Here is a list of quantum codes that have notable decoders.
Name | Decoder(s) |
---|---|
2D color code | Projection decoder of \(O(n^4)\) complexity [1], modified to account for syndrome errors [2].Concatenated MPWM decoder [3].Syndrome extraction circuits based on superdense coding and a middle-out strategy [4]. |
2D hyperbolic surface code | Due to the symmetries of hyperbolic surface codes, optimal measurement schedules of the stabilizers can be found [5].Bounds on code capacity thresholds using ML decoding can be obtained by mapping the effect of noise on the code to a statistical mechanical model [6]. |
2D lattice stabilizer code | Renormalization group (RG) decoder [7].Tensor-network based decoder for 2D codes subject to correlated noise [8].Standard stabilizer-based error correction can be performed even in the presence of perturbations to the codespace [9,10]; see also Refs. [11,12]. |
3D color code | Decoder that maps 3D color code to three copies of the 3D surface code [13]. |
3D surface code | Flip decoder and its modification p-flip [14].Tensor-network decoder [15].Efficient MWPM decoder for 3D toric and 3D welded surface codes [16].Generalization of linear-time ML erasure decoder [17] to 3D surface codes [16]. |
Abelian LP code | Ensemble BP decoder for codes without short cycles of length 4 [18]. |
Abelian quantum-double stabilizer code | Efficient decoder correcting below the code distance [19]. |
Alternant code | Variation of the Berlekamp-Welch algorithm [20].Euclidean algorithm; see [21; Ch. 12] for more details.Guruswami-Sudan list decoder [22,23]. |
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 [24]. |
Approximate quantum error-correcting code (AQECC) | Given an encoding and a noise channel, a decoder that yields the optimal entanglement fidelity can be obtained by solving a semi-definite program [25–28]. This optimal decoder is robust to unexpected variations in the noise channel [29].The decoupling approach a.k.a. the Uhlmann decoder [30–32].Quantum machine-learning based decoders such as quantum convolutional neural networks [33] and quantum autoencoders [34].The Leung recovery map [35] for a noise channel whose Kraus operators \(E_j\) yield a diagonal QEC matrix, \(c_{ij}\propto\delta_{ij}\), has Kraus operators \(\Pi V_j^{\dagger}\), where \(\Pi\) is the codespace projection, and where \(V_j\) is the unitary from the polar decomposition of \(E_j \Pi\). This is the recovery used in the proof of the Knill-Laflamme conditions [36; Thm. 10.1].The Cafaro recovery map [37] can be obtained for noise Kraus operators if there exists a basis of error words with respect to which the uncorrectable piece in the Knill-Laflamme conditions is diagonal; see Ref. [38]. The map recovers information perfectly for strictly correctable noise.The Petz recovery map a.k.a. the transpose map [39–41], a quantum channel determined by the codespace and noise channel, yields an infidelity of recovery that is at most twice away from the infidelity of the best possible recovery [42]. The fidelity can be expressed exactly as a function of the Knill-Laflamme conditions [43; Thm. 1], and it can be used to derive a generalization of the Knill-Laflamme conditions for approximate QECCs [44]. The infidelity of a modified Petz recovery map under erasure can be bounded using the conditional mutual information [45–47]. In the case of topological codes, the Petz infidelity is related to the topological entanglement entropy [48]. Modifications include the Petz-like decoder [49].The Yoshida-Kitaev decoder for the Hayden-Preskill protocol [50] can be extended to general QECCs [49]. |
Approximate secret-sharing code | Decoding is analagous to reconstruction in a secret sharing scheme and is done in polynomial time. The only required operations are verification of quantum authentication, which is a pair of polynomial-time quantum algorithms that check if the fidelity of the received state is close to 1, and erasure correction for a stabilizer code, which involves solving a system of linear equations. |
B-code | Efficient decoding algorithm against erasures [51]. |
BPSK c-q code | Linear-optical quantum receiver [52].Kennedy receiver [53,54].Photon-number resolving detector [55].Non-Gaussian near-optimal receiver [54].Multi-stage quantum receiver [56]. |
