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 subsystem color code | Phenomenological noise: \(0.31\%\) under clustering decoder [2]. |
3D surface code | Phenomenological noise model for the 3D toric code: \(2.90(2)\%\) under BP-OSD decoder [3], \(7.1\%\) under improved BP-OSD [4], \(7.3\%\) under RG [5], and \(2.6\%\) under flip decoder [6]. For 3D surface code: \(3.08(4)\%\) under flip decoder [3]. |
Bacon-Shor code | Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise [7].The Bacon-Shor code has a measurement threshold of zero [8]. |
Concatenated Steane code | Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise [7]; see also [9].A measurement threshold of one [8]. |
Concatenated qubit code | The first methods to achieve a fault-tolerant computational threshold use concatenated qubit stabilizer codes [10–16]; see the book [17]. Such methods require constant-space and polylogarithmic time overhead, but concatenations using quantum Hamming codes improve this to quasi-polylogarithmic time [18]. |
Dual-rail quantum code | Between \(1.78\%\) and \(11.5\%\) with faulty photon detectors when repeatedly concatenating with the Steane code [19]. |
Fibonacci string-net code | Between \(10^{-2}\%\) and \(5\cdot 10^{-2}\%\) for pair-creation and measurement noise [20]. |
Five-qubit perfect code | Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise [7]. |
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 [21]. |
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 [22]. Anti-squeezing does not affect the threshold [23]. |
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 [24]. The GKP-toric decoder yields a threshold displacement standard deviation of \(\sigma = 0.243\) [24], but this noise model did not properly take into account error propagation [25].\(11.2\)dB of squeezing under displacement noise using MWPM decoding for GKP-rotated-surface codes [25,26]. |
Guth-Lubotzky code | Phenomenological noise: data consisted with a threshold of \(4\%\) with BP-OSD or cellular automaton decoders [27]. |
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 bits to scale as \(k \sim L\). |
Haar-random qubit code | Haar-random qubit codes have a measurement threshold of one [8]. |
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. |
Heptagon holographic code | \(~33\%\) under erasures using optimal erasure decoder for the finite-rate family, and \(50\%\) for the zero-rate family [28].Depolarizing noise: \(9.4\%\) using tensor-network decoder, and \(\sim 7\%\) using integer optimization decoder [29]. |
Hierarchical code | Threshold exists for the locally decaying error model; see [30; 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 | Circuit-level noise: \(0.2\%\) using two flag qubits per stabilizer generator and the restriction decoder [31], and \(0.46\%\) under concatenated MWPM decoder [32].A measurement threshold of one [8]. |
Honeycomb Floquet code | \(0.2\%-0.3\%\) in a controlled-not circuit model with a correlated minimum-weight perfect-matching decoder [33].\(1.5\%<p<2.0\%\) in a circuit model with native weight-two measurements and a correlated minimum-weight perfect-matching decoder [33]. 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) [34,35]. |
Hyperbolic Floquet code | \(0.1\%\) under standard circuit-level depolarizing noise [36].\(0.1\%\) under phenomenological error model including depolarizing and measurement errors for the octagonal codes [37]. |
Hypergraph product (HGP) code | Circuit-level noise: \(0.1\%\) with all-to-all connected syndrome extraction circuits [38]. 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 CNOT-gate errors and single-qubit-gate depolarizing noise [39] with optimal decoder [40], and \(0.35\%\) under independent \(X,Z\) noise with optimal decoder [40]. Also, \(0.57\%\) for depolarizing noise on data and syndrome qubits as well initialization, gate, and measurement errors under MPWM decoding [41]. 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 \(10^{-4}\) [42]. Thresholds of \(0.5-2.9\%\) have been observed for various noise models [40,43–48]. A threshold of \(0.41\%\) when concatenated with the \([[4,2,2]]\) code [49].Phenomenological noise: \(3.3\%\) for square tiling [50], and \(2.93(2)\%\) using several rounds of syndrome measurement [43].Quasistatic phase damping and readout noise: \(2.85\%\) [51].Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise [7]. |
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 [5], and \(4.3\%\) under improved BP-OSD decoder [4].Gate-based depolarizing noise: \(0.31\%\) with RG decoder for 4D loop toric code [5].\(1.59\%\) for independent \(X,Z\) noise and faulty syndrome measurements using the Hastings decoder [52]. |
