The Error Correction Zoo

Victor V. Albert; Philippe Faist

Here is a list of all classical and quantum codes that have notable decoders.

Name	Decoder(s)
3D surface code	Flip decoder and its modification p-flip [1].Tensor-network decoder [2].
Alternant code	Variation of the Berlekamp-Welch algorithm [3].Guruswami-Sudan list decoder [4].
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 [5].
Approximate quantum error-correcting code (AQECC)	Given an encoding, a decoder that yields the optimal entanglement fidelity can be obtained by solving a semi-definite program [6,7] (see also Ref. [8]).The Petz recovery map (a.k.a. the transpose map) [9,10], a quantum channel determined by the codespace and noise channel, recovers information perfectly for strictly correctable noise and yields an infidelity of recovery that is at most twice away from the infidelity of the best possible recovery [11]. The infidelity of a modified Petz recovery map under erasure can be bounded using the conditional mutual information [12–14].
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.
B-code	Efficient decoding algorithm against erasures [15].
BPSK c-q code	Linear-optical quantum receiver [16].Kennedy receiver [17,18].Photon-number resolving detector [19].Non-Gaussian near-optimal receiver [18].Multi-stage quantum receiver [20].
Bacon-Casaccino subsystem code	Efficient decoder [21].
Bacon-Shor code	Utilizing the mapping of the effect of the noise to a statistical mechanical model [22,23] yields several copies of the 1D Ising model [24; 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 [25,26].
Balanced code	Efficient decoder [27–29].
Balanced product (BP) code	BP-OSD decoder [30].
Binary BCH code	Peterson decoder with runtime of order \(O(n^3)\) [31,32] (see exposition in Ref. [33]).Berlekamp-Massey decoder with runtime of order \(O(n^2)\) [34,35] and modification by Burton [36]; see also [37,38].Sugiyama et al. modification of the extended Euclidean algorithm [39,40].Guruswami-Sudan list decoder [4].
Binary Varshamov-Tenengolts (VT) code	Decoder based on checksums \(\sum_{i=1}^n i~x_i^{\prime}\) of corrupted codewords \(x_i^{\prime}\) [41,42].
Binary code	For few-bit 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 decoder.Given a received string \(x\) and an error bound \(e\), a list decoder returns a list of all codewords that are at most \(e\) from \(x\) in Hamming distance. The number of codewords in a neighborhood of \(x\) has to be polynomial in \(n\) in order for this decoder to run in time polynomial in \(n\).
Binary quantum Goppa code	Farran algorithm [43].
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 [44]. 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 [44,45].
Bivariate bicycle code	Syndrome extraction circuit requires seven layers of CNOT gates regardless of code length. BP-OSD decoder [30] has been extended [46] to account for measurement errors (i.e., the circuit-based noise model [47]).Random and optimized syndrome extraction schedules from Ref. [46] are not distance preserving.
Block code	Decoding an error-correcting code is equivalent to finding the ground state of some statistical mechanical model [48].
Bose–Chaudhuri–Hocquenghem (BCH) code	Berlekamp-Massey decoder with runtime of order \(O(n^2)\) [34,35,49] and modification by Burton [36]; see also [37,38].Gorenstein-Peterson-Zierler decoder with runtime of order \(O(n^3)\) [31,50] (see exposition in Ref. [33]).Sugiyama et al. modification of the extended Euclidean algorithm [39,40].Guruswami-Sudan list decoder [4] and modification by Koetter-Vardy for soft-decision decoding [51].
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 [52].
Braunstein five-mode code	Error correction can be done using linear-optical elements and feedback [53].
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.
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.
Classical Goppa code	Algebraic decoding algorithms [54]. If \( \text{deg} G(x) = 2t \) , then there exists a \(t\)-correcting algebraic decoding algorithm for \( \Gamma(L,G) \).Sugiyama et al. modification of the extended Euclidean algorithm [39,40].Guruswami-Sudan list decoder [4].Binary Goppa codes can be decoded using a RS-based decoder [55].
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.Structured error recovery [56], which reduces to syndrome-based recovery for additive (i.e., stabilizer) CWS codes.
