Here is a list of quantum codes that have notable decoders.
Name Decoder(s)
2D color code Projection decoder of \(O(n^4)\) complexity [1], modified to account for syndrome errors [2].Concatenated MPWM decoder [3].Syndrome extraction circuits based on superdense coding and a middle-out strategy [4].
2D hyperbolic surface code Due to the symmetries of hyperbolic surface codes, optimal measurement schedules of the stabilizers can be found [5].Bounds on code capacity thresholds using ML decoding can be obtained by mapping the effect of noise on the code to a statistical mechanical model [6].Two flag-based decoders [7].
2D lattice stabilizer code Renormalization group (RG) decoder [8].Tensor-network based decoder for 2D codes subject to correlated noise [9].Standard stabilizer-based error correction can be performed even in the presence of perturbations to the codespace [10–12]; see also Refs. [13,14].
3D color code Decoder that maps 3D color code to three copies of the 3D surface code [15].
3D surface code Flip decoder and its modification p-flip [16].Tensor-network decoder [17].Efficient MWPM decoder for 3D toric and 3D welded surface codes [18].Generalization of linear-time ML erasure decoder [19] to 3D surface codes [18].
Abelian LP code Ensemble BP decoder for codes without short cycles of length 4 [20].Efficient decoder correcting order \(\Theta(n/\log n)\) errors [21].
Abelian quantum-double stabilizer code Efficient decoder correcting below the code distance [22].
Analog stabilizer code Homodyne measurement of nullifiers yields real-valued syndromes, and recovery can be performed by displacements conditional on the syndromes.
Analog surface code Shift-based decoder [23].
Approximate quantum error-correcting code (AQECC) Given an encoding and a noise channel, a decoder that yields the optimal entanglement fidelity can be obtained by solving a semi-definite program [24–27]. This optimal decoder is robust to unexpected variations in the noise channel [28].The decoupling approach a.k.a. the Uhlmann decoder [29–31].Quantum machine-learning based decoders such as quantum convolutional neural networks [32] and quantum autoencoders [33].The Leung recovery map [34] for a noise channel whose Kraus operators \(E_j\) yield a diagonal QEC matrix, \(c_{ij}\propto\delta_{ij}\), has Kraus operators \(\Pi V_j^{\dagger}\), where \(\Pi\) is the codespace projection, and where \(V_j\) is the unitary from the polar decomposition of \(E_j \Pi\). This is the recovery used in the proof of the Knill-Laflamme conditions [35; Thm. 10.1].The Cafaro recovery map [36] can be obtained for noise Kraus operators if there exists a basis of error words with respect to which the uncorrectable piece in the Knill-Laflamme conditions is diagonal; see Ref. [37]. The map recovers information perfectly for strictly correctable noise.The Petz recovery map a.k.a. the transpose map [38–40], a quantum channel determined by the codespace and noise channel, yields an infidelity of recovery that is at most twice away from the infidelity of the best possible recovery [41]. The fidelity can be expressed exactly as a function of the Knill-Laflamme conditions [42; Thm. 1], and it can be used to derive a generalization of the Knill-Laflamme conditions for approximate QECCs [43]. Satisfaction of the Knill-Laflamme conditions is sufficient but not necessary for the Petz recovery map to be the optimal recovery, and a necessary and sufficient condition has been derived [44]. The infidelity of a modified Petz recovery map under erasure can be bounded using the conditional mutual information via the approximate Petz theorem [45–47]. In the case of topological codes, the Petz infidelity is related to the topological entanglement entropy [48]. Modifications include the Petz-like decoder [49].The Yoshida-Kitaev decoder for the Hayden-Preskill protocol [50] can be extended to general QECCs [49].
Approximate secret-sharing code Decoding is analagous to reconstruction in a secret sharing scheme and is done in polynomial time. The only required operations are verification of quantum authentication, which is a pair of polynomial-time quantum algorithms that check if the fidelity of the received state is close to 1, and erasure correction for a stabilizer code, which involves solving a system of linear equations.
BPSK c-q code Linear-optical quantum receiver [51].Kennedy receiver [52,53].Photon-number resolving detector [54].Non-Gaussian near-optimal receiver [53].Multi-stage quantum receiver [55].Quantum receiver attaining the Helstrom bound in the low-photon regime [56].
Bacon-Shor code Both Steane error correction and Shor error correction can be used for syndrome extraction, with the former outperforming the latter [57].Utilizing the mapping of the effect of the noise to a statistical mechanical model [58,59] yields several copies of the 1D Ising model [60; Sec. V.B].While check operators are few-body, stabilizer weights scale with the number of qubits, and stabilizer expectation values are obtained by taking products of gauge-operator expectation values. It is thus not clear how to extract stabilizer values in a fault-tolerant manner [61,62].Continuous-time QEC [63].
Balanced product (BP) code BP-OSD decoder [64].
Binomial code Photon loss and dephasing errors can be detected by measuring the phase-space rotation \(\exp\left(2\pi\mathrm{i} \hat{n} / (S+1)\right)\) and the check operator \(J_x/J\) in the spin-coherent state language, where \(J\) is the total angular momentum and \(J_x\) is the angular momentum in the \(x\) direction [65]. This type of error correction fails for errors that are products of photon loss/gain and dephasing errors. However, for certain \((N,S)\) instances of the binomial code, detection of these types of errors can be done.Recovery can be done via projective measurements and unitary operations in a version of the Cafaro recovery map [65,66].Fault-tolerant scheme that converts the required POVM into binary measurements whose redundancy is guaranteed by a classical code [67].
