootton, “Using Detector Likelihood for Benchmarking Quantum Error Correction”, (2024) arXiv:2408.02082
[61]: N. P. Breuckmann, K. Duivenvoorden, D. Michels, and B. M. Terhal, “Local Decoders for the 2D and 4D Toric Code”, (2016) arXiv:1609.00510
[62]: W. van Dam and M. Howard, “Noise thresholds for higher-dimensional systems using the discrete Wigner function”, Physical Review A 83, (2011) arXiv:1011.2497 DOI
[63]: M. J. Gullans and D. A. Huse, “Dynamical Purification Phase Transition Induced by Quantum Measurements”, Physical Review X 10, (2020) arXiv:1905.05195 DOI
[64]: L. Hanggli and R. Konig, “Oscillator-to-Oscillator Codes Do Not Have a Threshold”, IEEE Transactions on Information Theory 68, 1068 (2022) arXiv:2102.05545 DOI
[65]: S. Antonini, G. Bentsen, C. Cao, J. Harper, S.-K. Jian, and B. Swingle, “Holographic measurement and bulk teleportation”, Journal of High Energy Physics 2022, (2022) arXiv:2209.12903 DOI
[66]: D. Gottesman, “Fault-Tolerant Quantum Computation with Constant Overhead”, (2014) arXiv:1310.2984
[67]: O. Fawzi, A. Grospellier, and A. Leverrier, “Efficient decoding of random errors for quantum expander codes”, Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing 521 (2018) arXiv:1711.08351 DOI
[68]: A. J. F. Hayes, H. L. Haselgrove, A. Gilchrist, and T. C. Ralph, “Fault tolerance in parity-state linear optical quantum computing”, Physical Review A 82, (2010) arXiv:0908.3932 DOI
[69]: A. M. Steane, “Overhead and noise threshold of fault-tolerant quantum error correction”, Physical Review A 68, (2003) arXiv:quant-ph/0207119 DOI
[70]: B. M. Terhal and G. Burkard, “Fault-tolerant quantum computation for local non-Markovian noise”, Physical Review A 71, (2005) arXiv:quant-ph/0402104 DOI
[71]: D. Aharonov, A. Kitaev, and J. Preskill, “Fault-Tolerant Quantum Computation with Long-Range Correlated Noise”, Physical Review Letters 96, (2006) arXiv:quant-ph/0510231 DOI
[72]: P. Aliferis and B. M. Terhal, “Fault-Tolerant Quantum Computation for Local Leakage Faults”, (2006) arXiv:quant-ph/0511065
[73]: M. Raginsky, “Scaling and Renormalization in Fault-Tolerant Quantum Computers”, Quantum Information Processing 2, 249 (2003) arXiv:quant-ph/0307166 DOI
[74]: H. Yamasaki and M. Koashi, “Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation”, Nature Physics 20, 247 (2024) arXiv:2207.08826 DOI
[75]: S. Yoshida, S. Tamiya, and H. Yamasaki, “Concatenate codes, save qubits”, (2024) arXiv:2402.09606
[76]: S. Tamiya, M. Koashi, and H. Yamasaki, “Polylog-time- and constant-space-overhead fault-tolerant quantum computation with quantum low-density parity-check codes”, (2024) arXiv:2411.03683
[77]: E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
[78]: S. H. Choe and R. Koenig, “How to fault-tolerantly realize any quantum circuit with local operations”, (2024) arXiv:2402.13863
[79]: W. van Dam and M. Howard, “Tight Noise Thresholds for Quantum Computation with Perfect Stabilizer Operations”, Physical Review Letters 103, (2009) arXiv:0907.3189 DOI
[80]: M. Bilokur, S. Gopalakrishnan, and S. Majidy, “Thermodynamic limitations on fault-tolerant quantum computing”, (2024) arXiv:2411.12805
[81]: O. Oreshkov, T. A. Brun, and D. A. Lidar, “Fault-Tolerant Holonomic Quantum Computation”, Physical Review Letters 102, (2009) arXiv:0806.0875 DOI
[82]: O. Oreshkov, T. A. Brun, and D. A. Lidar, “Scheme for fault-tolerant holonomic computation on stabilizer codes”, Physical Review A 80, (2009) arXiv:0904.2143 DOI
[83]: P. Zanardi and M. Rasetti, “Holonomic quantum computation”, Physics Letters A 264, 94 (1999) arXiv:quant-ph/9904011 DOI
