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].
Alternant code Variation of the Berlekamp-Welch algorithm [2].Guruswami-Sudan list decoder [3].
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 [4].
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 [5][6] (see also Ref. [7]).The Petz recovery map (a.k.a. the transpose map) [8][9], 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 [10]. The infidelity of a modified Petz recovery map can be bounded using relative entropies between uncorrupted and corrupted code states on countably infinite Hilbert spaces [11].
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.
Bacon-Shor code 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 [17][18].
Balanced code Efficient decoder [19][20][21].
Balanced product code BP-OSD decoder [22].
Binary BCH code Peterson decoder with runtime of order $$O(n^3)$$ [23][24] (see exposition in Ref. [25]).Berlekamp-Massey decoder with runtime of order $$O(n^2)$$ [26][27] and modification by Burton [28]; see also [29][30].Sugiyama et al. modification of the extended Euclidean algorithm [31][32].Guruswami-Sudan list decoder [3].
Binary Varshamov-Tenengolts (VT) code Decoder based on checksums $$\sum_{i=1}^n i~x_i^{\prime}$$ of corrupted codewords $$x_i^{\prime}$$ [33][34].
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 [35].
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 [36]. 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 [37][36].
Bose–Chaudhuri–Hocquenghem (BCH) code Berlekamp-Massey decoder with runtime of order $$O(n^2)$$ [38][26][27] and modification by Burton [28]; see also [29][30].Gorenstein-Peterson-Zierler decoder with runtime of order $$O(n^3)$$ [23][39] (see exposition in Ref. [25]).Sugiyama et al. modification of the extended Euclidean algorithm [31][32].Guruswami-Sudan list decoder [3] and modification by Koetter-Vardy for soft-decision decoding [40].
Bosonic c-q code See review [41] for details on receivers used for bosonic c-q codes.
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 [42].
Braunstein five-mode code Error correction can be done using linear-optical elements and feedback [43].
Calderbank-Shor-Steane (CSS) stabilizer code Coherent decoders allow for measurement-free error correction [44]. One method is table/multi-control decoding [45], 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 [46][47] such that the decoding problem maps to preparation of the ground state of an Ising model.Decoders based on neural networks [48].
Cat code Measuring the Fock-state number modulo $$2S$$ can be used to determine if photon loss or excitation errors occurred.
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 [49]. 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 [31][32].Guruswami-Sudan list decoder [3].Binary Goppa codes can be decoded using a RS-based decoder [50].
Codeword stabilized (CWS) code There is no known efficient algorithm to decode non-additive (non-stabilizer) CWS codes.
Color code Projection decoder [56].Matching decoder gives low logical failure rate [57].Integer-program-based decoder [58].Restriction decoder [59].Cellular-automaton decoder for the $$XYZ$$ color code [60].
Concatenated code Generalized minimum-distance decoder [61].
Convolutional code Decoders based on the Viterbi algorithm (trellis decoding) were developed first, which result in the most-likely codeword for the encoded bits [62]. Following, other trellis decoders such as the BCJR decoding algorithm [63] were developed later.
Cyclic linear $$q$$-ary code Meggitt decoder [64].Information set decoding (ISD) [65], 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 [66].
Cyclic linear binary code Meggitt decoder [64].Information set decoding (ISD) [65], 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 [66].
Cyclic quantum code Adapted from the Berlekamp decoding algorithm for classical BCH codes [67].
Dinur-Hsieh-Lin-Vidick (DHLV) code Linear-time decoder utilizing the small set-flip decoder [68] for $$Z$$ errors and a reconstruction procedure for $$X$$ errors [69].
EA qubit stabilizer code Optical implementation of a minimal code using hyper-entangled states [70].
Eigenstate thermalization hypothesis (ETH) code An explicit universal recovery channel for the ETH code is given in [71].
Evaluation AG code Generalization of plane-curve decoder [72][73]. Another decoder [74] was later showed to be equivalent in Ref. [75]. Application of several algorthims in parallel can be used to decode up to half the minimum distance [76][77]. Computational procedure implementing these decoders is based on an extension of the Berlekamp-Massey algorithm by Sakata [78][79][80].Decoder based on majority voting of unknown syndromes [81] decodes up to half of the minimum distance [82].List decoders generalizing Sudan's RS decoder by Shokrollahi-Wasserman [83] and Guruswami-Sudan [3].
Expander code Decoding can be done in $$O(n)$$ runtime using a greedy flip decoder [84]. 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.
Expander lifted-product code Linear-time decoder [85].Logarithmic-time subroutine [86].
Fiber-bundle code Greedy algorithm can be used to efficiently decode $$X$$ errors, but no known efficient decoding of $$Z$$ errors yet [17].
Fibonacci code An efficient algorithm base on minimum-weight perfect matching [87], which can correct high-weight errors that span the lattice, with failure rate decaying super-exponentially with $$L$$.
Fibonacci string-net code Clustering decoder (provides best known threshold for this code) [88][89][90].Fusion-aware iterative minimum-weight perfect matching decoder. Note that ordinary MWPM decoders do not produce a threshold with this code [90].Cellular automaton decoder [91].
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 an encoder, which does not correct errors.Quantum machine-learning based decoders such as quantum convolutional neural networks [94] and quantum autoencoders [95].
Five-qubit perfect code Combined dynamical decoupling and error correction protocol on individually-controlled qubits with always-on Ising couplings [96].Symmetric decoder correcting all weight-one Pauli errors. The resulting logical error channel after coherent noise has been explicitly derived [97].
Folded RS (FRS) code Guruswami and Rudra [98][99] achieved list-decoding up to $$1-\frac{k}{n}-\epsilon$$ fraction of errors using the Parvaresh-Vardy algorithm [100]; see Ref. [101] for a randomized construction.Folded RS codes, concatenated with suitable inner codes, can be efficiently list-decoded up to the Blokh-Zyablov bound [98][102].
Folded quantum Reed-Solomon (FQRS) code Quantum list decodable [103].
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 [104].
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 [105].
Gabidulin code Fast decoder based on a transform-domain approach [106].
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 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 $$O(n^2)$$ [38][26][27].Guruswami-Sudan list decoder [3] and modification by Koetter-Vardy for soft-decision decoding [40].
Golay code Majority decoding for the extended Golay code [107].Decoder for the extended Golay code using the hexacode [108].Both Golay codes have a trellis representation and can thus be decoded using trellis decoding [109][110].Bounded-distance decoder requiring at most 121 real operations [111].
Gold code General decoding is done by building a sparse parity check matrix, followed by applying an iterative message passing alogirithm. [112].
Gottesman-Kitaev-Preskill (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 [113].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 [113].Reinforcement learning decoder that uses only one ancilla qubit [114].
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 decoder [115].
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 [78].
Hexacode Bounded-distance decoder requiring at most 34 real operations [111].
Higher-dimensional surface code Improved BP-OSD decoder [116].
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. [117] calls the QND Gate and which is defined as $$QND_c | x , y \rangle = |x + c y, y \rangle$$.
Homological product code Union-find [118].
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.
Hypergraph product code ReShape decoder that uses minimum weight decoders for the classical codes used in the hypergraph construction [119].Erasure-correction can be implemented approximately with $$O(n^2)$$ operations, with a probabilistic version running in $$O(n^{1.5})$$ operations [120].Improved BP-OSD decoder [116].
Interleaved RS (IRS) code Decoder that corrects up to $$1-\frac{2k+n}{3n}$$ fraction of random errors [121].Decoder that corrects up to $$1-(\frac{k}{n})^{2/3}$$ fraction of random errors [122].
Justesen code Generalized minimum distance decoding [123].