Bivariate bicycle (BB) code Syndrome extraction circuit requires seven layers of CNOT gates regardless of code length. BP-OSD decoder [64] has been extended [68] to account for measurement errors (i.e., the circuit-based noise model [69]).Random and optimized syndrome extraction schedules from Ref. [68] are not distance-preserving.Some long-range check operators can be measured less frequently than others [70].Syndrome extraction circuits called morphing circuits [71], generalizing circuits for the color code [4].
Bosonic rotation code One can distinguish (destructively) the codewords by performing a Fock-state number measurement. If a Fock state state \(|n\rangle\) is measured, then one rounds to the nearest integer of the form \((kq+j)/N\), and deduces that the true state was \(|\overline{j}\rangle\).One can distinguish states in the dual basis by performing phase estimation on \(\mathrm{e}^{\mathrm{i} \theta \hat n}\). One then rounds the resulting \(\theta\) to the nearest number \(2\pi j / qN\) in order to determine which dual basis state \(j \in \mathbb Z_q\) it came from.Autonomous quantum error correction schemes for \(S=1\) codes [72].Decoder [73] based on measuring in the phase-state basis and using Knill error correction (a.k.a. telecorrection [74]), which is based on teleportation [75,76].
Brown-Fawzi random Clifford-circuit code Minimum-weight decoding via using tropical tensor networks [77].
Camara-Ollivier-Tillich code Iterative error estimation based on the MIN-SUM and SUM-PRODUCT algorithms [78].
Cat code Measuring the Fock-state number modulo \(2S\) can be used to determine if photon loss or excitation errors occurred. For \(S=1\), this is the occupation number parity.
Chamon model code Repetition-based decoder, based on the three underlying repetition codes and improved by pre-treatment with a probabilistic greedy local algorithm [79].
Checkerboard model code Parallelized matching decoder [80].
Chuang-Leung-Yamamoto (CLY) code Destructive decoding with a photon number measurement on each mode.State can be decoded with a network of beamsplitters, phase shifters, and Kerr media.
Circuit-to-Hamiltonian approximate code Local detection of Pauli errors can be done using circuits of depth \(O(\text{polylog}(n))\) based on exact decoders for the Brown-Fawzi code [81; Lemma 3.2].
Classical-quantum (c-q) code Unambiguous state discrimination (USD) [82].
Cluster-state code MBQC syndrome extraction is performed by multiplying certain single-qubit \(X\)-type measurements, which yield syndrome values.
Codeword stabilized (CWS) code There is no known efficient algorithm to decode non-additive (non-stabilizer) CWS codes.Clustered bounded-distance decoder [83–85].Structured error recovery [86], which reduces to syndrome-based recovery for additive (i.e., stabilizer) CWS codes.
Coherent FSK (CFSK) c-q code Bondurant receiver [87].Cyclic receiver [88].Time-resolving receiver [89–91].Bayesian inference [89].
Coherent-state c-q code Optimal receiver performance in ambiguous state discrimination is determined using the Yuen-Kennedy-Lax (YKL) conditions [92]. See review [93] for details on receivers used for coherent-state c-q codes.Joint-detection receiver that can attain channel capacity [94].Various near-optimal receiver designs that can handle arbitrary constellations of coherent states with possible degeneracies [95].The square-root measurement is optimal for geometrically uniform [96–98], direct sums of geometrically uniform [99], and compound geometrically uniform [100] constellations.
Color code In contrast to the surface code, the color code can suffer from unremovable hook errors due to the specifics of its syndrome extraction circuits. Fault-tolerant decoders thus have to utilize additional flag qubits.
Compass code Asymmetrically-weighed variant of the union-find decoder [101].
Concatenated Steane code There exist fault-tolerant syndrome extraction protocols for the concatenated Steane code [102].Randomized compiling helps reduce logical error rate for some noise models [103].
Concatenated bosonic code Decoder exploiting analog information from the outer code for bosonic codes concatenated with qubit QLDPC codes [104].
Concatenated quantum code Standard decoding proceeds by first decoding the outer code and then using the resulting data to decode the inner code.
Concatenated qubit code The effective channel for a concatenation of codes is the composition of the codes' effective channels [105].Message passing algorithm for concatenated codes can be equivalent to ML decoding [106].
Cubic theory code Probabilistic local cellular-automaton decoder [107].
Cyclic quantum code Adapted from the Berlekamp decoding algorithm for classical BCH codes [108].
Dinur-Hsieh-Lin-Vidick (DHLV) code Linear-time decoder utilizing the small set-flip decoder [109] for \(Z\) errors and a reconstruction procedure for \(X\) errors [110].
Distance-balanced code The effective distance of single-ancilla syndrome extraction QLDPC code circuits can be preserved under weight reduction [111]. The distance balancing technique of Ref. [112] preserves the effective distance of single-ancilla syndrome extraction circuits [111].
Doubled color code ML decoder that can utilize a history of syndromes, based on the Walsh-Hadamard transform [113].
EA Galois-qudit stabilizer code Syndrome extraction and computation based on classical additive codes [114].
EA QC-QLDPC code Sum-product algorithm (SPA) decoder [115].
EA QLDPC code Decoder adapted for an all-optical impelementation [116].
EA qubit code Decoding algorithm [117].
EA qubit stabilizer code Optical implementation of a minimal code using hyper-entangled states [118].
Eigenstate thermalization hypothesis (ETH) code An explicit universal recovery channel for the ETH code is given in [119].