Bacon-Shor code | Message passing for \([[9,1,3,3]]\) Bacon-Shor code [57].Both Steane error correction and Shor error correction can be used for syndrome extraction, with the former outperforming the latter [58].Utilizing the mapping of the effect of the noise to a statistical mechanical model [59,60] yields several copies of the 1D Ising model [61; 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 [62,63].Continuous-time QEC [64]. |
Balanced code | Efficient decoder [65–67]. |
Balanced product (BP) code | BP-OSD decoder [68]. |
Binary BCH code | Peterson decoder with runtime of order \(O(n^3)\) [69,70] (see exposition in Ref. [71]).Berlekamp-Massey decoder with runtime of order \(O(n^2)\) [72,73] and modification by Burton [74]; see also [75,76].Sugiyama et al. modification of the extended Euclidean algorithm [77,78].Guruswami-Sudan list decoder [22,23]. |
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 quadratic-residue (QR) code | Algebraic decoder [79]. |
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 [80]. This type of error correction fails for errors that are products of photon loss/gain and dephasing errors. However, for certain \((N,S)\) instances of the binomial code, detection of these types of errors can be done.Recovery can be done via projective measurements and unitary operations in a version of the Cafaro recovery map [80,81].Fault-tolerant scheme that converts the required POVM into binary measurements whose redundancy is guaranteed by a classical code [82]. |
Bivariate bicycle (BB) code | Syndrome extraction circuit requires seven layers of CNOT gates regardless of code length. BP-OSD decoder [68] has been extended [83] to account for measurement errors (i.e., the circuit-based noise model [84]).Random and optimized syndrome extraction schedules from Ref. [83] are not distance preserving.Some long-range check operators can be measured less frequently than others [85].Syndrome extraction circuits called morphing circuits [86], generalizing circuits for the color code [4]. |
Block code | Decoding an error-correcting code is equivalent to finding the ground state of some statistical mechanical model [87]. |
Bose–Chaudhuri–Hocquenghem (BCH) code | Berlekamp-Massey decoder with runtime of order \(O(n^2)\) [72,73,88] and modification by Burton [74]; see also [75,76].Gorenstein-Peterson-Zierler decoder with runtime of order \(O(n^3)\) [69,89] (see exposition in Ref. [71]).Sugiyama et al. modification of the extended Euclidean algorithm [77,78]. |
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 [90].Decoder [91] based on measuring in the phase-state basis and using Knill error correction (a.k.a. telecorrection [92]), which is based on teleportation [93,94]. |
Brown-Fawzi random Clifford-circuit code | Minimum-weight decoding via using tropical tensor networks [95]. |
Camara-Ollivier-Tillich code | Iterative error estimation based on the MIN-SUM and SUM-PRODUCT algorithms [96]. |
Cat code | Measuring the Fock-state number modulo \(2S\) can be used to determine if photon loss or excitation errors occurred. For \(S=1\), this is the occupation number parity. |
Chamon model code | Repetition-based decoder, based on the three underlying repetition codes and improved by pre-treatment with a probabilistic greedy local algorithm [97]. |
Checkerboard model code | Parallelized matching decoder [98]. |
Chuang-Leung-Yamamoto (CLY) code | Destructive decoding with a photon number measurement on each mode.State can be decoded with a network of beamsplitters, phase shifters, and Kerr media. |
Circuit-to-Hamiltonian approximate code | Local detection of Pauli errors can be done using circuits of depth \(O(\text{polylog}(n))\) based on exact decoders for the Brown-Fawzi code [99; Lemma 3.2]. |
Classical Goppa code | Algebraic decoding algorithms [100]. 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 [77,78].Binary Goppa codes can be decoded using an RS-based decoder [101].List decoder for binary Goppa codes [102]. |
Cluster-state code | MBQC syndrome extraction is performed by multiplying certain single-qubit \(X\)-type measurements, which yield syndrome values. |
Codeword stabilized (CWS) code | There is no known efficient algorithm to decode non-additive (non-stabilizer) CWS codes.Clustered bounded-distance decoder [103–105].Structured error recovery [106], which reduces to syndrome-based recovery for additive (i.e., stabilizer) CWS codes. |
Coherent FSK (CFSK) c-q code | Bondurant receiver [107].Cyclic receiver [108].Time-resolving receiver [109–111].Bayesian inference [109]. |