Coherent FSK (CFSK) c-q code	Bondurant receiver [57].Cyclic receiver [58].Time-resolving receiver [59–61].Bayesian inference [59].
Coherent-state c-q code	Joint-detection receiver that can attain channel capacity [62].Various near-optimal receiver designs that can handle arbitrary constellations of coherent states with possible degeneracies [63].
Color code	Projection decoder [64].Matching decoder gives low logical failure rate [65].Integer-program-based decoder [66].Restriction decoder [67].Cellular-automaton decoder for the \(XYZ\) color code [68].MaxSAT-based decoder [69].
Concatenated bosonic code	Decoder exploiting analog information from the inner code for bosonic codes concatenated with qubit QLDPC codes [70].
Concatenated code	Generalized minimum-distance decoder [71].
Concatenated quantum code	The effective channel for a concatenation of codes is the composition of the codes' effective channels [72].
Convolutional code	Decoders based on the Viterbi algorithm (trellis decoding) were developed first, which result in the most-likely codeword for the encoded bits [73]. Following, other trellis decoders such as the BCJR decoding algorithm [74] were developed later.
Cyclic linear \(q\)-ary code	Meggitt decoder [75].Information set decoding (ISD) [76], a probabilistic decoding strategy that essentially tries to guess \(k\) correct positions in the received word, where \(k\) is the size of the code. Then, an error vector is constructed to map the received word onto the nearest codeword, assuming the \(k\) positions are error free. When the Hamming weight of the error vector is low enough, that codeword is assumed to be the intended transmission.Permutation decoding [77].
Cyclic linear binary code	Meggitt decoder [75].Information set decoding (ISD) [76], a probabilistic decoding strategy that essentially tries to guess \(k\) correct positions in the received word, where \(k\) is the size of the code. Then, an error vector is constructed to map the received word onto the nearest codeword, assuming the \(k\) positions are error free. When the Hamming weight of the error vector is low enough, that codeword is assumed to be the intended transmission.Permutation decoding [77].
Cyclic quantum code	Adapted from the Berlekamp decoding algorithm for classical BCH codes [78].
Dinur-Hsieh-Lin-Vidick (DHLV) code	Linear-time decoder utilizing the small set-flip decoder [79] for \(Z\) errors and a reconstruction procedure for \(X\) errors [80].
Doubled color code	ML decoder that can utilize a history of syndromes, based on the Walsh-Hadamard transform [81].
EA qubit stabilizer code	Optical implementation of a minimal code using hyper-entangled states [82].
Eigenstate thermalization hypothesis (ETH) code	An explicit universal recovery channel for the ETH code is given in [83].
Evaluation AG code	Generalization of plane-curve decoder [84,85]. Another decoder [86] was later showed to be equivalent in Ref. [87]. Application of several algorthims in parallel can be used to decode up to half the minimum distance [88,89]. Computational procedure implementing these decoders is based on an extension of the Berlekamp-Massey algorithm by Sakata [90–92].Decoder based on majority voting of unknown syndromes [93] decodes up to half of the minimum distance [94].List decoders generalizing Sudan's RS decoder by Shokrollahi-Wasserman [95] and Guruswami-Sudan [4].
Expander LP code	Linear-time decoder [96].Logarithmic-time subroutine [97].
Expander code	Decoding can be done in \(O(n)\) runtime using a greedy flip decoder [98]. The algorithm consists of flipping a bit of the received word if it will result in a greater number of satisfied parity checks. This is repeated until a codeword is reached.
Fiber-bundle code	Greedy algorithm can be used to efficiently decode \(X\) errors, but no known efficient decoding of \(Z\) errors yet [25].
Fibonacci code	An efficient algorithm base on minimum-weight perfect matching [99], which can correct high-weight errors that span rows and columns of the 2D lattice, with failure rate decaying super-exponentially with \(L\).
Fibonacci string-net code	Clustering decoder (provides best known threshold for this code) [100–102].Fusion-aware iterative minimum-weight perfect matching decoder. Note that ordinary MWPM decoders do not produce a threshold with this code [102].Cellular automaton decoder [103].