Expander LP code Linear-time decoder [120].Logarithmic-time subroutine [121].
Fiber-bundle code Greedy algorithm can be used to efficiently decode \(X\) errors, but no known efficient decoding of \(Z\) errors yet [61].
Fibonacci string-net code Clustering decoder (provides best known threshold for this code) [122–124].Fusion-aware iterative minimum-weight perfect matching decoder. Note that ordinary MWPM decoders do not produce a threshold with this code [124].Cellular automaton decoder [125].
Finite-dimensional quantum error-correcting code The operation \(\cal{D}\) in the definition of this code is called the decoder. However, the term decoder can sometimes be used for the unencoder \(\cal{U}\) (i.e., the inverse of the encoder), which does not correct errors.There are several recovery maps which work for noise that is not exactly correctable; see AQECC entry.Protection can be implemented via continuous-time QEC [126–130] via, e.g., reservoir engineering [131]; see review [132]. There are analogues of the Knill-Laflamme conditions for continuous-time QEC [133], and it has been adapted to non-Markovian noise [134].
Five-qubit perfect code Syndrome extraction circuit using only CNOT-SWAP gates [135].Combined dynamical decoupling and error correction protocol on individually-controlled qubits with always-on Ising couplings [136].Symmetric decoder correcting all weight-one Pauli errors. The resulting logical error channel after coherent noise has been explicitly derived [137].Inspired by the honeycomb Floquet code, various weight-two measurement schemes have been designed [138].
Floquet color code Period-six measurement sequence utilizing two-qubit measurements [139].
Folded quantum RS (FQRS) code Quantum list decodable [140].
Fracton Floquet code Period-six measurement sequence utilizing two- and three-qubit measurements [139].
Frobenius code Adapted from the Berlekamp decoding algorithm for classical BCH codes. There exists a polynomial time quantum algorithm to correct errors of weight at most \(\tau\), where \(\delta=2\tau+1\) is the BCH distance of the code [141].
GKP CV-cluster-state code GKP error correction can be naturally combined with CV MBQC protocols since the performance of both is quantified by a squeezing parameter [142].
GKP-surface code Decoder for GKP-toric code [23].MWPM closest point decoder [143].
GNU PI code For a family of shifted gnu codes, decoding can be done using projection, probability amplitude rebalancing, and gate teleportation in time \(O(n^2)\) [144].
Galois-qudit code For few-qudit codes (\(n\) is small), decoding can be based on a lookup table. For infinite code families, the size of such a table scales exponentially with \(n\), so approximate decoding algorithms scaling polynomially with \(n\) have to be used. The decoder determining the most likely error given a noise channel is called the maximum-likelihood (ML) decoder.
Galois-qudit stabilizer code Syndrome extraction and computation based on classical additive codes [114].
Generalized 2D color code Chromöbius, an open-source implementation of the Möbius decoder works for many 2D color codes [4].
Generalized Shor code Efficient decoder [145].
Generalized bicycle (GB) code BP-OSD decoder [64].
Generalized five-squares code Decoding of five-squares codes leads to a mapping of these codes to two copies of the surface code [146,147].
Gottesman-Kitaev-Preskill (GKP) code The MLD decoder for Gaussian displacement errors is realized by evaluating a lattice theta function, and in general the decision can be approximated by either solving (approximating) the closest vector problem (CVP) [148] (a.k.a. closest lattice point problem) or by using other effective iterative schemes when, e.g., the lattice represents a concatenated GKP code [23,149–151]. While the decoder time scales exponentially with number of modes \(n\) generically, the time can be polynomial in \(n\) for certain codes [143].Babai's nearest plane algorithm [152] can be used for bounded-distance decoding [143].Combining AD noise with amplification yields displacement noise, the noise that GKP codes are designed to correct [153].ML decoder for correcting shift errors in GKP two-qubit gates [154].
Haah cubic code (CC) Hard-decisions RG decoder [155].
Hayden-Nezami-Salton-Sanders bosonic code Decoding requires a different circuit for each possible erasure error, with no general circuit decoding any possible erasure error. Every circuit relies on a generalized conditional rotation, which Ref. [156] calls the QND Gate and which is defined as \(QND_c | x , y \rangle = |x + c y, y \rangle\).
Heavy-hexagon code Any graph-based decoder can be used, such as MWPM and Union Find. However, edge weights must be dynamically renormalized using flag-qubit measurement outcomes after each syndrome measurement round.Machine-learning [157] and neural-network [158] decoders.
Heptagon holographic code Optimal erasure decoder [159].
Hierarchical code Decoding is performed as in a standard concatenated code using a decoder for the inner code and outer code. The syndrome extraction circuit depth for the outer code is optimized using a permutation routing algorithm [160]. The bilayer architecture allows for logical entangling gates between logical surface-code patches.Soft output decoding [161].
High-dimensional expander (HDX) code For 2D simplicial complexes, cycle code decoder admitting a polynomial-time decoding algorithm can be used [112].
Homological code Sweep decoder [162,163].Improved BP-OSD decoder [164].Renormalization group (RG) decoder [165].
Homological product code Union-find [166].
Honeycomb (6.6.6) color code Distance-three measurement schedule based on detector error models [167].Message-passing decoder [168].Adaptation of the restriction decoder [169].Neural-network decoder [170].Möbius matching decoder gives low logical failure rate [171] and has an open-source implementation called Chromöbius [4].AMBP4, a quaternary version [172] of the MBP decoder [173].MaxSAT-based decoder [174].Most likely error (MLE) decoder [175].Neural network decoder [175].