Kitaev surface code Maximum-likelihood (ML) [46], which takes time of order $$O(n^2)$$ for independent $$X,Z$$ noise [124].Minimum weight perfect-matching (MWPM) [46][125] (based on work by Edmonds on finding a matching in a graph [126][127]). Pipeline MWPM [128][129] - a modification accounting for correlations between events. 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) [130]. Correlated matching modifies MWPM to include correlations between $$X$$ and $$Z$$-type errors [128]. Belief perfect matching is a combination of belief-propagation and MWPM [131].Renormalization group (RG) [132][133][134].Markov-chain Monte Carlo [135].Tensor network [124].Cellular automaton [136][137].Neural network [138][139][140][141] and reinforcement learning [142].Union-find [143]. 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) [130]. Belief union find is a combination of belief-propagation and union-find [131].Decoders can be augmented with a pre-decoder [144][145], which can allow for some processing to be done inside the cryogenic environment of the quantum system [146].Sliding-window [147][148] and parallel-window [147] parallelizable decoders can be combined with many inner decoders, such as MWPM or union-find.Generalized belief propagation (GBP) [149] based on a classical version [150].
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 [151].Information set decoding (ISD) [152], 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 [61].Soft-decision maximum-likelihood trellis-based decoder [153].Random linear codes over large fields are list-recoverable and list-decodable up to near-optimal rates [154].Extensions of algebraic-geometry decoders to linear codes [155][156].
Linear binary code Decoding an arbitary linear binary code is $$NP$$-complete [157].Slepian's standard-array decoding [158].Recursive maximum likelihood decoding [159].Transformer neural net for soft decoding [160].
Low-density parity-check (LDPC) code Message-passing algorithm called belief propagation (BP) [161][162].Linear programming [163].Reinforcement learning [164].
Low-rank parity-check (LRPC) code Efficient probabilistic decoder [165].Mixed decoder [166].
Luby transform (LT) code Sum-product algorithm, often called a peeling decoder [167][168], similar to belief propagation [169].
Matrix-product code Decoder up to half of the minimum distance for NSC codes [170].
Melas code Algebraic decoder [171].
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 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 $$\mathsf{N(S)}$$ [172].
Modular-qudit surface code Renormalization-group decoder [173].
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) [174]
Multi-mode 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) or by using other effective iterative schemes when e.g. the lattice represents a concatenated GKP code [4][175][176][177].
Niset-Andersen-Cerf code Optical decoder using three beam splitters, electronic gain detectors, and two phase-insensitive amplifiers as described in Ref. [178].
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 phase operator [179]. Approximate number-phase codes are characterized by vanishing phase uncertainty. Such measurements can be utilized for Knill error correction (a.k.a. telecorrection [180]), which is based on teleportation [181][182]. This type of error correction substitutes the complicated correction procedures typical in Fock-state codes for necessity of clean codewords [183].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 [184][185] on an ancillary mode. See Section 4.B.1 of Ref. [183].
On-off keyed (OOK) c-q code Dolinar receiver [186].Superconducting transition edge sensor (TES) photon-number resolving detector [187].
Orthogonal Spacetime Block Code (OSTBC) Maximum-likelihood decoding can be achieved with only linear processing [188].
PPM c-q code Conditional pulse nulling (CPN) receiver [189].
PSK c-q code Multi-stage quantum receivers [190][191][192][193][194][195].Bayesian inference [53].
Pair-cat code Lindbladian-based dissipative encoding utilizing two-mode two-photon absorption [196]. 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. [197].
Permutation spherical code Efficient maximum-likelihood decoder determining the Voronoi region of an error word.
Permutation-invariant code For a family of codes, using projection, probability amplitude rebalancing, and gate teleportation can be done in $$O(N^2)$$ [198].
Plane-curve code Generalization of the Peterson algorithm for BCH codes [199].
Polar c-q code Quantum-limited successive-cancellation (SC) joint-detection receiver [200].
Polar code Successive cancellation (SC) decoder [201].Successive cancellation list (SCL) decoder [202] and a modification utilizing sequence repetition (SR-List) [203].Soft cancellation (SCAN) decoder [204][205].Belief propagation (BP) decoder [206].Noisy quantum gate-vased decoder [207].
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 [208].Tensor-network-based decoder when the encoding unitary is known [209].
Quantum Tanner code Linear-time decoder similar to the small-set-flip decoder for quantum expander codes [68].Linear-time decoder [85].Logarithmic-time decoder [86].
Quantum convolutional code ML decoder [210].
Quantum expander code Small set-flip linear-time decoder, which corrects $$\Omega(n^{1/2})$$ adversarial errors [211].Log-time decoder [212].Constant-time decoder [213].
Quantum low-density parity-check (QLDPC) code Belief-propagation (BP) decoder [214] and neural BP decoder [215] for qubit codes.Non-binary decoding algorithm for CSS-type QLDPC codes [216].BP-OSD decoder adds a post-processing step based on ordered statistics decoding (OSD) to the belief propogation (BP) decoder [22].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) [217].Message-passing decoder utilizing stabilizer inactivation (MP-SI) for CSS-type QLDPC qubit codes [218].Extension of the union-find decoder for qubit QLDPC codes, as well as a related heuristic decoder [219].
Quantum polar code Constructed using classical polar decoders for the amplitude and phase channels.
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 [220][221]. An automaton by Gacs yields a decoder for a 1D lattice [222].
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 [223].
Qubit code The decoder determining the most likely error given a noise channel is called the maximum-likelihood decoder. For few-qubit codes ($$n$$ is small), maximum-likelihood 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.
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. Maximum-likelihood decoding is $$NP$$-complete in general [224][225]. Degenerate maximum-likelihood decoding is $$\#P$$-complete in general [226].Trellis decoder, which builds a compact representation of the algebraic structure of the normalizer $$\mathsf{N(S)}$$ [227].Quantum extension of GRAND decoder [228].Deep neural-network probabilistic decoder [229].Generalized belief propagation (GBP) [149] based on a classical version [150].
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 [5][6] (see also Ref. [7]). 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 [36].
Ramanujan-complex product code For 2D simplicial complexes, cycle code decoder admitting a polynomial-time decoding algorithm can be used [230].
Random code Ball-collision decoding [231].Information set decoding (ISD) [232] and Finiasz and Sendrier (FS-ISD) decoding [233].
Rank-metric code Polynomial-reconstruction Berlekamp-Welch based decoder [234].Berlekamp-Massey based decoder [235].
Raptor (RAPid TORnado) code Raptor codes can be decoded using inactivation decoding [236], a combination of belief-propogation and Gaussian elimination decoding.
Reed-Muller (RM) code Reed decoder with $$r+1$$-step majority decoding corrects $$\frac{1}{2}(2^{m-r}-1)$$ errors [237] (see also Ch. 13 of Ref. [238]).Sequential code-reduction decoding [239].First-order ($$r=1$$) RM codes admit specialized decoders [240].
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 $$O(n^2)$$ [26][27].Gorenstein-Peterson-Zierler decoder with runtime of order $$O(n^3)$$ [23][39] (see exposition in Ref. [25]).Berlekamp-Welch decoder with runtime of order $$O(n^3)$$ [241] (see exposition in Ref. [242]), assuming that $$t \geq (n+k)/2$$.Gao decoder using extended Euclidean algorithm [243].Fast-Fourier-transform decoder with runtime of order $$O(n \text{polylog}n)$$ [244].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 [245]. Roth and Ruckenstein proposed a modified key equation that allows for correction of more than $$\left\lfloor (n-k)/2 \right\rfloor$$ errors [246]. The Guruswami-Sudan algorithm improved the Sudan algorithm to $$1-\sqrt{R}$$ [3]; see Ref. [247] for bounds. A further modification by Koetter and Vardy is used for soft-decision decoding [40] (see also Ref. [248]).
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 [220][221]. An automaton by Gacs yields a decoder for a 1D lattice [222].
Rotated surface code Local neural-network using 3D convolutions, combined with a separate global decoder [249].