Finite-dimensional error-correcting code (ECC)	Capacity-achieving Guessing Random Additive Noise Decoding (GRAND) [104] (see also [105]).
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 inverse of the encoder \(\cal{U}\), which does not correct errors.Quantum machine-learning based decoders such as quantum convolutional neural networks [106] and quantum autoencoders [107].
Five-qubit perfect code	Combined dynamical decoupling and error correction protocol on individually-controlled qubits with always-on Ising couplings [108].Symmetric decoder correcting all weight-one Pauli errors. The resulting logical error channel after coherent noise has been explicitly derived [109].
Folded RS (FRS) code	Guruswami and Rudra [110,111] achieved list-decoding up to \(1-\frac{k}{n}-\epsilon\) fraction of errors using the Parvaresh-Vardy algorithm [112]; see Ref. [113] for a randomized construction.Folded RS codes, concatenated with suitable inner codes, can be efficiently list-decoded up to the Blokh-Zyablov bound [110,114].
Folded quantum Reed-Solomon (FQRS) code	Quantum list decodable [115].
Fountain code	Invert the fragment generator matrix resulting from the continuous encoding process. If exactly \(K\) packets are received, then the probability of decoding correctly is \(0.289\). Extra packets increase this probability exponentially. The decoding runtime is dominated by the matrix inversion step, which takes order \(O(n^3)\) time.
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 [116].
GKP-stabilizer 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 [117].
GNU permutation-invariant code	For a family of shifted gnu codes, decoding can be done using projection, probability amplitude rebalancing, and gate teleportation in time \(O(n^2)\) [118].
Gabidulin code	Fast decoder based on a transform-domain approach [119].
Galois-field \(q\)-ary code	For small \(n\), 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 decoder.Given a received string \(x\) and an error bound \(e\), a list decoder returns a list of all codewords that are at most \(e\) from \(x\). The number of codewords in a neighborhood of \(x\) has to be polynomial in \(n\) in order for this decoder to run in time polynomial in \(n\).
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 non-stabilizer code	The decoding circuit involves the application of phase estimation.
Generalized RS (GRS) code	The decoding process of GRS codes reduces to the solution of a polynomial congruence equation, usually referred to as the key equation. Decoding schemes are based on applications of the Euclid algorithm to solve the key equation.Berlekamp-Massey decoder with runtime of order \(O(n^2)\) [34,35,49].Guruswami-Sudan list decoder [4] and modification by Koetter-Vardy for soft-decision decoding [51].
Generalized Shor code	Efficient decoder [21].
Generalized bicycle (GB) code	BP-OSD decoder [30].
Generalized surface code	Improved BP-OSD decoder [120].
Golay code	Majority decoding for the extended Golay code [121].Decoder for the extended Golay code using the hexacode [122].Both Golay codes have a trellis representation and can thus be decoded using trellis decoding [123,124].Bounded-distance decoder requiring at most 121 real operations [125].
Gold code	General decoding is done by building a sparse parity check matrix, followed by applying an iterative message passing alogirithm. [126].
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) [127] (a.k.a. closest lattice point problem) or by using other effective iterative schemes when e.g. the lattice represents a concatenated GKP code [5,128–130]. While the decoder time scales exponentially with number of modes \(n\) generically, the time can be polynomial in \(n\) for certain codes [131].
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 [132] and neural-network [133] decoders.
Hermitian code	Unique decoding using syndromes and error locator ideals for polynomial evaluations. Note that Hermitian codes are linear codes so we can compute the syndrome of a received vector. Moreover, akin to the error-locator ideals found in decoding Reed-Solomon codes, for the multivariate case we must define an error locator ideal \(\Lambda \) such that the variety of this ideal over \(\mathbb{F}^{2}_q\) is exactly the set of errors. The Sakata algorithm uses these two ingredients to get a unique decoding procedure [90].
Hexacode	Bounded-distance decoder requiring at most 34 real operations [125].
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 [134]. The bilayer architecture allows for logical entangling gates between logical surface-code patches.