Honeycomb Floquet code The ISG has a static subgroup for all time steps \(r\geq 3\) – that is, a subgroup which remains a subgroup of the ISG for all future times – given by so-called plaquette stabilizers. These are stabilizers consisting of products of check operators around homologically trivial paths. The syndrome bits correspond to the eigenvalues of the plaquette stabilizers. Because of the structure of the check operators, only one-third of all plaquettes are measured each round. The syndrome bits must therefore be represented by a lattice in spacetime, to reflect when and where the outcome was obtained.
Hyperbolic Floquet code Syndrome structure allows for MWPM decoding.
Hyperbolic color code Two flag-based decoders [7].
Hyperbolic surface code Hastings decoder [176].
Hypergraph product (HGP) code Single-ancilla syndrome extraction circuits do not admit hook errors [111].ReShape decoder that uses minimum weight decoders for the classical codes used in the hypergraph construction [177].2D geometrically local syndrome extraction circuits with depth order \(O(\sqrt{n})\) using order \(O(n)\) ancilla qubits [178].Improved BP-OSD decoder [164].Erasure-correction can be implemented approximately with \(O(n^2)\) operations with quantum generalizations [179] of the peeling and pruned peeling decoders [180], with a probabilistic version running in \(O(n^{1.5})\) operations.Syndrome measurements are distance-preserving because syndrome extraction circuits can be designed to avoid hook errors [181].Generalization [182] of Viderman's algorithm for expander codes [183].
Kitaev surface code Using data from multiple syndrome measurements prior to decoding allows for correcting syndrome measurement errors. The surface code requires order \(O(d)\) extraction rounds in order to gain a reliable estimate. Syndrome measurements are distance-preserving because syndrome extraction circuits can be designed to avoid hook errors [58].Syndrome extraction circuits consist of CNOT gates and ancillary measurements since this is a stabilizer code [184]. Measurement schedules can be optimized using spacetime circuit codes to yield what is known as the 3CX surface code [185]. Schedules can also be optimized via ZX calculus [186,187]. Inspired by the honeycomb Floquet code, various weight-two measurement schemes have been designed [188–190], with the scheme in Ref. [189] being a special case of DWR.Expanding diamonds decoder correcting errors of some maximum fractal dimension [191]. The sub-threshold failure probability scales as \((p/p_{\text{th}})^{d^\beta}\), where \(p_{\text{th}}\) is the threshold and \(\beta = \log_3 2\).Minimum weight perfect-matching (MWPM) [58,192] (based on work by Edmonds on finding a matching in a graph [193,194]), which takes time up to polynomial in \(n\) for the surface code. For the case of the surface code, minimum-weight decoding reduces to MWPM [58,193,195]. MWPM solves the MPE decoding problem exactly for independent \(X\) and \(Z\) noise. MPE decoding is \(NP\)-hard in general for the surface code [196].Bravyi-Suchara-Vargo (BSV) tensor network decoder [197] approximately solves the ML decoding problem under independent \(X,Z\) noise for the surface code and takes time of order \(O(n^2)\) [197]. ML decoding [58] is \(\#P\)-hard in general for the surface code [196].Union-find decoder [198] uses the union-find data structure [199–201], solving the MPE decoding problem exactly for low-weight errors under depolarizing noise. A subsequent modification utilizes the continuous signal obtained in the physical implementation of the stabilizer measurement (as opposed to discretizing the signal into a syndrome bit) [202]. Belief union find is a combination of belief-propagation and union-find [203]. Strictly local (as opposed to partially local) union find [204] has a worst-case runtime of order \(O(d^3)\) in the distance \(d\).Modified MWPM decoders: pipeline MWPM (accounting for correlations between events) [205,206]; modification tailored to asymmetric noise [207]; parity blossom MWPM and fusion blossom MWPM [208], a modification utilizing the continuous signal obtained in the physical implementation of the stabilizer measurement (as opposed to discretizing the signal into a syndrome bit) [202]; belief perfect matching (a combination of belief-propagation and MWPM) [203]; spanning tree matching (STM) and rapid-fire (RFire) decoders [209]; ordered decoding based on MWPM [210]. Combining, or harmonizing, various decoders can improve performance [211]. One such example is the Libra decoder [212], a combination of MWPM decoders combined with matching synthesis.Renormalization group (RG) [213–215]; see Ref. [216] for the planar surface code.Linear-time ML erasure decoder [19].Markov-chain Monte Carlo [217].Cellular automaton decoders [218–220]; see also [221].Neural network [222–224], reinforcement learning [225–227], and transformer-based [228,229] decoders.Lightweight low-latency look-up table (LILLIPUT) decoder for small surface codes [230].Decoders can be augmented with a pre-decoder [231,232], which can allow for some processing to be done inside the cryogenic environment of the quantum system [233].Sliding-window [234,235], parallel-window [234], and predictive-window [236] parallelizable decoders, designed to overcome the backlog problem, can be combined with many inner decoders, such as MWPM or union-find.Modifications of BP: generalized belief propagation (GBP) [237], based on a classical version [238]; AMBP4, a quaternary version [172] of the MBP decoder [173] of complexity \(O(n\log\log n)\); blockBP, a combination of BP and tensor-network decoders [239]; machine-learning inspired modifications [240]. See Ref. [241] for a review of BP decoders. The min-sum decoder, a simple variant of BP, cannot be used to attain the benefits of codes with distance greater than 9 [242].A color-code decoder can be used for the surface code [243].Progressive-Proximity Bit-Flipping (PPBF) decoder [244].Collision clustering decoder [245].Quasi-local Lindbladian decoder based on the approximate Petz theorem [246].Exclusive decoder family incorporating post-selection on decoding instances deemed not too difficult [247].