Simplex code Due to the small size, it can be decoded according to maximum likelihood.Some faster decoders for the $$q=2$$ case: [250][251]A quantum decoder for the $$q=2$$ case: [252].
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 $$n$$ [253] (see [254; Sec. 29.7.2] for a description).
Singleton-bound approaching AQECC Quantum list decodable [103].
Skew-cyclic code Only given for skew-BCH codes, adapted froom standard BCH codes.
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.
String-net code Fusing non-Abelian anyons cannot be done in one step [255].Syndrome measurement circuits analyzed in Ref. [256].Clustering decoder [89].
Subsystem qubit stabilizer code Syndrome measurements are obtained by first measuring gauge operators of the code and taking their products, which give the stabilizer measurement outcomes.
Surface-17 code Lookup table [257].
Ta-Shma zigzag code Unique and list decoders [258].
Tanner code Parallel decoding algorithm corrects a fraction $$\delta_0^2/48$$ of errors for Tanner codes [84]. A modification of said algorithm improves the fraction to $$\delta_0^2/4$$ with no extra cost to complexity [259].
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 [61] 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 [260].
Ternary Golay Code Decoder for the extended ternary Golay code using the tetracode [108].
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.
Torus-layer spherical code (TLSC) Efficiently decodable [261].
Translationally invariant stabilizer code Clustering decoder [136][262].
Two-component cat code All-optical decoder [263] based on Knill error correction (a.k.a. telecorrection [180]), which is based on teleportation [181][182].
Two-dimensional hyperbolic surface code Due to the symmetries of hyperbolic surface codes, optimal measurement schedules of the stabilizers can be found [264].
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 $$\sigma_x (a+a^\dagger)$$ [265]. This results in an $$X$$-dependent shift of the readout cavity resonance which can be measured.
Wasilewski-Banaszek code Destructive measurement with photon number measurements on each mode.
XYZ$$^2$$ hexagonal stabilizer code Maximum-likelihood decoding using the EWD decoder [266].
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. [267].
Zetterberg code Kallquist first described an algebraic decoding theorem [268]. A faster version was later provided in Ref. [269] and further modified in Ref. [270].
$$[[9,1,3]]$$ Shor code Bit- and phase-flip circuits utilize CNOT and Hadamard gates ([271], Fig. 10.6).
$$\chi^{(2)}$$ code Linear optics and $$\chi^{(2)}$$ interactions.

## References

[1]
T. R. Scruby and K. Nemoto, “Local Probabilistic Decoding of a Quantum Code”. 2212.06985
[2]
H. Helgert, “Decoding of alternant codes (Corresp.)”, IEEE Transactions on Information Theory 23, 513 (1977). DOI
[3]
V. Guruswami and M. Sudan, “Improved decoding of Reed-Solomon and algebraic-geometric codes”, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280). DOI
[4]
C. Vuillot et al., “Quantum error correction with the toric Gottesman-Kitaev-Preskill code”, Physical Review A 99, (2019). DOI; 1810.00047
[5]
K. Audenaert and B. De Moor, “Optimizing completely positive maps using semidefinite programming”, Physical Review A 65, (2002). DOI; quant-ph/0109155
[6]
M. Reimpell and R. F. Werner, “Iterative Optimization of Quantum Error Correcting Codes”, Physical Review Letters 94, (2005). DOI; quant-ph/0307138
[7]
Andrew S. Fletcher, “Channel-Adapted Quantum Error Correction”. 0706.3400
[8]
D. Petz, “Sufficient subalgebras and the relative entropy of states of a von Neumann algebra”, Communications in Mathematical Physics 105, 123 (1986). DOI
[9]
D. PETZ, “SUFFICIENCY OF CHANNELS OVER VON NEUMANN ALGEBRAS”, The Quarterly Journal of Mathematics 39, 97 (1988). DOI
[10]
H. Barnum and E. Knill, “Reversing quantum dynamics with near-optimal quantum and classical fidelity”. quant-ph/0004088
[11]
M. Junge et al., “Universal Recovery Maps and Approximate Sufficiency of Quantum Relative Entropy”, Annales Henri Poincaré 19, 2955 (2018). DOI; 1509.07127
[12]
K. Tsujino et al., “Quantum Receiver beyond the Standard Quantum Limit of Coherent Optical Communication”, Physical Review Letters 106, (2011). DOI; 1103.5592
[13]
Kennedy, Robert S. "A near-optimum receiver for the binary coherent state quantum channel." Quarterly Progress Report 108 (1973): 219-225.
[14]
M. Takeoka and M. Sasaki, “Discrimination of the binary coherent signal: Gaussian-operation limit and simple non-Gaussian near-optimal receivers”, Physical Review A 78, (2008). DOI; 0706.1038
[15]
M. T. DiMario and F. E. Becerra, “Robust Measurement for the Discrimination of Binary Coherent States”, Physical Review Letters 121, (2018). DOI; 1807.05199
[16]
D. Sych and G. Leuchs, “Practical Receiver for Optimal Discrimination of Binary Coherent Signals”, Physical Review Letters 117, (2016). DOI; 1404.5033
[17]
Matthew B. Hastings, Jeongwan Haah, and Ryan O'Donnell, “Fiber Bundle Codes: Breaking the $N^{1/2} \operatorname{polylog}(N)$ Barrier for Quantum LDPC Codes”. 2009.03921
[18]
M. B. Hastings and J. Haah, “Dynamically Generated Logical Qubits”, Quantum 5, 564 (2021). DOI; 2107.02194
[19]
D. Knuth, “Efficient balanced codes”, IEEE Transactions on Information Theory 32, 51 (1986). DOI
[20]
S. Al-Bassam and B. Bose, “On balanced codes”, IEEE Transactions on Information Theory 36, 406 (1990). DOI
[21]
K. A. Schouhamer Immink and J. H. Weber, “Very Efficient Balanced Codes”, IEEE Journal on Selected Areas in Communications 28, 188 (2010). DOI
[22]
P. Panteleev and G. Kalachev, “Degenerate Quantum LDPC Codes With Good Finite Length Performance”, Quantum 5, 585 (2021). DOI; 1904.02703
[23]
W. Peterson, “Encoding and error-correction procedures for the Bose-Chaudhuri codes”, IEEE Transactions on Information Theory 6, 459 (1960). DOI
[24]
S. Arimoto, "Encoding and decoding of p-ary group codes and the correction system," Information Processing in Japan (in Japanese), vol. 2, pp. 320-325, Nov. 1961.
[25]
R.E. Blahut, Theory and practice of error-control codes, Addison-Wesley 1983.
[26]
J. Massey, “Shift-register synthesis and BCH decoding”, IEEE Transactions on Information Theory 15, 122 (1969). DOI
[27]
E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, 1968
[28]
H. Burton, “Inversionless decoding of binary BCH codes”, IEEE Transactions on Information Theory 17, 464 (1971). DOI
[29]
W. W. Peterson and E. J. Weldon, Error-correcting codes. MIT press 1972.
[30]
R. Gallager, Information Theory and Reliable Communication (Springer Vienna, 1972). DOI
[31]
Y. Sugiyama et al., “A method for solving key equation for decoding goppa codes”, Information and Control 27, 87 (1975). DOI
[32]
R. McEliece, The Theory of Information and Coding (Cambridge University Press, 2002). DOI
[33]
V. I. Levenshtein, Binary codes capable of correcting deletions, insertions and reversals (translated to English), Soviet Physics Dokl., 10(8), 707-710 (1966).
[34]
V. I. Levenshtein, Binary codes capable of correcting spurious insertions and deletions of one (translated to English), Prob. Inf. Transmission, 1(1), 8-17 (1965).