Coherent-state c-q code | Optimal receiver performance in ambiguous state discrimination is determined using the Yuen-Kennedy-Lax (YKL) conditions [112]. See review [113] for details on receivers used for coherent-state c-q codes.Joint-detection receiver that can attain channel capacity [114].Various near-optimal receiver designs that can handle arbitrary constellations of coherent states with possible degeneracies [115].The square-root measurement is optimal for geometrically uniform constellations [116–118]. |
Color code | In contrast to the surface code, the color code can suffer from unremovable hook errors due to the specifics of its syndrome extraction circuits. Fault-tolerant decoders thus have to utilize additional flag qubits. |
Compass code | Asymmetrically-weighed variant of the union-find decoder [119]. |
Concatenated Steane code | There exist fault-tolerant syndrome extraction protocols for the concatenated Steane code [120].Randomized compiling helps reduce logical error rate for some noise models [121]. |
Concatenated bosonic code | Decoder exploiting analog information from the outer code for bosonic codes concatenated with qubit QLDPC codes [122]. |
Concatenated code | Generalized minimum-distance decoder [123]. |
Concatenated quantum code | Standard decoding proceeds by first decoding the outer code and then using the resulting data to decode the inner code. |
Concatenated qubit code | The effective channel for a concatenation of codes is the composition of the codes' effective channels [124].Message passing algorithm for concatenated codes can be equivalent to ML decoding [125]. |
Convolutional code | Decoders based on the Viterbi algorithm (trellis decoding) were developed first, which result in the most-likely codeword for the encoded bits [126].BCJR decoder, also a trellis-based decoder [127]. |
Cubic theory code | Probabilistic local cellular-automaton decoder [128]. |
Cyclic linear \(q\)-ary code | Meggitt decoder [129].Information set decoding (ISD) [130], 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 [131]. |
Cyclic linear binary code | Meggitt decoder [129].Information set decoding (ISD) [130], 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 [131]. |
Cyclic quantum code | Adapted from the Berlekamp decoding algorithm for classical BCH codes [132]. |
Cyclic redundancy check (CRC) code | GRAND [133]. |
Delsarte-Goethals (DG) code | Since the equivalent \(\mathbb{Z_4}\) codes are extended cyclic codes, efficient encoding and decoding is possible [134]. |
Determinant code | For exact repair, the interior points of the storage-bandwidth trade-off curve can be shown to be the convex hull of \(k\) corner points described by \((\alpha_m,\beta_m)= (\binom{k}{m},\binom{k-1}{m-1})\) for \(m\in\{1,2,\cdots,k\}\). |
Dinur-Hsieh-Lin-Vidick (DHLV) code | Linear-time decoder utilizing the small set-flip decoder [135] for \(Z\) errors and a reconstruction procedure for \(X\) errors [136]. |
Doubled color code | ML decoder that can utilize a history of syndromes, based on the Walsh-Hadamard transform [137]. |
EA Galois-qudit stabilizer code | Syndrome extraction and computation based on classical additive codes [138]. |
EA QLDPC code | Decoder adapted for an all-optical impelementation [139]. |
EA quasi-cyclic QLDPC code | Sum-product algorithm (SPA) decoder [140]. |
EA qubit code | Decoding algorithm [141]. |
EA qubit stabilizer code | Optical implementation of a minimal code using hyper-entangled states [142]. |
EVENODD code | Efficient decoding algorithm against two erasures [143]. |
Eigenstate thermalization hypothesis (ETH) code | An explicit universal recovery channel for the ETH code is given in [144]. |
Error-correcting output code (ECOC) | Standard Hamming-distance decoding [145].Inverse Hamming decoding [146].Euclidean-distance decoding or attenuated Euclidean decoding [147].Loss-based decoding [148].Probabilistic-based decoding [149]. |
Evaluation AG code | Generalization of plane-curve decoder [150]. Another decoder [151] was later showed to be equivalent in Ref. [152]. Application of several algorithms in parallel can be used to decode up to half the minimum distance [153,154]. Computational procedure implementing these decoders is based on an extension of the Berlekamp-Massey algorithm by Sakata [155–157].Decoder based on majority voting of unknown syndromes [158] decodes up to half of the minimum distance [159].List decoders generalizing Sudan's RS decoder by Shokrollahi-Wasserman [160] and Guruswami-Sudan [23]. |
Expander LP code | Linear-time decoder [161].Logarithmic-time subroutine [162]. |