Ladder Floquet code Period-four measurement sequence utilizing two-qubit measurements [62].
Lattice stabilizer code A local decoder (a.k.a. automaton decoder) applies local rules to each small region of sites in a lattice geometry. Such decoders do not require any potentially non-local classical post-processing of error syndromes.Clustering decoder [155,218].Quantum neural-network (QNN) decoder [11].Almost linear-time decoder [248].
Lechner-Hauke-Zoller (LHZ) code BP decoder [249].
Loop toric code Measurement-free local error correction circuit (LEC) using reinforcement learning [250].
Majorana box qubit Qubit readout can be done by charge sensing [251–254].
Modular-qudit code For few-qudit codes (\(n\) is small), decoding can be based on a lookup table. For infinite code families, the size of such a table scales exponentially with \(n\), so approximate decoding algorithms scaling polynomially with \(n\) have to be used. The decoder determining the most likely error given a noise channel is called the maximum-likelihood (ML) decoder.
Modular-qudit stabilizer code Trellis decoder for prime-dimensional qudits, which builds a compact representation of the algebraic structure of the normalizer \(\mathsf{N(S)}\) [255].
Modular-qudit surface code Renormalization group decoder [215,256].
Monitored random-circuit code The recovery operation is the reverse unitary transformation with access to the measurement record (for dynamically generated codes with a strong purification transition) [257]
NTRU-GKP code Babai's nearest plane algorithm [152] can be used for bounded-distance decoding.An NTRU-based decoder against stochastic displacement noise is efficient because the decoding problem is equivalent to decrypting the NTRU cryptosystem with knowledge of the encoder.
Niset-Andersen-Cerf code Optical decoder using three beam splitters, electronic gain detectors, and two phase-insensitive amplifiers as described in Ref. [258].
Number-phase code Modular phase measurement done in the logical \(X\), or dual, basis has zero uncertainty in the case of ideal number phase codes. This is equivalent to a quantum measurement of the spectrum of the Susskind–Glogower phase operator. Approximate number-phase codes are characterized by vanishing phase uncertainty. Such measurements can be utilized for Knill error correction (a.k.a. telecorrection [74]), which is based on teleportation [75,76]. This type of error correction avoids the complicated correction procedures typical in Fock-state codes, but requires a supply of clean codewords [73]. Performance of this method was analyzed in Ref. [259], and it was extended in Ref. [260].Number measurement can be done by extracting modular number information using a CROT gate \(\mathrm{e}^{(2\pi \mathrm{i} / NM) \hat n \otimes \hat n}\) and performing phase measurements [261,262] on an ancillary mode. See Section 4.B.1 of Ref. [73].
Numerically optimized four-qubit AD code Analytical recovery channel [263].
On-off keyed (OOK) c-q code Dolinar receiver [264].Superconducting transition edge sensor (TES) photon-number resolving detector [265].
Oscillator-into-oscillator GKP code Syndromes can be read off using ancilla modes, yielding partial information about noise in the logical modes that can then be used in an efficient ML decoding procedure [266].
PI qubit code Schur-Weyl-transform based decoder [267]. Here, one first measures the total angular momentum of consecutive pairs of qubits, and then its projection modulo some spacing. Recovery can be performed by applying geometric phase gates [268] and the quantum Schur transform.
PPM c-q code Conditional pulse nulling (CPN) receiver [269].
PSK c-q code Multi-stage quantum receivers [270–275].Bayesian inference [89].
Pair-cat code Lindbladian-based dissipative encoding utilizing two-mode two-photon absorption [276]. Encoding passively protects against cavity dephasing, suppressing dephasing noise exponentially with \(\gamma^2\).
Pastawski-Yoshida-Harlow-Preskill (HaPPY) code Hierarchical recovery model [277].Greedy decoder [277].
Polar c-q code Quantum-limited successive-cancellation (SC) joint-detection receiver [278].
Quantum Goppa code Farran algorithm [279].
Quantum LDPC (QLDPC) code Iterative error estimation based on the MIN-SUM and SUM-PRODUCT algorithms [78].Quantum belief propagation (BP) decoder [280–282] is a quantum version of the classical BP decoder, but performance suffers due to degeneracy [283]. Various post-processing algorithms have been proposed (see below and also Refs. [284,285]).BP-OSD decoder, scaling as \(O(n^3)\), adds a post-processing step based on ordered statistics decoding (OSD) to the belief propogation (BP) decoder [64]. For an open-source implementation, see [286,287].Neural BP decoder [288,289] and GNN decoders [290,291] for qubit codes.Partially and fully decoupled BP decoders, which use the decoupling representation, yield improvements against depolarizing noise [292].Message-passing decoder utilizing stabilizer inactivation (MP-SI a.k.a. BP-SI) for CSS-type QLDPC qubit codes [293].BP localized statistics decoding (BP-LSD) that exploits error clustering [294].Syndrome-based linear programming (SB-LP) algorithm can be applied as a post-processing step after syndrome-based min-sum (SM-MS) decoding [295].BP guided decimation (BPGD) decoder [296].SymBreak decoder, which adaptively modifies the decoding graph to break the degeneracy of the BP decoder [297].Ambiguity clustering (AC) decoder, in which measurement data is divided into clusters and decoded independently [298].Non-binary decoding algorithm for CSS-type QLDPC codes [299].2D geometrically local syndrome extraction circuits with bounded depth using order \(O(n^2)\) ancilla qubits [178]. For CSS codes, syndrome extraction can be implemented in constant depth [300].Soft (i.e., analog) syndrome iterative BP for CSS-type QLDPC codes, utilizing the continuous signal obtained in the physical implementation of the stabilizer measurement (as opposed to discretizing the signal into a syndrome bit) [301].The MWPM decoder for surface codes may be generalizable to QLDPC codes [302].Extensions of the union-find decoder for qubit QLDPC codes [303,304].Sliding-window decoding [305].Closed-branch decoder [306].BP with guided decimation guessing (GDG) sliding-window decoder for CSS qubit codes [307].Performing \(d\) syndrome extraction rounds obtains an effective distance of \(d\) for a qubit QLDPC code [308].Fault-tolerant constant-depth encoder and unencoder [309].BP plus ordered Tanner forest (BP+OTF) almost-linear time decoder [310].Cluster decoder [311].