[84]: C. M. Dawson, H. L. Haselgrove, and M. A. Nielsen, “Noise Thresholds for Optical Quantum Computers”, Physical Review Letters 96, (2006) arXiv:quant-ph/0509060 DOI
[85]: D. A. Herrera-Martí, A. G. Fowler, D. Jennings, and T. Rudolph, “Photonic implementation for the topological cluster-state quantum computer”, Physical Review A 82, (2010) arXiv:1005.2915 DOI
[86]: S. D. Barrett and T. M. Stace, “Fault Tolerant Quantum Computation with Very High Threshold for Loss Errors”, Physical Review Letters 105, (2010) arXiv:1005.2456 DOI
[87]: Z. Li, I. Kim, and P. Hayden, “Concatenation Schemes for Topological Fault-tolerant Quantum Error Correction”, Quantum 7, 1089 (2023) arXiv:2209.09390 DOI
[88]: S. Gu, Y. Vaknin, A. Retzker, and A. Kubica, “Optimizing quantum error correction protocols with erasure qubits”, (2024) arXiv:2408.00829
[89]: J. F. Kam, S. Gicev, K. Modi, A. Southwell, and M. Usman, “Detrimental non-Markovian errors for surface code memory”, (2024) arXiv:2410.23779
[90]: J. Kim, E. Altman, and J. Y. Lee, “Error Threshold of SYK Codes from Strong-to-Weak Parity Symmetry Breaking”, (2024) arXiv:2410.24225
[91]: A. J. Landahl, J. T. Anderson, and P. R. Rice, “Fault-tolerant quantum computing with color codes”, (2011) arXiv:1108.5738
[92]: A. M. Stephens, “Efficient fault-tolerant decoding of topological color codes”, (2014) arXiv:1402.3037
[93]: A. J. Landahl and C. Ryan-Anderson, “Quantum computing by color-code lattice surgery”, (2014) arXiv:1407.5103
[94]: D. S. Wang, A. G. Fowler, C. D. Hill, and L. C. L. Hollenberg, “Graphical algorithms and threshold error rates for the 2d colour code”, (2009) arXiv:0907.1708
[95]: O. Higgott and N. P. Breuckmann, “Subsystem Codes with High Thresholds by Gauge Fixing and Reduced Qubit Overhead”, Physical Review X 11, (2021) arXiv:2010.09626 DOI
[96]: S. Bravyi, G. Duclos-Cianci, D. Poulin, and M. Suchara, “Subsystem surface codes with three-qubit check operators”, (2013) arXiv:1207.1443
[97]: A. Kubica, M. E. Beverland, F. Brandão, J. Preskill, and K. M. Svore, “Three-Dimensional Color Code Thresholds via Statistical-Mechanical Mapping”, Physical Review Letters 120, (2018) arXiv:1708.07131 DOI
[98]: L. H. English, D. J. Williamson, and S. D. Bartlett, “Thresholds for post-selected quantum error correction from statistical mechanics”, (2024) arXiv:2410.07598
[99]: T. J. Yoder and I. H. Kim, “The surface code with a twist”, Quantum 1, 2 (2017) arXiv:1612.04795 DOI
[100]: Q. Xu, N. Mannucci, A. Seif, A. Kubica, S. T. Flammia, and L. Jiang, “Tailored XZZX codes for biased noise”, (2022) arXiv:2203.16486
[101]: D. K. Tuckett, S. D. Bartlett, S. T. Flammia, and B. J. Brown, “Fault-Tolerant Thresholds for the Surface Code in Excess of 5% Under Biased Noise”, Physical Review Letters 124, (2020) arXiv:1907.02554 DOI
[102]: J. C. Magdalena de la Fuente, J. Old, A. Townsend-Teague, M. Rispler, J. Eisert, and M. Müller, “XYZ Ruby Code: Making a Case for a Three-Colored Graphical Calculus for Quantum Error Correction in Spacetime”, PRX Quantum 6, (2025) arXiv:2407.08566 DOI
[103]: Y. Wu, S. Kolkowitz, S. Puri, and J. D. Thompson, “Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays”, Nature Communications 13, (2022) arXiv:2201.03540 DOI
[104]: O. Higgott, T. C. Bohdanowicz, A. Kubica, S. T. Flammia, and E. T. Campbell, “Improved decoding of circuit noise and fragile boundaries of tailored surface codes”, (2023) arXiv:2203.04948
[105]: A. Paetznick and B. W. Reichardt, “Fault-tolerant ancilla preparation and noise threshold lower bounds for the 23-qubit Golay code”, (2013) arXiv:1106.2190
[106]: B. Reichardt and Y. Ouyang. Unpublished (2006).