High-dimensional expander (HDX) code	For 2D simplicial complexes, cycle code decoder admitting a polynomial-time decoding algorithm can be used [135].
Homological 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. [136] calls the QND Gate and which is defined as \(QND_c \| x , y \rangle = \|x + c y, y \rangle\).
Homological product code	Union-find [137].
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.
Hypergraph product (HGP) code	ReShape decoder that uses minimum weight decoders for the classical codes used in the hypergraph construction [138].2D geometrically local syndrome extraction circuits with depth order \(O(\sqrt{n})\) using order \(O(n)\) ancilla qubits [139].Improved BP-OSD decoder [120].Erasure-correction can be implemented approximately with \(O(n^2)\) operations with quantum generalizations [140] of the peeling and pruned peeling decoders [141], 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 [142].
Interleaved RS (IRS) code	Decoder that corrects up to \(1-\frac{2k+n}{3n}\) fraction of random errors [143].Decoder that corrects up to \(1-(\frac{k}{n})^{2/3}\) fraction of random errors [144].
Irregular repeat-accumulate (IRA) code	Linear-time decoder [145].
Justesen code	Generalized minimum distance decoding [146].
Kerdock code	Soft decision decoding involves extending the Fast Hadamard Transform decoding algorithm for the binary first-order Reed-Muller code to Kerdock code [147].Complexity of soft decision decoding algorithm: \(4^m\) multiplications and \(m4^m\) additions [147,148].
Kitaev surface code	Using data from multiple syndrome measurements prior to decoding allows for correcting syndrome measurement errors. Syndrome measurements are distance-preserving because syndrome extraction circuits can be designed to avoid hook errors [22].Expanding diamonds decoder correcting errors of some maximum fractal dimension [149]. 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) [22,150] (based on work by Edmonds on finding a matching in a graph [151,152]), 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 [22,151,153]. MWPM solves the MPE decoding problem exactly for independent \(X\) and \(Z\) noise. MPE decoding is \(NP\)-hard in general for the surface code [154].Tensor network decoder [155] solves the ML decoding problem under independent \(X,Z\) noise for the surface code and takes time of order \(O(n^2)\) [155]. ML decoding [22] is \(\#P\)-hard in general for the surface code [154].Union-find decoder [156] uses the union-find data structure [157–159], 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) [160]. Belief union find is a combination of belief-propagation and union-find [161]. Strictly local (as opposed to partially local) union find [162] has a worst-case runtime of order \(O(d^3)\) in the distance \(d\).Modified MWPM decoders: pipeline MWPM (accounting for correlations between events) [163,164], parity blossom MWPM and fusion blossom MWPM [165], and 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) [160].Belief perfect matching is a combination of belief-propagation and MWPM [161].Renormalization group (RG) [166–168].Markov-chain Monte Carlo [169].Cellular automaton [170,171].Neural network [172–175], reinforcement learning [176], and transformer-based [177] decoders.Decoders can be augmented with a pre-decoder [178,179], which can allow for some processing to be done inside the cryogenic environment of the quantum system [180].Sliding-window [181,182] and parallel-window [181] parallelizable decoders, designed to overcome the backlog problem, can be combined with many inner decoders, such as MWPM or union-find.Generalized belief propagation (GBP) [183] based on a classical version [184]. See Ref. [185] for a review of BP decodes.Color-code decoder [186].
Linear \(q\)-ary code	Maximum likelihood (ML) decoding. This algorithm decodes a received word to the most likely sent codeword based on the received word. ML decoding of reduced complexity is possible for virtually all \(q\)-ary linear codes [187].Optimal symbol-by-symbol decoding rule [188].Information set decoding (ISD) [189], a probabilistic decoding strategy that essentially tries to guess \(k\) correct positions in the received word, where \(k\) is the size of the code. Then, an error vector is constructed to map the received word onto the nearest codeword, assuming the \(k\) positions are error free. When the Hamming weight of the error vector is low enough, that codeword is assumed to be the intended transmission.Generalized minimum-distance decoder [71].Soft-decision maximum-likelihood trellis-based decoder [190].Random linear codes over large fields are list-recoverable and list-decodable up to near-optimal rates [191].Extensions of algebraic-geometry decoders to linear codes [192,193].