Quantum Tamo-Barg (QTB) code Polynomially efficient decoder for QTB codes against errors acting on a number of subsystems that can go up to half of their conjectured distance [312; Thm. 8]. The decoder is based on decoding RS codes, and its runtime is independent of the locality \(r\).Polynomially efficient decoder for FQTB codes against errors acting on a number of subsystems that can go up to half of their conjectured distance [312; Thm. 7]. The runtime depends on the locality \(r\).
Quantum Tanner code Linear-time potetial-based decoder similar to the small-set-flip decoder for quantum expander codes [109].Linear-time decoder [120].Logarithmic-time mismatch decomposition decoder [121].
Quantum convolutional code ML decoder [313].
Quantum data-syndrome (QDS) code Syndrome errors are decoded using redundant stabilizer measurements.
Quantum error-correcting code (QECC) The effect of an error is a mapping of the code subspace into another, potentially overlapping, subspace. To determine, or diagnose, the effect of the error in what is known as syndrome-based decoding, one can measure one or more operators called check operators, which resolve code and error spaces without collapsing the quantum information inside the spaces. The eigenvalues of check operators are called error syndromes. One round or cycle of quantum error correction proceeds by extracting syndromes and performing correcting operations to map the error space containing the logical information back into the codespace. For some codes, correcting operations are not necessary because one can instead track which error space contains the logical information.
Quantum expander code Small set-flip linear-time decoder, which corrects order \(\Omega(n^{1/2})\) adversarial errors [314].Log-time decoder [315].Constant-time decoder [316].2D geometrically local syndrome extraction circuits acting on a patch of \(N\) physical qubits have to be of depth of order \(\Omega(n/\sqrt{N})\) or deeper. More generally, there is a tradeoff between the depth \(D\) and width \(W\) of a syndrome extraction circuit, namely, \(D \geq n/\sqrt{W}\) [178].
Quantum locally recoverable code (QLRC) Codes constructed with the help of AEL distance amplification [317,318] admit efficient decoders [312].
Quantum parity code (QPC) Teleportation-based QEC [319].
Quantum polar code Quantum successive-cancellation list decoder (SCL-E) for quantum polar codes that do not need entanglement assistance [320].
Quantum repetition code Automaton-like decoders for the repetition code on a 2D lattice, otherwise known as the classical 2D Ising model, were developed by Toom [321,322]. An automaton by Gacs [323,324] and Tsirelson [325] yield local decoders for a 1D lattice.Machine learning algorithm to implement continuous error-correction for the three-qubit quantum repetition code [326].
Quantum spherical code (QSC) Lindbladian scheme stabilizing all points in the constellation and protecting from the AD operator \(E_{0}^{\otimes n}\) [327].
Quantum turbo code Turbo decoder [328; Sec. V].Modified decoder yield improvement over the memoryless depolarizing channel [329].
Quantum-double code For any solvable group \(G\), topological charge measurements can be done with an adaptive constant-depth circuit with geometrically local gates and measurements throughout [330].
Qubit CSS code Coherent decoders allow for measurement-free error correction [331]. One method is table/multi-control decoding [332], which scales exponentially with the number of ancillas used in syndrome measurement. A fault-tolerant measurement-free scheme for low-distance CSS codes is formulated in Ref. [333]. Another method, the Ising-based decoder, utilizes the mapping of the effect of the noise to a statistical mechanical model [58,59] such that the decoding problem maps to preparation of the ground state of an Ising model. See [334; Table 3.2] for a Rosetta stone comparing statistical mechanical models, CSS codes, and chain complexes.Decoders based on neural networks [335].MaxSAT decoder [336].
Qubit c-q code BP with quantum messages (BPQM) decoder [241,337–340].
Qubit code Incorporating faulty syndrome measurements can be done using the phenomenological noise model, which simulates errors during syndrome extraction by flipping some of the bits of the measured syndrome bitstring. In the more involved circuit-level noise model, every component of the syndrome extraction circuit can be faulty.The decoder determining the most likely error given a noise channel is called the maximum probability error (MPE) decoder. For few-qubit codes (\(n\) is small), MPE decoding can be based by creating a lookup table. For infinite code families, the size of such a table scales exponentially with \(n\), so approximate decoding algorithms scaling polynomially with \(n\) have to be used.