Oscillator-into-oscillator GKP code | Thresholds against displacement noise cannot be obtained without ideal (i.e., non-normalizable) codewords [53]. |
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 [55].A single-qubit HaPPY code has a measurement threshold of one [8] (see also [56]). |
Quantum Golay code | \(1.32\times 10^{-3}\)-per gate error rate for depolarizing noise upon recursive concatenation [57], improving previous lower bounds [7,58,59]. The first numerical study [58] found that the Golay code achieved the highest threshold among a dozen well-known codes at the time [7]. |
Quantum expander code | Locally stochastic noise: \(2.7 \cdot 10^{-16}\) [60]. |
Quantum parity code (QPC) | All optical scheme using QPCs concatenated with either Steane or Golay codes [61]. |
Quantum repetition code | Phenomenological noise: \(11\%\) and \(17.2\%\) with RG decoder for quantum repetition code arranged on a 1D and 2D lattice, respectively [5]. |
Qubit BCH code | Semi-analytical estimates of concatenated thresholds [58]. Qubit BCH codes are difficult to study numerically [7]. |
Qubit code | Computational threshold: A fault-tolerant computational threshold is the maximum noise rate in a 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 concatenated stabilizer codes [10–16]. Such methods require constant-space and polylogarithmic-time overhead, but concatentions using quantum Hamming codes improve this to quasi-polylogarithmic time [18]. Fault-tolerant computations with no notion of locality can be made local on a 2D or 3D geometry with minimial overhead [62]. Measurement threshold: One can derive conditions quantifying how many random single-qubit measurements can be made without destroying the logical information [8]. 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 [8; Thm. 4]. |
Qubit stabilizer code | Computational thresholds against stochastic local noise can be achieved through repeated use of concatenatenation, and can rely on the same small code in every level [10,11,13,15]. The resulting code is highly degenerate, with all but an exponentially small fraction of generators having small weights. Circuit and measurement designs have to take case of the few stabilizer generators with large weights in order to be fault tolerant. |
Raussendorf-Bravyi-Harrington (RBH) cluster-state code | Various thresholds for optical quantum computing scheme with RBH codes [63,64].\(0.75\%\) for preparation, gate, storage, and measurement errors [44].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 [65]. |
Square-octagon (4.8.8) color code | Phenomenological noise: \(3.05(4)\%\) under IP decoder [66; Table I] and \(2.08(1)\%\) under projection decoder [67].Circuit-level noise: \(0.082(3)\%\) under IP decoder, \(0.143(1)\%\) under projection decoder [67], \(0.143\%\) under matching decoder [68], and an analytic lower bound of \(\sim 0.1\%\) [69] (see [66; 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 [70], improving the \(0.67\%\) threshold for standard measurement schedules [71].\(2.22\%\) threshold for circuit-level infinitely biased noise under a variant of MWPM and using gauge-fixing and specific measurement schedules [70], improving the \(0.52\%\) threshold with standarn measurement schedules. |
Tetrahedral color code | \(0.46\%\) with clustering decoder [72].\(1.9\%\) 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 [72]. |
Toric code | \(0.133\%\) for independent \(X,Z\) noise and faulty syndrome measurements using a cellular automaton decoder [52]. |
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 [73].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 \(36\) [73]. |
Twisted XZZX toric code | Phenomenological noise: between \(3\%\) and \(10\%\) at noise bias ranging from 1 to 4 under MWPM [74; Fig. 5]. |
XY surface code | \(6.32(3)\%\) for infinite \(Z\) bias, and thresholds of \(\sim 5\%\) for \(Z\) bias around \(\eta = 100\) using a variant of the minimum-weight perfect matching decoder [75]. |
XYZ ruby Floquet code | Circuit-level noise: \(\approx 0.18\%\) using BP-OSD decoder [76]. |
XZZX surface code | \(\sim 4.5\%\) using minimum-weight perfect matching decoder for depolarizing noise (bias \(\eta=0.5\)); \(\sim 10\%\) 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 [77].\(0.817\%\) and \(0.940\%\) with minimum-weight perfect matching and belief-matching decoder, respectively, for biased circuit-level noise [78]. |
\([[15,1,3]]\) quantum Reed-Muller code | Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise [7]. |
\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code | Concatenated thresholds requiring constant-space and quasi-polylogarithmic time overhead [18]. |
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