Linear binary code	Decoding an arbitary linear binary code is \(NP\)-complete [194].Slepian's standard-array decoding [195].Recursive maximum likelihood decoding [196].Transformer neural net for soft decoding [197].
Locally decodable code (LDC)	LDCs admit local decoders, i.e., decoders whose runtime scales polylogarithmically with \(n\).
Low-density parity-check (LDPC) code	Message-passing algorithm called belief propagation (BP) [198–201].Soft-decision Sum-Product Algorithm (SPA) [198,199,202] and its simplification the Min-Sum Algorithm (MSA) [203].Linear programming [204–206].Iterative LDPC decoders can get stuck at stopping sets of their Tanner graphs [207], with decoder performance improving with the size of the smallest stopping set; see [208; Sec. 21.3.1] for more details. The smallest stopping set size can reach the minimum distance of the code [209].Ensembles of random LDPC codes under iterative decoders are subject to the concentration theorem [199,210]; see [208; Thm. 21.7.1] for the case of the BEC.Reinforcement learning [211].
Low-depth random Clifford-circuit qubit code	Minimum-weight decoding via using tropical tensor networks [212].
Low-rank parity-check (LRPC) code	Efficient probabilistic decoder [213].Mixed decoder [214].
Luby transform (LT) code	Sum-Product Algorithm (SPA), often called a peeling decoder [215,216], similar to belief propagation [217].
MacKay-Neal LDPC (MN-LDPC) code	Free-energy minimization and a BP decoder [218].
Matrix-product code	Decoder up to half of the minimum distance for NSC codes [219].
Melas code	Algebraic decoder [220].
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	The structure of stabilizer codes allows for syndrome-based decoding, where errors are corrected based on the results of stabilizer measurements (syndromes).Trellis decoder for prime-dimensional qudits, which builds a compact representation of the algebraic structure of the normalizer \(\mathsf{N(S)}\) [221].
Modular-qudit surface code	Renormalization-group decoder [222].
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) [223]
NTRU-GKP code	Efficient decoder against stochastic displacement noise because the decoding problem is equivalent to decrypting the NTRU cryptosystem.
Newman-Moore code	Efficient decoder [224].
Niset-Andersen-Cerf code	Optical decoder using three beam splitters, electronic gain detectors, and two phase-insensitive amplifiers as described in Ref. [225].
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 [226]), which is based on teleportation [227,228]. This type of error correction avoids the complicated correction procedures typical in Fock-state codes, but requires a supply of clean codewords [229]. Performance of this method was analyzed in Ref. [230].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 [231,232] on an ancillary mode. See Section 4.B.1 of Ref. [229].
On-off keyed (OOK) c-q code	Dolinar receiver [233].Superconducting transition edge sensor (TES) photon-number resolving detector [234].
Orthogonal Spacetime Block Code (OSTBC)	Maximum-likelihood decoding can be achieved with only linear processing [235].
PPM c-q code	Conditional pulse nulling (CPN) receiver [236].
PSK c-q code	Multi-stage quantum receivers [237–242].Bayesian inference [59].
Pair-cat code	Lindbladian-based dissipative encoding utilizing two-mode two-photon absorption [243]. Encoding passively protects against cavity dephasing, suppressing dephasing noise exponentially with \(\gamma^2\).
Parvaresh-Vardy (PV) code	PV codes can be list-decoded up to \(1-(t k/n)^{1/(t+1)}\) fraction of errors. This result improves over the Guruswami-Sudan algorithm for ordinary RS codes, which list-decodes up to \(1-\sqrt{k/n}\) fraction of errors.
Pastawski-Yoshida-Harlow-Preskill (HaPPY) code	Greedy algorithm for decoding specified in Ref. [244].
Permutation spherical code	Efficient maximum-likelihood decoder determining the Voronoi region of an error word.
Permutation-invariant code	Schur-Weyl-transform based decoder for qubit permutation-invariant codes [245]. 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 [246] and the quantum Schur transform.