[35]
J. I. Farran, “Decoding Algebraic Geometry codes by a key equation”. math/9910151
[36]
V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018). DOI; 1708.05010
[37]
M. H. Michael et al., “New Class of Quantum Error-Correcting Codes for a Bosonic Mode”, Physical Review X 6, (2016). DOI; 1602.00008
[38]
E. Berlekamp, “Nonbinary BCH decoding (Abstr.)”, IEEE Transactions on Information Theory 14, 242 (1968). DOI
[39]
D. Gorenstein and N. Zierler, “A Class of Error-Correcting Codes in $p^m$ Symbols”, Journal of the Society for Industrial and Applied Mathematics 9, 207 (1961). DOI
[40]
R. Koetter and A. Vardy, “Algebraic soft-decision decoding of reed-solomon codes”, IEEE Transactions on Information Theory 49, 2809 (2003). DOI
[41]
I. A. Burenkov, M. V. Jabir, and S. V. Polyakov, “Practical quantum-enhanced receivers for classical communication”, AVS Quantum Science 3, 025301 (2021). DOI
[42]
S. Kwon, S. Watabe, and J.-S. Tsai, “Autonomous quantum error correction in a four-photon Kerr parametric oscillator”, npj Quantum Information 8, (2022). DOI; 2203.09234
[43]
S. L. Braunstein, “Quantum error correction for communication with linear optics”, Nature 394, 47 (1998). DOI
[44]
Toshiaki Inada et al., “Measurement-Free Ultrafast Quantum Error Correction by Using Multi-Controlled Gates in Higher-Dimensional State Space”. 2109.00086
[45]
G. A. Paz-Silva, G. K. Brennen, and J. Twamley, “Fault Tolerance with Noisy and Slow Measurements and Preparation”, Physical Review Letters 105, (2010). DOI; 1002.1536
[46]
E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002). DOI; quant-ph/0110143
[47]
Albert T. Schmitz, “Thermal Stability of Dynamical Phase Transitions in Higher Dimensional Stabilizer Codes”. 2002.11733
[48]
Yoni Choukroun and Lior Wolf, “Deep Quantum Error Correction”. 2301.11930
[49]
N. Patterson, “The algebraic decoding of Goppa codes”, IEEE Transactions on Information Theory 21, 203 (1975). DOI
[50]
Daniel J. Bernstein, "Understanding binary-Goppa decoding." Cryptology ePrint Archive (2022).
[51]
R. S. Bondurant, “Near-quantum optimum receivers for the phase-quadrature coherent-state channel”, Optics Letters 18, 1896 (1993). DOI
[52]
M. V. Jabir et al., “Experimental demonstration of the near-quantum optimal receiver”, OSA Continuum 3, 3324 (2020). DOI
[53]
I. A. Burenkov, O. V. Tikhonova, and S. V. Polyakov, “Quantum receiver for large alphabet communication”, Optica 5, 227 (2018). DOI; 1802.08287
[54]
I. A. Burenkov et al., “Time-Resolving Quantum Measurement Enables Energy-Efficient, Large-Alphabet Communication”, PRX Quantum 1, (2020). DOI
[55]
M. V. Jabir et al., “Energy and bandwidth efficiency optimization of quantum-enabled optical communication channels”, npj Quantum Information 8, (2022). DOI
[56]
A. M. Kubica, The Abcs of the Color Code: A Study of Topological Quantum Codes as Toy Models for Fault-tolerant Quantum Computation and Quantum Phases of Matter, California Institute of Technology, 2018. DOI
[57]
K. Sahay and B. J. Brown, “Decoder for the Triangular Color Code by Matching on a Möbius Strip”, PRX Quantum 3, (2022). DOI; 2108.11395
[58]
Ashley M. Stephens, “Efficient fault-tolerant decoding of topological color codes”. 1402.3037
[59]
C. Chamberland et al., “Triangular color codes on trivalent graphs with flag qubits”, New Journal of Physics 22, 023019 (2020). DOI; 1911.00355
[60]
Jonathan F. San Miguel, Dominic J. Williamson, and Benjamin J. Brown, “A cellular automaton decoder for a noise-bias tailored color code”. 2203.16534
[61]
G. Forney, “Generalized minimum distance decoding”, IEEE Transactions on Information Theory 12, 125 (1966). DOI
[62]
A. Viterbi, “Error bounds for convolutional codes and an asymptotically optimum decoding algorithm”, IEEE Transactions on Information Theory 13, 260 (1967). DOI
[63]
L. Bahl et al., “Optimal decoding of linear codes for minimizing symbol error rate (Corresp.)”, IEEE Transactions on Information Theory 20, 284 (1974). DOI
[64]
J. Meggitt, “Error correcting codes and their implementation for data transmission systems”, IEEE Transactions on Information Theory 7, 234 (1961). DOI
[65]
E. Prange, “The use of information sets in decoding cyclic codes”, IEEE Transactions on Information Theory 8, 5 (1962). DOI
[66]
J. Macwilliams, “Permutation Decoding of Systematic Codes”, Bell System Technical Journal 43, 485 (1964). DOI
[67]
Sagarmoy Dutta and Piyush P Kurur, “Quantum Cyclic Code”. 1007.1697
[68]
Shouzhen Gu, Christopher A. Pattison, and Eugene Tang, “An efficient decoder for a linear distance quantum LDPC code”. 2206.06557
[69]
Irit Dinur et al., “Good Quantum LDPC Codes with Linear Time Decoders”. 2206.07750
[70]
M. M. Wilde and D. B. Uskov, “Linear-optical hyperentanglement-assisted quantum error-correcting code”, Physical Review A 79, (2009). DOI; 0807.4906
[71]
N. Bao and N. Cheng, “Eigenstate thermalization hypothesis and approximate quantum error correction”, Journal of High Energy Physics 2019, (2019). DOI; 1906.03669
[72]
A. N. Skorobogatov and S. G. Vladut, “On the decoding of algebraic-geometric codes”, IEEE Transactions on Information Theory 36, 1051 (1990). DOI
[73]
