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]. 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]. |
| Qubit stabilizer code | Certain operations can be implemented in a fault-tolerant version [83,84] of holonomic quantum computation [85]. |
| Raussendorf-Bravyi-Harrington (RBH) cluster-state code | Various thresholds for optical quantum computing scheme with RBH codes [86,87].\(0.75\%\) for preparation, gate, storage, and measurement errors [51].\(24.9\%\) under erasure noise [88].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 [89]. |
| Rotated surface code | Thresholds for various amounts of erasure, Pauli, correlated, and measurement noise are known [90,91]. |
| SYK code | The coherent information of noise channels that either break or conserve fermion parity has been calculated for the SYK thermofield double state [92]. |
| 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 [93; Table I] and \(2.08(1)\%\) under projection decoder [94].Circuit-level noise: \(0.082(3)\%\) under IP decoder, \(0.143(1)\%\) under projection decoder [94], \(0.143\%\) under matching decoder [95], and an analytic lower bound of \(\approx 0.1\%\) [96] (see [93; 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 [97], improving the \(0.67\%\) threshold for standard measurement schedules [98].\(2.22\%\) threshold for circuit-level infinitely biased noise under a variant of MWPM and using gauge-fixing and specific measurement schedules [97], improving the \(0.52\%\) threshold with standarn measurement schedules. |
| Tetrahedral color code | \(0.46\%\) with clustering decoder [99].\(1.9\%\) 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 [99]. |
| 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 [100]. |
| 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 [101].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 \(36\) [101]. |
| Twisted XZZX toric code | Phenomenological noise: between \(3\%\) and \(10\%\) at noise bias ranging from 1 to 4 under MWPM [102; 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 [103]. |
| XYZ ruby Floquet code | Circuit-level noise: \(\approx 0.18\%\) using BP-OSD decoder [104]. |
| 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. The dominant source of noise in neutral atom arrays 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 [105].\(0.817\%\) and \(0.940\%\) with minimum-weight perfect matching and belief-matching decoder, respectively, for biased circuit-level noise [106]. |
| \([[15,1,3]]\) quantum RM 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 [107], improving previous lower bounds [9,69,108]. 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]. |
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