Plane-curve code	Generalization of the Peterson algorithm for BCH codes [247].
Polar c-q code	Quantum-limited successive-cancellation (SC) joint-detection receiver [248].
Polar code	Successive cancellation (SC) decoder [249].Successive cancellation list (SCL) decoder [250] and a modification utilizing sequence repetition (SR-List) [251].Soft cancellation (SCAN) decoder [252,253].Belief propagation (BP) decoder [254].Noisy quantum gate-vased decoder [255].
Preparata code	Preparata Codes can be decoded using a syndrome calculation based algorithm to correct all error patterns of Lee weight atmost 2 and detect all/ some error patterns of Lee weight 3/ 4 [147,148].
Quantum Lego 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 [256].Tensor-network-based decoder when the encoding unitary is known [257].
Quantum Tanner code	Linear-time potetial-based decoder similar to the small-set-flip decoder for quantum expander codes [79].Linear-time decoder [96].Logarithmic-time mismatch decomposition decoder [97].
Quantum convolutional code	ML decoder [258].
Quantum expander code	Small set-flip linear-time decoder, which corrects \(\Omega(n^{1/2})\) adversarial errors [259].Log-time decoder [260].Constant-time decoder [261].2D geometrically local syndrome extraction circuits acting on a patch of \(N\) physical qubits have to be of depth at least \(\Omega(n/\sqrt{N})\). More generally, there is a tradeoff between the depth \(D\) and width \(W\) of a syndrome extraction circuit, namely, \(D \geq n/\sqrt{W}\) [139].
Quantum low-density parity-check (QLDPC) code	Belief-propagation (BP) decoder [262] and neural BP decoder [263] for qubit codes. Partially and fully decoupled BP decoders, which uses the decoupling representation, yield improvements against depolarizing noise [264].Non-binary decoding algorithm for CSS-type QLDPC codes [265].BP-OSD decoder adds a post-processing step based on ordered statistics decoding (OSD) to the belief propogation (BP) decoder [30].2D geometrically local syndrome extraction circuits with bounded depth using order \(O(n^2)\) ancilla qubits [139].Soft (i.e., analog) syndrome iterative belief propagation 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) [266].Message-passing decoder utilizing stabilizer inactivation (MP-SI) for CSS-type QLDPC qubit codes [267].Extension of the union-find decoder for qubit QLDPC codes, as well as a related heuristic decoder [268].Sliding-window decoding [269].
Quantum polar code	Quantum successive-cancellation list decoder (SCL-E) for quantum polar codes that do not need entanglement assistance [270].
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 [271,272]. An automaton by Gacs yields a decoder for a 1D lattice [273].Machine learning algorithm to implement continuous error-correction for the three-qubit quantum repetition code [274].
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 [275].
Qubit CSS code	Coherent decoders allow for measurement-free error correction [276]. One method is table/multi-control decoding [277], which scales exponentially with the number of ancillas used in syndrome measurement. Another method, the Ising-based decoder, utilizes the mapping of the effect of the noise to a statistical mechanical model [22,23] such that the decoding problem maps to preparation of the ground state of an Ising model.Decoders based on neural networks [278].
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 bit string. In the more involved circuit-level noise model, every component of the syndrome extraction circuit can be faulty.Hook errors are syndrome measurement circuit faults that cause more than one data-qubit error [22]. 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 to detect hook errors may be necessary to yield fault-tolerant decoders.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.Decoders are characterized by an effective distance or 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.
Qubit stabilizer code	The structure of stabilizer codes allows for syndrome-based decoding, where errors are corrected based on the results of stabilizer measurements (syndromes). The size of the circuit extracting the syndrome depends on the weight of its corresponding stabilizer generator. MPE decoding, i.e., the process of finding the most likely error, is \(NP\)-complete in general [279,280]. 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 [281].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 [282], slowing down the computation.Splitting decoders [283].Trellis decoder, which builds a compact representation of the algebraic structure of the normalizer \(\mathsf{N(S)}\) [284].Quantum extension of GRAND decoder [285].Deep neural-network probabilistic decoder [286].Generalized belief propagation (GBP) [183] based on a classical version [184].