V. Yu. Krachkovskii, "Decoding of codes on algebraic curves," (in Russian), Conference Odessa, 1988.
[74]
S. C. Porter, B.-Z. Shen, and R. Pellikaan, “Decoding geometric Goppa codes using an extra place”, IEEE Transactions on Information Theory 38, 1663 (1992). DOI
[75]
D. Ehrhard, “Decoding Algebraic-Geometric Codes by solving a key equation”, Lecture Notes in Mathematics 18 (1992). DOI
[76]
R. Pellikaan, “On a decoding algorithm for codes on maximal curves”, IEEE Transactions on Information Theory 35, 1228 (1989). DOI
[77]
S. Vladut, “On the decoding of algebraic-geometric codes over F/sub q/ for q&amp;lt;or=16”, IEEE Transactions on Information Theory 36, 1461 (1990). DOI
[78]
S. Sakata, “Finding a minimal set of linear recurring relations capable of generating a given finite two-dimensional array”, Journal of Symbolic Computation 5, 321 (1988). DOI
[79]
S. Sakata, “Extension of the Berlekamp-Massey algorithm to N dimensions”, Information and Computation 84, 207 (1990). DOI
[80]
S. Sakata, “Decoding binary 2-D cyclic codes by the 2-D Berlekamp-Massey algorithm”, IEEE Transactions on Information Theory 37, 1200 (1991). DOI
[81]
G.-L. Feng and T. R. N. Rao, “Decoding algebraic-geometric codes up to the designed minimum distance”, IEEE Transactions on Information Theory 39, 37 (1993). DOI
[82]
D. Ehrhard, “Achieving the designed error capacity in decoding algebraic-geometric codes”, IEEE Transactions on Information Theory 39, 743 (1993). DOI
[83]
M. A. Shokrollahi and H. Wasserman, “List decoding of algebraic-geometric codes”, IEEE Transactions on Information Theory 45, 432 (1999). DOI
[84]
M. Sipser and D. A. Spielman, “Expander codes”, IEEE Transactions on Information Theory 42, 1710 (1996). DOI
[85]
Anthony Leverrier and Gilles Zémor, “Efficient decoding up to a constant fraction of the code length for asymptotically good quantum codes”. 2206.07571
[86]
Anthony Leverrier and Gilles Zémor, “Decoding quantum Tanner codes”. 2208.05537
[87]
G. M. Nixon and B. J. Brown, “Correcting Spanning Errors With a Fractal Code”, IEEE Transactions on Information Theory 67, 4504 (2021). DOI; 2002.11738
[88]
S. Burton, C. G. Brell, and S. T. Flammia, “Classical simulation of quantum error correction in a Fibonacci anyon code”, Physical Review A 95, (2017). DOI; 1506.03815
[89]
G. Dauphinais and D. Poulin, “Fault-Tolerant Quantum Error Correction for non-Abelian Anyons”, Communications in Mathematical Physics 355, 519 (2017). DOI; 1607.02159
[90]
Alexis Schotte et al., “Quantum error correction thresholds for the universal Fibonacci Turaev-Viro code”. 2012.04610
[91]
Alexis Schotte, Lander Burgelman, and Guanyu Zhu, “Fault-tolerant error correction for a universal non-Abelian topological quantum computer at finite temperature”. 2301.00054
[92]
K. R. Duffy, J. Li, and M. Medard, “Capacity-Achieving Guessing Random Additive Noise Decoding”, IEEE Transactions on Information Theory 65, 4023 (2019). DOI; 1802.07010
[93]
K. R. Duffy, J. Li, and M. Medard, “Guessing noise, not code-words”, 2018 IEEE International Symposium on Information Theory (ISIT) (2018). DOI
[94]
I. Cong, S. Choi, and M. D. Lukin, “Quantum convolutional neural networks”, Nature Physics 15, 1273 (2019). DOI; 1810.03787
[95]
David F. Locher, Lorenzo Cardarelli, and Markus Müller, “Quantum Error Correction with Quantum Autoencoders”. 2202.00555
[96]
A. De and L. P. Pryadko, “Universal set of dynamically protected gates for bipartite qubit networks: Soft pulse implementation of the [[5,1,3]] quantum error-correcting code”, Physical Review A 93, (2016). DOI; 1509.01239
[97]
Chaobin Liu, “Exact performance of the five-qubit code with coherent errors”. 2203.01706
[98]
Venkatesan Guruswami and Atri Rudra, “Explicit Codes Achieving List Decoding Capacity: Error-correction with Optimal Redundancy”. cs/0511072
[99]
Atri Rudra. List Decoding and Property Testing of Error Correcting Codes. PhD thesis, University of Washington, 8 2007.
[100]
F. Parvaresh and A. Vardy, “Correcting Errors Beyond the Guruswami-Sudan Radius in Polynomial Time”, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05). DOI
[101]
V. Guruswami, “Linear-Algebraic List Decoding of Folded Reed-Solomon Codes”, 2011 IEEE 26th Annual Conference on Computational Complexity (2011). DOI; 1106.0436
[102]
V. Guruswami and A. Rudra, “Better Binary List Decodable Codes Via Multilevel Concatenation”, IEEE Transactions on Information Theory 55, 19 (2009). DOI
[103]
Thiago Bergamaschi, Louis Golowich, and Sam Gunn, “Approaching the Quantum Singleton Bound with Approximate Error Correction”. 2212.09935
[104]
Sagarmoy Dutta and Piyush P Kurur, “Quantum Cyclic Code of length dividing $p^{t}+1$”. 1011.5814
[105]
Yijia Xu et al., “Qubit-oscillator concatenated codes: decoding formalism & code comparison”. 2209.04573
[106]
D. Silva and F. R. Kschischang, “Fast encoding and decoding of Gabidulin codes”, 2009 IEEE International Symposium on Information Theory (2009). DOI; 0901.2483
[107]
J.-M. Goethals, “On the Golay perfect binary code”, Journal of Combinatorial Theory, Series A 11, 178 (1971). DOI
[108]
V. Pless, “Decoding the Golay codes”, IEEE Transactions on Information Theory 32, 561 (1986). DOI
[109]
A. J. VITERBI, “Error Bounds for Convolutional Codes and an Asymptotically Optimum Decoding Algorithm”, The Foundations of the Digital Wireless World 41 (2009). DOI
[110]
B. Honary and G. Markarian, “New simple encoder and trellis decoder for Golay codes”, Electronics Letters 29, 2170 (1993). DOI
[111]
A. Vardy, “Even more efficient bounded-distance decoding of the hexacode, the Golay code, and the Leech lattice”, IEEE Transactions on Information Theory 41, 1495 (1995). DOI
[112]
O. W. Yeung and K. M. Chugg, “An Iterative Algorithm and Low Complexity Hardware Architecture for Fast Acquisition of Long PN Codes in UWB Systems”, Journal of VLSI signal processing systems for signal, image and video technology 43, 25 (2006). DOI
[113]
A. L. Grimsmo and S. Puri, “Quantum Error Correction with the Gottesman-Kitaev-Preskill Code”, PRX Quantum 2, (2021). DOI; 2106.12989
[114]
V. V. Sivak et al., “Real-time quantum error correction beyond break-even”. 2211.09116
[115]
Debasmita Bhoumik et al., “Efficient Machine-Learning-based decoder for Heavy Hexagonal QECC”. 2210.09730
[116]
Oscar Higgott and Nikolas P. Breuckmann, “Improved single-shot decoding of higher dimensional hypergraph product codes”. 2206.03122
[117]
P. Hayden et al., “Spacetime replication of continuous variable quantum information”, New Journal of Physics 18, 083043 (2016). DOI; 1601.02544
[118]
N. Delfosse and M. B. Hastings, “Union-Find Decoders For Homological Product Codes”, Quantum 5, 406 (2021). DOI; 2009.14226
[119]
Armanda O. Quintavalle and Earl T. Campbell, “ReShape: a decoder for hypergraph product codes”. 2105.02370
[120]
Nicholas Connolly et al., “Fast erasure decoder for a class of quantum LDPC codes”. 2208.01002
[121]
D. Bleichenbacher, A. Kiayias, and M. Yung, “Decoding interleaved Reed–Solomon codes over noisy channels”, Theoretical Computer Science 379, 348 (2007). DOI
[122]
D. Coppersmith and M. Sudan, “Reconstructing curves in three (and higher) dimensional space from noisy data”, Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03 (2003). DOI
[123]
J. Justesen, “Class of constructive asymptotically good algebraic codes”, IEEE Transactions on Information Theory 18, 652 (1972). DOI
[124]
S. Bravyi, M. Suchara, and A. Vargo, “Efficient algorithms for maximum likelihood decoding in the surface code”, Physical Review A 90, (2014). DOI; 1405.4883
[125]
Austin G. Fowler, “Minimum weight perfect matching of fault-tolerant topological quantum error correction in average $O(1)$ parallel time”. 1307.1740
[126]
J. Edmonds, “Paths, Trees, and Flowers”, Canadian Journal of Mathematics 17, 449 (1965). DOI
[127]
J. Edmonds, “Maximum matching and a polyhedron with 0,1-vertices”, Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics 69B, 125 (1965). DOI
[128]
Austin G. Fowler, “Optimal complexity correction of correlated errors in the surface code”. 1310.0863