Effective distance and hook errors: Decoders are characterized by an effective distance (a.k.a. circuit-level distance), the minimum number of faulty operations during syndrome measurement that is required to make an undetectable error. A code is distance-preserving if it admits a decoder whose circuit-level distance is equal to the code distance. A particularly dangerous class of syndrome measurement circuit faults are hook errors, which are ancilla faults that cause more than one data-qubit error [58]. Hook errors occur at specific places in a syndrome extraction circuit and can sometimes be removed by re-ordering the gates of the circuit. If not, the use of flag qubits (see [341]) to detect hook errors may be necessary to yield fault-tolerant decoders.
Qubit stabilizer code The size of the circuit extracting the syndrome depends on the weight of its corresponding stabilizer generator. Syndrome extraction circuits can be simulated efficiently using dedicated software (e.g., STIM [342]) and there are many general schemes for generating them [343] (see also [67]). Decoding of qubit stabilizer codes is an approximately optimal strategy for various quantum lights-out (QLO) games that can be played on the codes' encoder-respecting form [344].DiVincenzo-Aliferis syndrome extraction circuits [345].Greedy syndrome measurement schedule [7].Dynamical weight reduction (DWR) scheme in which measurements of smaller-weight Paulis yield the outcome of a larger-weight Pauli via the use of ZX calculus and ancillary qubits [346].MPE decoding, i.e., the process of finding the most likely error, is \(NP\)-complete in general [347,348]. If the noise model is such that the most likely error is the lowest-weight error, then ML decoding is called minimum-weight decoding. Maximum-likelihood (ML) decoding (a.k.a. degenerate maximum-likelihood decoding), i.e., the process of finding the most likely error class (up to degeneracy of errors), is \(\#P\)-complete in general [349].Incorporating faulty syndrome measurements can be done by performing spacetime decoding, i.e., using data from past rounds for decoding syndromes in any given round. If a decoder does not process syndrome data sufficiently quickly, it can lead to the backlog problem [350], slowing down the computation.Splitting decoders [351].Trellis decoder, which builds a compact representation of the algebraic structure of the normalizer \(\mathsf{N(S)}\) [352].Quantum extension of GRAND decoder [353].Deep neural-network probabilistic decoder [354].Generalized belief propagation (GBP) [237] based on a classical version [238].Integer optimization decoder [355].Autonomous Lindbladian based decoders for codes encoding a single logical qubit [356].For codes encoding a single logical qubit, logical information can be extracted by single-qubit operations and classical communication [357].Correlated decoding can improve performance of Clifford and non-Clifford entangling gates [358].Detector graphs [342,359] and detector error models [167] can be used to design syndrome extraction circuits and logical measurements.Fault-tolerant constant-depth unencoder transforming logical states into physical states using single-qubit measurements [309].Degenerate erasure decoder showing near ML decoding for various codes [360].
Qudit-into-oscillator code Given an encoding of a finite-dimensional code, a decoder that yields the optimal entanglement fidelity can be obtained by solving a semi-definite program [24,25] (see also Ref. [26]). This approximate QEC technique can be adapted to bosonic codes as long as they are restricted to a finite-dimensional subspace of the oscillator Hilbert space [65].
Raussendorf-Bravyi-Harrington (RBH) cluster-state code MBQC syndrome extraction consists of single-qubit measurements and classical post-processing. The six \(X\)-measurements of qubits on the faces of a cube of the bcc lattice multiply to the product of the six cluster-state stabilizers whose vertices are on the faces of the cube. Such measurements, if done on a 2D slice, also yield \(Z\)-type syndromes on the next slice.Minimum weight perfect-matching (MWPM) [58,192] (based on work by Edmonds on finding a matching in a graph [193,194]).
Rotated surface code Only certain syndrome extraction schedules are distance-preserving [361].Local neural-network using 3D convolutions, combined with a separate global decoder [362].Iterative CNOT decoder [363].
Singleton-bound approaching AQECC Quantum list decodable [140].
Spacetime circuit code Efficient decoders can be constructed for some circuits [364].
Spin cat code Measurement-free error correction protocol [365].
Square-lattice GKP code Syndrome measurement can be done by applying a controlled-displacement controlled by an ancilla qubit. The syndrome information can be obtained by measuring the ancilla qubit after controlled-displacement opearation. See Section. 2D in [366].Decoder [367] based on Knill error correction (a.k.a. telecorrection [74]), which is based on teleportation [75,76].Pauli \(X\),\(Y\) and \(Z\) measurements can be performed by measuring \(-\hat{p},\hat{x}-\hat{p}\) and \(\hat{x}\) repectively. If the measurement outcome is closed to an even multiple of \(\sqrt{\pi}\), then the outcome is +1. If the measurement outcome is closed to an odd multiple of \(\sqrt{\pi}\), then the outcome is -1. See Section. 2D in [366].Reinforcement learning decoder that uses only one ancilla qubit [368]. It has been extended to utilize previously measured syndrome information [369].
Square-octagon (4.8.8) color code Fault-tolerant syndrome extraction circuits [370].Matching decoder [371,372].Integer-program (IP) decoder [370].Two-copy surface-code decoder [373].
Stabilizer code The structure of stabilizer codes allows for straightforward syndrome-based decoding because the stabilizer generators serve as the code's check operators, and their eigenvalues serve as the error syndromes. The error correction process involves measuring the stabilizer generators and applying correcting Pauli-type operators based on the measurement outcomes.
String-net code Syndrome measurement circuits analyzed in Ref. [374].Clustering decoder [123].