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 [6,7] (see also Ref. [8]). 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 [44].
Random code	Ball-collision decoding [287].Information set decoding (ISD) [288] and Finiasz and Sendrier (FS-ISD) decoding [289].
Rank-metric code	Polynomial-reconstruction Berlekamp-Welch based decoder [290].Berlekamp-Massey based decoder [291].
Raptor (RAPid TORnado) code	Raptor codes can be decoded using inactivation decoding [292], a combination of belief-propogation and Gaussian elimination decoding.
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) [22,150] (based on work by Edmonds on finding a matching in a graph [151,152]).
Reed-Muller (RM) code	Reed decoder with \(r+1\)-step majority decoding corrects \(\frac{1}{2}(2^{m-r}-1)\) errors [293] (see also Ch. 13 of Ref. [294]).Sequential code-reduction decoding [295].First-order (\(r=1\)) RM codes admit specialized decoders [296].
Reed-Solomon (RS) code	Although using iFFT has its counterpart iNNT for finite fields, the decoding is usually standard polynomial interpolation in \(k=O(n\log^2 n)\). However, in erasure decoding, encoded values are only erased in \(r\) points, which is a specific case of polynomial interpolation and can be done in \(O(n\log n)\) by computing product of the received polynomial and an erasure locator polynomial and using long division to find an original polynomial. The long division step can be omitted to increase speed further by only dividing the derivative of the product polynomial, and derivative of erasure locator polynomial evaluated at erasure locations.Berlekamp-Massey decoder with runtime of order \(O(n^2)\) [34,35].Gorenstein-Peterson-Zierler decoder with runtime of order \(O(n^3)\) [31,50] (see exposition in Ref. [33]).Berlekamp-Welch decoder with runtime of order \(O(n^3)\) [297] (see exposition in Ref. [298]), assuming that \(t \geq (n+k)/2\).Gao decoder using extended Euclidean algorithm [299].Fast-Fourier-transform decoder with runtime of order \(O(n \text{polylog}n)\) [300].List decoders try to find a low-degree bivariate polynomial \(Q(x,y)\) such that evaluation of \(Q\) at \((\alpha_i,y_i)\) is zero. By choosing proper degrees, it can be shown such polynomial exists by drawing an analogy between evaluation of \(Q(\alpha_i,y_i)\) and solving a homogenous linear equation (interpolation). Once this is done, one lists roots of \(y\) that agree at \(\geq t\) points. The breakthrough Sudan list-decoding algorithm corrects up to \(1-\sqrt{2R}\) fraction of errors [301]. Roth and Ruckenstein proposed a modified key equation that allows for correction of more than \(\left\lfloor (n-k)/2 \right\rfloor\) errors [302]. The Guruswami-Sudan algorithm improved the Sudan algorithm to \(1-\sqrt{R}\) [4]; see Ref. [303] for bounds. A further modification by Koetter and Vardy is used for soft-decision decoding [51] (see also Ref. [304]).
Regular binary Tanner code	Parallel decoding algorithm corrects a fraction \(\delta_0^2/48\) of errors for Tanner codes [98]. A modification of said algorithm improves the fraction to \(\delta_0^2/4\) with no extra cost to complexity [305].
Repetition code	Calculate the Hamming weight \(d_H\) of the code. If \(d_H\leq \frac{n-1}{2}\), decode the code as 0. If \(d_H\geq \frac{n+1}{2}\), decode the code as 1.Automaton-like decoders for the repetition code on a 2D lattice, otherwise known as the classical 2D Ising model, were developed by Toom [271,272]. An automaton by Gacs yields a decoder for a 1D lattice [273].
Rotated surface code	Only certain syndrome extraction schedules are distance-preserving [306].Local neural-network using 3D convolutions, combined with a separate global decoder [307].
Simplex code	Due to the small size, it can be decoded according to maximum likelihood.Some faster decoders for the \(q=2\) case: [308,309]A quantum decoder for the \(q=2\) case: [310].