[129]
Alexandru Paler and Austin G. Fowler, “Pipelined correlated minimum weight perfect matching of the surface code”. 2205.09828
[130]
Christopher A. Pattison et al., “Improved quantum error correction using soft information”. 2107.13589
[131]
Oscar Higgott et al., “Fragile boundaries of tailored surface codes and improved decoding of circuit-level noise”. 2203.04948
[132]
G. Duclos-Cianci and D. Poulin, “Fast Decoders for Topological Quantum Codes”, Physical Review Letters 104, (2010). DOI; 0911.0581
[133]
Guillaume Duclos-Cianci and David Poulin, “Fault-Tolerant Renormalization Group Decoder for Abelian Topological Codes”. 1304.6100
[134]
F. H. E. Watson, H. Anwar, and D. E. Browne, “Fast fault-tolerant decoder for qubit and qudit surface codes”, Physical Review A 92, (2015). DOI; 1411.3028
[135]
A. Hutter, J. R. Wootton, and D. Loss, “Efficient Markov chain Monte Carlo algorithm for the surface code”, Physical Review A 89, (2014). DOI; 1302.2669
[136]
J. W. Harrington, Analysis of Quantum Error-correcting Codes: Symplectic Lattice Codes and Toric Codes, California Institute of Technology, 2004. DOI
[137]
M. Herold et al., “Cellular automaton decoders of topological quantum memories in the fault tolerant setting”, New Journal of Physics 19, 063012 (2017). DOI; 1511.05579
[138]
G. Torlai and R. G. Melko, “Neural Decoder for Topological Codes”, Physical Review Letters 119, (2017). DOI; 1610.04238
[139]
C. Chamberland and P. Ronagh, “Deep neural decoders for near term fault-tolerant experiments”, Quantum Science and Technology 3, 044002 (2018). DOI; 1802.06441
[140]
R. Sweke et al., “Reinforcement learning decoders for fault-tolerant quantum computation”, Machine Learning: Science and Technology 2, 025005 (2020). DOI; 1810.07207
[141]
Yosuke Ueno et al., “NEO-QEC: Neural Network Enhanced Online Superconducting Decoder for Surface Codes”. 2208.05758
[142]
Elisha Siddiqui Matekole et al., “Decoding surface codes with deep reinforcement learning and probabilistic policy reuse”. 2212.11890
[143]
N. Delfosse and N. H. Nickerson, “Almost-linear time decoding algorithm for topological codes”, Quantum 5, 595 (2021). DOI; 1709.06218
[144]
Nicolas Delfosse, “Hierarchical decoding to reduce hardware requirements for quantum computing”. 2001.11427
[145]
Samuel C. Smith, Benjamin J. Brown, and Stephen D. Bartlett, “A local pre-decoder to reduce the bandwidth and latency of quantum error correction”. 2208.04660
[146]
Gokul Subramanian Ravi et al., “Better Than Worst-Case Decoding for Quantum Error Correction”. 2208.08547
[147]
Xinyu Tan et al., “Scalable surface code decoders with parallelization in time”. 2209.09219
[148]
Luka Skoric et al., “Parallel window decoding enables scalable fault tolerant quantum computation”. 2209.08552
[149]
Josias Old and Manuel Rispler, “Generalized Belief Propagation Algorithms for Decoding of Surface Codes”. 2212.03214
[150]
J. S. Yedidia, W. T. Freeman, and Y. Weiss, Generalized belief propagation, in NIPS, Vol. 13 (2000) pp. 689–695.
[151]
I. Dumer, “Maximum likelihood decoding with reduced complexity”, Proceedings of IEEE International Symposium on Information Theory. DOI
[152]
C. Peters, “Information-Set Decoding for Linear Codes over F q”, Post-Quantum Cryptography 81 (2010). DOI
[153]
J. Wolf, “Efficient maximum likelihood decoding of linear block codes using a trellis”, IEEE Transactions on Information Theory 24, 76 (1978). DOI
[154]
Atri Rudra and Mary Wootters, “Average-radius list-recovery of random linear codes: it really ties the room together”. 1704.02420
[155]
R. Kotter. A unified description of an error locating procedure for linear codes. In D. Yorgov, editor, Proc. 3rd International Workshop on Algebraic and Combinatorial Coding Theory, pages 113–117, Voneshta Voda, Bulgaria, June 1992. Hermes.
[156]
R. Pellikaan, “On decoding by error location and dependent sets of error positions”, Discrete Mathematics 106-107, 369 (1992). DOI
[157]
E. Berlekamp, R. McEliece, and H. van Tilborg, “On the inherent intractability of certain coding problems (Corresp.)”, IEEE Transactions on Information Theory 24, 384 (1978). DOI
[158]
D. Slepian, “Some Further Theory of Group Codes”, Bell System Technical Journal 39, 1219 (1960). DOI
[159]
Yunghsiang S. Han et al., “Maximum-likelihood Soft-decision Decoding for Binary Linear Block Codes Based on Their Supercodes”. 1408.1310
[160]
Yoni Choukroun and Lior Wolf, “Error Correction Code Transformer”. 2203.14966
[161]
R. Gallagher, Low-density parity check codes. 1963. PhD thesis, MIT Cambridge, MA.
[162]
S. Lin and D. J. Costello, Error Control Coding, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 2004.
[163]
J. Feldman, “LP Decoding”, Encyclopedia of Algorithms 1177 (2016). DOI
[164]
Salman Habib, Allison Beemer, and Joerg Kliewer, “RELDEC: Reinforcement Learning-Based Decoding of Moderate Length LDPC Codes”. 2112.13934
[165]
Gaborit, P., Murat, G., Ruatta, O., & Zemor, G. (2013, April). Low rank parity check codes and their application to cryptography. In Proceedings of the Workshop on Coding and Cryptography WCC (Vol. 2013).
[166]
P. Gaborit et al., “RankSign: An Efficient Signature Algorithm Based on the Rank Metric”, Post-Quantum Cryptography 88 (2014). DOI
[167]
T. Richardson and R. Urbanke, Modern Coding Theory (Cambridge University Press, 2008). DOI
[168]
David J. C. MacKay. 2002. Information Theory, Inference & Learning Algorithms. Cambridge University Press, USA
[169]
J. Pearl, “Reverend Bayes on Inference Engines: A Distributed Hierarchical Approach”, Probabilistic and Causal Inference 129 (2022). DOI
[170]
F. Hernando, K. Lally, and D. Ruano, “Construction and decoding of matrix-product codes from nested codes”, Applicable Algebra in Engineering, Communication and Computing 20, 497 (2009). DOI
[171]
A. Alahmadi et al., “On the lifted Melas code”, Cryptography and Communications 8, 7 (2015). DOI
[172]
Eric Sabo, Arun B. Aloshious, and Kenneth R. Brown, “Trellis Decoding For Qudit Stabilizer Codes And Its Application To Qubit Topological Codes”. 2106.08251
[173]
H. Anwar et al., “Fast decoders for qudit topological codes”, New Journal of Physics 16, 063038 (2014). DOI; 1311.4895
[174]
M. J. Gullans and D. A. Huse, “Dynamical Purification Phase Transition Induced by Quantum Measurements”, Physical Review X 10, (2020). DOI; 1905.05195
[175]
K. Noh and C. Chamberland, “Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code”, Physical Review A 101, (2020). DOI; 1908.03579
[176]
J. Conrad, J. Eisert, and F. Arzani, “Gottesman-Kitaev-Preskill codes: A lattice perspective”, Quantum 6, 648 (2022). DOI; 2109.14645
[177]
N. Raveendran et al., “Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound”, Quantum 6, 767 (2022). DOI; 2111.07029
[178]
J. Niset, U. L. Andersen, and N. J. Cerf, “Experimentally Feasible Quantum Erasure-Correcting Code for Continuous Variables”, Physical Review Letters 101, (2008). DOI; 0710.4858
[179]
L. Susskind and J. Glogower, “Quantum mechanical phase and time operator”, Physics Physique физика 1, 49 (1964). DOI
[180]
C. M. Dawson, H. L. Haselgrove, and M. A. Nielsen, “Noise thresholds for optical cluster-state quantum computation”, Physical Review A 73, (2006). DOI; quant-ph/0601066
[181]
E. Knill, “Quantum computing with realistically noisy devices”, Nature 434, 39 (2005). DOI; quant-ph/0410199
[182]
E. Knill, “Scalable Quantum Computation in the Presence of Large Detected-Error Rates”. quant-ph/0312190
[183]
A. L. Grimsmo, J. Combes, and B. Q. Baragiola, “Quantum Computing with Rotation-Symmetric Bosonic Codes”, Physical Review X 10, (2020). DOI; 1901.08071
[184]
C. W. Helstrom, “Quantum detection and estimation theory”, Journal of Statistical Physics 1, 231 (1969). DOI
[185]
A. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Edizioni della Normale, 2011). DOI
[186]
Dolinar, Samuel Joseph. "An optimum receiver for the binary coherent state quantum channel." Research Laboratory of Electronics, MIT, Quarterly Progress Report 11 (1973): 115-120.