Subsystem CSS code Steane-type decoder utilizing data from the underlying classical codes [375].
Subsystem color code Clustering decoder [376].Erasure decoder [377].Gauge-fixing decoders [377,378].
Subsystem hypergraph product (SHP) code Efficient decoder [145].
Subsystem modular-qudit CSS code Steane-type decoder utilizing data from the underlying classical codes [375].
Subsystem modular-qudit stabilizer code Syndrome measurements are obtained by first measuring gauge operators of the code and taking their products, which give the stabilizer measurement outcomes. The order in which gauge operators are measured is important since they do not commute. There is a sufficient condition for inferring the stabilizer syndrome from the measurements of the gauge generators [146; Appendix].Decoder for certain geometrically local subsystem codes from hypergraphs [147].
Surface-17 code Lookup table [361].
Tensor-network code The decoder is created by creating a decoding quantum circuit with dangling legs replaced with input/output wires, and tensors converted to unitary gates. Maximum likelihood decoding can be used when the tensors are stabilizer codes.Tensor-network decoder when the tensor network is contractible via stabilizer isometries [379]. Independent logical qubits can be decoded in parallel [380].Tensor-network-based decoder when the encoding unitary is known [381].
Three-qutrit code The quantum information (the secret) can be recovered from a unitary transformation acting on only two qutrits, \( U_{ij} \otimes I \), where \(U_{ij}\) acts on qutrits \(i,j\) and \(I\) is the identity on the remaining qutrit. By the cyclic structure of the codewords, this unitary transformation performs a permutation that recovers the information and stores it in one of the two qutrits involved in recovery.
Triangular surface code The decoding uses a single decoding graph since the triangle code is not a CSS code. Nodes of the graph are located at each stabilizer (center of the triangle graph) and have red or blue edges, where red associates with \(X\) errors and blue with \(Z\) errors. To take into account any errors from measuring the error syndrome, a three-dimensional stack of the decoding graphs is laid on top of the code with vertical edges connecting to qubits between layers [382].
Twisted XZZX toric code Fault-tolerant syndrome extraction circuits using flag qubits [383].AMBP4, a quaternary version [172] of the MBP decoder [173].
Two-component cat code All-optical decoder [384] based on Knill error correction (a.k.a. telecorrection [74]), which is based on teleportation [75,76].
Union stabilizer (USt) code Error-detection algorithm [83–85].
Very small logical qubit (VSLQ) code Logical qubit can be measured with physical qubit measurements along \(X\). Can be implemented by engineering a coupling of one of the qubits to a readout cavity via the interaction \(\sigma_x (a+a^\dagger)\) [385]. This results in an \(X\)-dependent shift of the readout cavity resonance which can be measured.Star-code autonomous correction scheme [386].
Wasilewski-Banaszek code Destructive measurement with photon number measurements on each mode.
X-cube Floquet code Period-six measurement sequence utilizing two-qubit measurements [387].
X-cube model code Parallelized matching decoder [80].
XYZ color code Efficient ML decoder at infinite bias [388].Cellular-automaton decoder [388].
XYZ ruby Floquet code Period-three and period-six measurement sequences utilizing two-qubit measurements [389].
XYZ\(^2\) hexagonal stabilizer code Maximum-likelihood decoding using the EWD decoder [390].
XZZX surface code MWPM decoder, which can be used for \(X\) and \(Z\) noise. For \(Y\) noise, a variant of the matching decoder could be used like it is used for the XY code in Ref. [391]. Decoding complexity scales as order \(O(n^3)\) because the code is non-CSS [172][391; Supplement].
Yoked surface code Soft information from the inner surface codes can be utilized via a message passing algorithm [106].Yokes can be measured using lattice surgery [392].
\(((9,12,3))\) qubit code Fault-tolerant scheme that converts the required POVM into 10 binary measurements whose redundancy is guaranteed by a classical code [67].
\([[12,2,4]]\) carbon code Syndrome extraction circuit based on Knill error correction (a.k.a. telecorrection [74]), but using only two code blocks instead of three [393; Fig. 5].
\([[144,12,12]]\) gross code The GDG sliding-window decoder [307], with a realization achieving a worst-case decoding latency of 3ms per window.AC decoder is faster than ordinary BP-OSD with no reduction of fidelity [298].
\([[16,6,4]]\) Tesseract color code Post-selected fault-tolerant syndrome extraction [394,395].
\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code Latin rectangle encoder [396].Efficient decoder [397].
\([[5,1,3]]_{\mathbb{R}}\) Braunstein five-mode code Error correction can be done using linear-optical elements and feedback [398].
\([[6,4,2]]\) error-detecting code Efficient decoder for the many-hypercube code [399].
\([[7,1,3]]\) Steane code Shor error correction fidelity calculation [400,401].
\([[7,1,3]]\) bare code Fault-tolerant syndrome extraction using only one ancilla per stabilizer generator measurement.
\([[9,1,3,3]]\) Nine-qubit Bacon-Shor code Message passing for \([[9,1,3,3]]\) Bacon-Shor code [402].
\([[9,1,3]]\) Shor code Bit- and phase-flip circuits utilize CNOT and Hadamard gates ([403], Fig. 10.6).
\([[9,1,3]]_{\mathbb{R}}\) Lloyd-Slotine code Syndromes are real-valued, and decoding is done by a continuous version of majority voting (a.k.a. triple modular redundancy).
\(\chi^{(2)}\) code Linear optics and \(\chi^{(2)}\) interactions.

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