Single parity-check (SPC) code	If the receiver finds that the parity information of a codeword disagrees with the parity bit, then the receiver will discard the information and request a resend.Wagner's rule yields a procedure that is linear in \(n\) [311] (see [312; Sec. 29.7.2] for a description).
Singleton-bound approaching AQECC	Quantum list decodable [115].
Skew-cyclic code	Only given for skew-BCH codes, adapted froom standard BCH codes.
Spacetime circuit code	Efficient decoders can be constructed for some circuits [313].
Sphere packing	Each signal point is assigned its own Voronoi cell, and a received point is mapped back to the center of the Voronoi cell that it is located upon reception.
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 [314].Decoder [315] based on Knill error correction (a.k.a. telecorrection [226]), which is based on teleportation [227,228].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 [314].Reinforcement learning decoder that uses only one ancilla qubit [316].
String-net code	Fusing non-Abelian anyons cannot be done in one step [317].Syndrome measurement circuits analyzed in Ref. [318].Clustering decoder [101].
Subsystem CSS code	Steane-type decoder utilizing data from the underlying classical codes [319].
Subsystem color code	Clustering decoder [320].Erasure decoder [321].
Subsystem modular-qudit CSS code	Steane-type decoder utilizing data from the underlying classical codes [319].
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 [322; Appendix].Decoder for certain geometrically local subsystem codes from hypergraphs [323].
Surface-17 code	Lookup table [306].
Ta-Shma zigzag code	Unique and list decoders [324].
Tamo-Barg code	Polynomial evaluation over \(r\) points [325].
Tanner code	Min-sum and sum-product iterative decoders for binary Tanner codes [326,327]; see also [40,328]. These decoders can be improved using a probabilistic message-passing schedule [329].Any code can be put into normal form without significantly altering the underlying graph or the decoding complexity [330]. For an algebraic viewpoint on decoding, see [331].Iterative decoding is optimal for Tanner graphs that are free of cycles [327]. However, codes that admit cycle-free representations have bounds on their distances [332,333]; see [208,334].
Tensor-product code	The simple decoding algorithm (first decode all columns with \(C_1\), then all rows with \(C_2\)) corrects up to \((d_A d_B-1)/4 \) errors.Algorithms such as generalized minimum-distance decoding [71] or the min-sum algorithm can decode all errors of weight up to \((d_A d_B-1)/2\). Error location may be coupled with Viterbi decoding for every faulty sub-block [335].
Ternary Golay code	Decoder for the extended ternary Golay code using the tetracode [122].
Tetron Majorana code	Qubit readout can be done by charge sensing [336–339].
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.
Tornado code	Linear-time peeling decoder [141]. This decoder either terminates when it has removed a given erasure pattern or when it is stuch in a stopping set.
Torus-layer spherical code (TLSC)	Efficiently decodable [340].
Translationally invariant stabilizer code	Clustering decoder [170,341].
Two-component cat code	All-optical decoder [342] based on Knill error correction (a.k.a. telecorrection [226]), which is based on teleportation [227,228].
Two-dimensional hyperbolic surface code	Due to the symmetries of hyperbolic surface codes, optimal measurement schedules of the stabilizers can be found [343].
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 \(\sigma_x (a+a^\dagger)\) [344]. This results in an \(X\)-dependent shift of the readout cavity resonance which can be measured.Star-code autonomous correction scheme [345].
Wasilewski-Banaszek code	Destructive measurement with photon number measurements on each mode.
X-cube model code	Parallelized matching decoder [346].
XYZ\(^2\) hexagonal stabilizer code	Maximum-likelihood decoding using the EWD decoder [347].
XZZX surface code	Minimum-weight perfect matching 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. [348].
Yoked surface code	Outer code stabilizer generators are measured using lattice surgery.
Zetterberg code	Kallquist first described an algebraic decoding theorem [349]. A faster version was later provided in Ref. [350] and further modified in Ref. [351].
\([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS code	Efficient decoder [352].
\([[9,1,3]]\) Shor code	Bit- and phase-flip circuits utilize CNOT and Hadamard gates ([353], Fig. 10.6).
\(\chi^{(2)}\) code	Linear optics and \(\chi^{(2)}\) interactions.

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