[187]
K. Tsujino et al., “Sub-shot-noise-limit discrimination of on-off keyed coherent signals via a quantum receiver with a superconducting transition edge sensor”, Optics Express 18, 8107 (2010). DOI
[188]
V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block coding for wireless communications: performance results”, IEEE Journal on Selected Areas in Communications 17, 451 (1999). DOI
[189]
S. J. Dolinar, Jr., “A near-optimum receiver structure for the detection of M-ary optical PPM signals”, The Telecommunications and Data Acquisition Progress Report 42 72: December 1982; NASA: Pasadena, CA, (1983)
[190]
M. Takeoka et al., “Implementation of projective measurements with linear optics and continuous photon counting”, Physical Review A 71, (2005). DOI; quant-ph/0410133
[191]
F. E. Becerra et al., “<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>M</mml:mi></mml:math>-ary-state phase-shift-keying discrimination below the homodyne limit”, Physical Review A 84, (2011). DOI
[192]
C. Wittmann, U. L. Andersen, and G. Leuchs, “Discrimination of optical coherent states using a photon number resolving detector”, Journal of Modern Optics 57, 213 (2010). DOI; 0905.2496
[193]
S. Izumi et al., “Displacement receiver for phase-shift-keyed coherent states”, Physical Review A 86, (2012). DOI; 1208.1815
[194]
S. Izumi et al., “Quantum receivers with squeezing and photon-number-resolving detectors for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>M</mml:mi></mml:math>-ary coherent state discrimination”, Physical Review A 87, (2013). DOI; 1302.2691
[195]
K. Li, Y. Zuo, and B. Zhu, “Suppressing the Errors Due to Mode Mismatch for $M$-Ary PSK Quantum Receivers Using Photon-Number-Resolving Detector”, IEEE Photonics Technology Letters 25, 2182 (2013). DOI; 1304.7316
[196]
G. S. Agarwal, “Generation of Pair Coherent States and Squeezing via the Competition of Four-Wave Mixing and Amplified Spontaneous Emission”, Physical Review Letters 57, 827 (1986). DOI
[197]
F. Pastawski et al., “Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence”, Journal of High Energy Physics 2015, (2015). DOI; 1503.06237
[198]
Y. Ouyang, “Permutation-invariant quantum coding for quantum deletion channels”, 2021 IEEE International Symposium on Information Theory (ISIT) (2021). DOI; 2102.02494
[199]
J. Justesen et al., “Construction and decoding of a class of algebraic geometry codes”, IEEE Transactions on Information Theory 35, 811 (1989). DOI
[200]
M. M. Wilde and S. Guha, “Polar Codes for Classical-Quantum Channels”, IEEE Transactions on Information Theory 59, 1175 (2013). DOI; 1109.2591
[201]
E. Arikan, “Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels”, IEEE Transactions on Information Theory 55, 3051 (2009). DOI
[202]
I. Tal and A. Vardy, “List Decoding of Polar Codes”, IEEE Transactions on Information Theory 61, 2213 (2015). DOI
[203]
Yuqing Ren et al., “A Sequence Repetition Node-Based Successive Cancellation List Decoder for 5G Polar Codes: Algorithm and Implementation”. 2205.08857
[204]
U. U. Fayyaz and J. R. Barry, “Low-Complexity Soft-Output Decoding of Polar Codes”, IEEE Journal on Selected Areas in Communications 32, 958 (2014). DOI
[205]
U. U. Fayyaz and J. R. Barry, “Polar codes for partial response channels”, 2013 IEEE International Conference on Communications (ICC) (2013). DOI
[206]
E. Arkan, “A performance comparison of polar codes and Reed-Muller codes”, IEEE Communications Letters 12, 447 (2008). DOI
[207]
Srikar Kasi et al., “Decoding Polar Codes via Noisy Quantum Gates: Quantum Circuits and Insights”. 2210.10854
[208]
T. Farrelly et al., “Tensor-Network Codes”, Physical Review Letters 127, (2021). DOI; 2009.10329
[209]
A. J. Ferris and D. Poulin, “Tensor Networks and Quantum Error Correction”, Physical Review Letters 113, (2014). DOI; 1312.4578
[210]
H. Ollivier and J.-P. Tillich, “Description of a Quantum Convolutional Code”, Physical Review Letters 91, (2003). DOI; quant-ph/0304189
[211]
A. Leverrier, J.-P. Tillich, and G. Zemor, “Quantum Expander Codes”, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (2015). DOI; 1504.00822
[212]
O. Fawzi, A. Grospellier, and A. Leverrier, “Constant Overhead Quantum Fault-Tolerance with Quantum Expander Codes”, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) (2018). DOI; 1808.03821
[213]
A. Grospellier. Constant time decoding of quantum expander codes and application to fault-tolerant quantum computation. PhD thesis, Inria Paris (2019).
[214]
David Poulin and Yeojin Chung, “On the iterative decoding of sparse quantum codes”. 0801.1241
[215]
Sisi Miao et al., “Neural Belief Propagation Decoding of Quantum LDPC Codes Using Overcomplete Check Matrices”. 2212.10245
[216]
Z. Babar et al., “Fifteen Years of Quantum LDPC Coding and Improved Decoding Strategies”, IEEE Access 3, 2492 (2015). DOI
[217]
Nithin Raveendran et al., “Soft Syndrome Decoding of Quantum LDPC Codes for Joint Correction of Data and Syndrome Errors”. 2205.02341
[218]
Julien du Crest, Mehdi Mhalla, and Valentin Savin, “Stabilizer Inactivation for Message-Passing Decoding of Quantum LDPC Codes”. 2205.06125
[219]
Lucas Berent, Lukas Burgholzer, and Robert Wille, “Software Tools for Decoding Quantum Low-Density Parity Check Codes”. 2209.01180
[220]
A. L. Toom, “Nonergodic Multidimensional System of Automata”, Probl. Peredachi Inf., 10:3 (1974), 70–79; Problems Inform. Transmission, 10:3 (1974), 239–246
[221]
L. F. Gray, “Toom’s Stability Theorem in Continuous Time”, Perplexing Problems in Probability 331 (1999). DOI
[222]
P. Gács, “[]”, Journal of Statistical Physics 103, 45 (2001). DOI
[223]
Sergey Bravyi et al., “Adaptive constant-depth circuits for manipulating non-abelian anyons”. 2205.01933
[224]
M.-H. Hsieh and F. Le Gall, “NP-hardness of decoding quantum error-correction codes”, Physical Review A 83, (2011). DOI; 1009.1319
[225]
Kuo, Kao-Yueh, and Chung-Chin Lu. "On the hardness of decoding quantum stabilizer codes under the depolarizing channel." 2012 International Symposium on Information Theory and its Applications. IEEE, 2012.
