Here is a list of all quantum codes that admit thresholds other than code capacity thresholds.
Name Threshold
3D surface code Phenomenological noise model for the 3D toric code: \(2.90(2)\%\) under BP-OSD decoder [1], \(7.1\%\) under improved BP-OSD [2], and \(2.6\%\) under flip decoder [3]. For 3D surface code: \(3.08(4)\%\) under flip decoder [1].
Bacon-Shor code The number of check operators scales sublinearly with system size, so the Bacon-Shor codes alone do not exhibit a threshold [4]. However, a threshold can be obtained from concatenated Bacon-Shor codes restricted to planar geometries, whose recovery circuit is a subset of a circuit used by a larger bona-fide Bacon-Shor code [5].A lower bound of \(1.94 \times 10^{-4}\) for the accuracy threshold was proved for Bacon-Shor code with 5 levels of concatenation, using Steane method of FTEC [6].The three dimensional version offers the possibility of being a self-correcting quantum memory [7].
Color code \(\geq 6.25\%\) threshold for 2D color codes with error-free syndrome extraction, and \(0.1\%\) with faulty syndrome extraction [8].\(0.46\%\) for 3D codes with clustering decoder [9].\(1.9\%\) 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 [9].\(0.31\%\) noise threshold error rate for gauge code using clustering decoder [10].\(0.2\%\) with depolarizing circuit-level noise using two flag-qubits per stabilizer generator and the restriction decoder [11].\(0.143\%\) with depolarizing circuit-level noise using perfect-matching decoder [12].\(>0\%\) threshold with sweep decoder [13].
Concatenated quantum code The first method to achieve a fault-tolerant computational threshold uses concatenated stabilizer codes [14–17]. Such methods require constant-space and polylogarithmic time overhead, but concatentions using quantum Hamming codes improve this to quasi-polylogarithmic time [18].
Fibonacci string-net code Between \(10^{-2}\%\) and \(5\cdot 10^{-2}\%\) for pair-creation and measurement noise [19].
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 [20].
GKP-stabilizer code Thresholds against displacement noise cannot be obtained without ideal (i.e., non-normalizable) codewords [21].
Generalized surface code Phenomenological noise model for the 4D toric code: \(4.3\%\) under improved BP-OSD decoder [2].
Haah cubic code The encoding rate depends on the code implemented, but code 0 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 \(k \sim L\).
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 [22; 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 Floquet code \(0.2\%-0.3\%\) in a controlled-not circuit model with a correlated minimum-weight perfect-matching decoder [23].\(1.5\%<p<2.0\%\) in a circuit model with native two-body measurements and a correlated minimum-weight perfect-matching decoder [23]. 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) [24,25].
Hypergraph product (HGP) code Circuit-level noise: \(0.1\%\) with all-to-all connected syndrome extraction circuits [26]. No threshold observed above physical noise rates at or above \(10^{-6}\) using 2D geometrically local syndrome extraction circuits.
Kitaev surface code \(1.8\%\) for circuit-level depolarizing noise under optimal decoder [27]. \(0.57\%\) for depolarizing noise on data and syndrome qubits as well initialization, gate, and measurement errors under MPWM decoding [28]. 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 \(10^{-4}\) [29].\(0.35\%\) for circuit-level independent \(X,Z\) noise under optimal decoder [27].Phenomenological noise: \(3.3\%\) for square tiling [30].Phenomenological noise model for the 2D toric code: \(2.93(2)\%\) using several rounds of syndrome measurement [31].\(0.5-2.9\%\) for various noise models [32] (see also Refs. [31,33]).
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 [34]
Pastawski-Yoshida-Harlow-Preskill (HaPPY) code \(26\%\) for boundary erasure errors on the the pentagon/hexagon HaPPY code, which has alternating layers of pentagons and hexagons in the tiling.\(\sim 50\%\) for boundary erasure errors on the single-qubit HaPPY code, which has a central pentagon encoding one bulk operator and hexagons tiling all other layers\(16.3\%\) for boundary Pauli errors on the single-qubit HaPPY code with 3 layers [35].There is no threshold for the pentagon HaPPY code as a constant number of errors (two) can make bulk recovery impossible.
Quantum Reed-Muller code Between \(10^{-3}\) and \(10^{-6}\) for depolarizing noise (assuming ideal decoders), see [36]
Quantum expander code Locally stochastic noise: \(2.7 \cdot 10^{-16}\) [37].
Quantum low-density parity-check (QLDPC) code QLDPC codes with a constant encoding rate can reduce the overhead of fault-tolerant quantum computation to be constant [38].
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 [14–17]. 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.
Repetition code Suppose each bit has probability \(p\) of being received correctly, independent for each bit. The probability that a repetition code is received correctly is \(\sum_{k=0}^{(n-1)/2}\frac{n!}{k!(n-k)!}p^{n-k}(1-p)^{k}\). If \(\frac{1}{2}\leq p\), then one can always increase the probability of success by increasing the number of physical bits \(n\); see section 2.2.1 Ref. [39] for a pedagogical explanation.
Subsystem surface code \(0.81\%\) threshold for circuit-level depolarizing noise under a variant of MWPM and using gauge-fixing and specific measurement schedules [40], improving the \(0.67\%\) threshold for standard measurement schedules [41].\(2.22\%\) threshold for circuit-level infinitely biased noise under a variant of MWPM and using gauge-fixing and specific measurement schedules [40], improving the \(0.52\%\) threshold with standarn measurement schedules.
Triorthogonal code Approximately \(\frac{1}{3k + 1}\) [42].
Two-dimensional hyperbolic surface code 1\(\%\) - 5\(\%\) for a \({5,4}\) tiling under minimum-weight decoding [43]. For larger tilings, the lower bound on the distance decreases, suggesting the threshold will also decrease.
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 [44].
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.
\([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS code Concatenated thresholds requiring constant-space and quasi-polylogarithmic time overhead [18].

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