[226]
Pavithran Iyer and David Poulin, “Hardness of decoding quantum stabilizer codes”. 1310.3235
[227]
H. Ollivier and J.-P. Tillich, “Trellises for stabilizer codes: Definition and uses”, Physical Review A 74, (2006). DOI; quant-ph/0512041
[228]
Diogo Cruz, Francisco A. Monteiro, and Bruno C. Coutinho, “Quantum Error Correction via Noise Guessing Decoding”. 2208.02744
[229]
S. Krastanov and L. Jiang, “Deep Neural Network Probabilistic Decoder for Stabilizer Codes”, Scientific Reports 7, (2017). DOI; 1705.09334
[230]
Shai Evra, Tali Kaufman, and Gilles Zémor, “Decodable quantum LDPC codes beyond the $\sqrt{n}$ distance barrier using high dimensional expanders”. 2004.07935
[231]
D. J. Bernstein, T. Lange, and C. Peters, “Smaller Decoding Exponents: Ball-Collision Decoding”, Advances in Cryptology – CRYPTO 2011 743 (2011). DOI
[232]
A. Becker et al., “Decoding Random Binary Linear Codes in 2 n/20: How 1 + 1 = 0 Improves Information Set Decoding”, Advances in Cryptology – EUROCRYPT 2012 520 (2012). DOI
[233]
M. Finiasz and N. Sendrier, “Security Bounds for the Design of Code-Based Cryptosystems”, Advances in Cryptology – ASIACRYPT 2009 88 (2009). DOI
[234]
P. Loidreau, “A Welch–Berlekamp Like Algorithm for Decoding Gabidulin Codes”, Coding and Cryptography 36 (2006). DOI
[235]
G. Richter and S. Plass, “Fast decoding of rank-codes with rank errors and column erasures”, International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings.. DOI
[236]
F. Lazaro, G. Liva, and G. Bauch, “Inactivation Decoding of LT and Raptor Codes: Analysis and Code Design”, IEEE Transactions on Communications 1 (2017). DOI; 1706.05814
[237]
D. E. Muller, “Application of Boolean algebra to switching circuit design and to error detection”, Transactions of the I.R.E. Professional Group on Electronic Computers EC-3, 6 (1954). DOI
[238]
F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
[239]
L. Rudolph and C. Hartmann, “Decoding by sequential code reduction”, IEEE Transactions on Information Theory 19, 549 (1973). DOI
[240]
E.C. Posner, Combinatorial Structures in Planetary Reconnaissance in Error Correcting Codes, ed. H.B. Mann, Wiley, NY 1968.
[241]
E. R. Berlekamp and L. Welch, Error Correction of Algebraic Block Codes. U.S. Patent, Number 4,633,470 1986.
[242]
P. Gemmell and M. Sudan, “Highly resilient correctors for polynomials”, Information Processing Letters 43, 169 (1992). DOI
[243]
S. Gao, “A New Algorithm for Decoding Reed-Solomon Codes”, Communications, Information and Network Security 55 (2003). DOI
[244]
I. Reed et al., “The fast decoding of Reed-Solomon codes using Fermat theoretic transforms and continued fractions”, IEEE Transactions on Information Theory 24, 100 (1978). DOI
[245]
M. Sudan, “Decoding of Reed Solomon Codes beyond the Error-Correction Bound”, Journal of Complexity 13, 180 (1997). DOI
[246]
R. M. Roth and G. Ruckenstein, “Efficient decoding of Reed-Solomon codes beyond half the minimum distance”, IEEE Transactions on Information Theory 46, 246 (2000). DOI
[247]
V. Guruswami and A. Rudra, “Limits to List Decoding Reed–Solomon Codes”, IEEE Transactions on Information Theory 52, 3642 (2006). DOI
[248]
A. Vardy and Y. Be'ery, “Bit-level soft-decision decoding of Reed-Solomon codes”, IEEE Transactions on Communications 39, 440 (1991). DOI
[249]
Christopher Chamberland et al., “Techniques for combining fast local decoders with global decoders under circuit-level noise”. 2208.01178
[250]
R. R. Green, "A serial orthogonal decoder," JPL Space Programs Summary, vol. 37–39-IV, pp. 247–253, 1966.
[251]
A. Ashikhmin and S. Litsyn, “Simple MAP decoding of first order Reed-Muller and Hamming codes”, Proceedings 2003 IEEE Information Theory Workshop (Cat. No.03EX674). DOI
[252]
A. Barg and S. Zhou, “A quantum decoding algorithm for the simplex code”, in Proceedings of the 36th Annual Allerton Conference on Communication, Control and Computing, Monticello, IL, 23–25 September 1998 (UIUC 1998) 359–365
[253]
R. Silverman and M. Balser, “Coding for constant-data-rate systems”, Transactions of the IRE Professional Group on Information Theory 4, 50 (1954). DOI
[254]
A. Lapidoth, A Foundation in Digital Communication (Cambridge University Press, 2017). DOI
[255]
D. Beckman et al., “Measurability of Wilson loop operators”, Physical Review D 65, (2002). DOI; hep-th/0110205
[256]
N. E. Bonesteel and D. P. DiVincenzo, “Quantum circuits for measuring Levin-Wen operators”, Physical Review B 86, (2012). DOI; 1206.6048
[257]
Y. Tomita and K. M. Svore, “Low-distance surface codes under realistic quantum noise”, Physical Review A 90, (2014). DOI; 1404.3747
[258]
Fernando Granha Jeronimo et al., “Unique Decoding of Explicit $ε$-balanced Codes Near the Gilbert-Varshamov Bound”. 2011.05500
[259]
G. Zemor, “On expander codes”, IEEE Transactions on Information Theory 47, 835 (2001). DOI
[260]
P. Chaichanavong and P. H. Siegel, “Tensor-product parity code for magnetic recording”, IEEE Transactions on Magnetics 42, 350 (2006). DOI
[261]
C. Torezzan, S. I. R. Costa, and V. A. Vaishampayan, “Spherical codes on torus layers”, 2009 IEEE International Symposium on Information Theory (2009). DOI
[262]
S. Bravyi and J. Haah, “Quantum Self-Correction in the 3D Cubic Code Model”, Physical Review Letters 111, (2013). DOI; 1112.3252
[263]
Jacob Hastrup and Ulrik Lund Andersen, “All-optical cat-code quantum error correction”. 2108.12225
[264]
O. Higgott and N. P. Breuckmann, “Subsystem Codes with High Thresholds by Gauge Fixing and Reduced Qubit Overhead”, Physical Review X 11, (2021). DOI; 2010.09626
[265]
N. Didier, J. Bourassa, and A. Blais, “Fast Quantum Nondemolition Readout by Parametric Modulation of Longitudinal Qubit-Oscillator Interaction”, Physical Review Letters 115, (2015). DOI
[266]
K. Hammar et al., “Error-rate-agnostic decoding of topological stabilizer codes”, Physical Review A 105, (2022). DOI; 2112.01977
[267]
D. K. Tuckett et al., “Fault-Tolerant Thresholds for the Surface Code in Excess of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>5</mml:mn><mml:mo>%</mml:mo></mml:math> Under Biased Noise”, Physical Review Letters 124, (2020). DOI; 1907.02554
[268]
P. Kallquist, "Decoding of Zetterberg codes," in Proc. Fourth Joint Swedish-Soviet Workshop on Inform. Theory, Gotland, Sweden, Aug. 27-Sept. 1, 1989, p. 305-300
[269]
S. M. Dodunekov and J. E. M. Nilsson, “Algebraic decoding of the Zetterberg codes”, IEEE Transactions on Information Theory 38, 1570 (1992). DOI
[270]
M.-H. Jing et al., “A Result on Zetterberg Codes”, IEEE Communications Letters 14, 662 (2010). DOI
[271]
M. Nakahara, “Quantum Computing”, [] (2008). DOI