Expander code | Decoding can be done in \(O(n)\) runtime using a greedy flip decoder [163] (see also [164]). 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.'Find erasures and Decode' a.k.a. Viderman's algorithm correcting order \(\Omega(n)\) errors in order \(O(n)\) time [165]. |
Fiber-bundle code | Greedy algorithm can be used to efficiently decode \(X\) errors, but no known efficient decoding of \(Z\) errors yet [62]. |
Fibonacci code | An efficient algorithm base on minimum-weight perfect matching [166], 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) [167–169].Fusion-aware iterative minimum-weight perfect matching decoder. Note that ordinary MWPM decoders do not produce a threshold with this code [169].Cellular automaton decoder [170]. |
Finite-dimensional error-correcting code (ECC) | Capacity-achieving Guessing Random Additive Noise Decoding (GRAND) [171] (see also [172]). |
Finite-dimensional quantum error-correcting code | The operation \(\cal{D}\) in the definition of this code is called the decoder. However, the term decoder can sometimes be used for the unencoder \(\cal{U}\) (i.e., the inverse of the encoder), which does not correct errors.There are several recovery maps which work for noise that is not exactly correctable; see AQECC entry.Protection can be implemented via continuous-time QEC [173–177] via, e.g., reservoir engineering [178]; see review [179]. There are analogues of the Knill-Laflamme conditions for continuous-time QEC [180], and it has been adapted to non-Markovian noise [181]. |
Five-qubit perfect code | Syndrome extraction circuit using only CNOT-SWAP gates [182].Combined dynamical decoupling and error correction protocol on individually-controlled qubits with always-on Ising couplings [183].Symmetric decoder correcting all weight-one Pauli errors. The resulting logical error channel after coherent noise has been explicitly derived [184]. |
Floquet color code | Period-six measurement sequence utilizing two-qubit measurements [185]. |
Folded RS (FRS) code | Guruswami and Rudra [186,187] achieved list-decoding up to \(1-\frac{k}{n}-\epsilon\) fraction of errors using the Parvaresh-Vardy algorithm [188]; see Ref. [189] for a randomized construction.List-decoding works up to the Johnson bound using the Guruswami-Sudan algorithm [190].Folded RS codes, concatenated with suitable inner codes, can be efficiently list-decoded up to the Blokh-Zyablov bound [186,191]. |
Folded quantum RS (FQRS) code | Quantum list decodable [192]. |
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. |
Fracton Floquet code | Period-six measurement sequence utilizing two- and three-qubit measurements [185]. |
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 [193]. |
GKP CV-cluster-state code | GKP error correction can be naturally combined with CV MBQC protocols since the performance of both is quantified by a squeezing parameter [194]. |
GKP-surface code | Decoder for GKP-toric code [24].MWPM closest point decoder [195]. |
GNU PI code | For a family of shifted gnu codes, decoding can be done using projection, probability amplitude rebalancing, and gate teleportation in time \(O(n^2)\) [196]. |
Gabidulin code | Fast decoder based on a transform-domain approach [197].Algebraic list decoder that decodes up to the Singleton bound [198]. |
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 stabilizer code | Syndrome extraction and computation based on classical additive codes [138]. |
Generalized 2D color code | Chromöbius, an open-source implementation of the Möbius decoder works for many 2D color codes [4]. |
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 \(O(n^2)\) [72,73,88].Guruswami-Sudan list decoder [22,23] and modification by Koetter-Vardy for soft-decision decoding [199].Hard-decision decoder for errors within the Singleton bound [200]. |
Generalized Shor code | Efficient decoder [201]. |
Generalized bicycle (GB) code | BP-OSD decoder [68]. |
Generalized five-squares code | Decoding of five-squares codes leads to a mapping of these codes to two copies of the surface code [202,203]. |
Golay code | Majority decoding for the extended Golay code [204].Decoder for the extended Golay code using the hexacode [205].Both Golay codes have a trellis representation and can thus be decoded using trellis decoding [206,207].Bounded-distance decoder requiring at most 121 real operations [208]. |
Gold code | General decoding is done by building a sparse parity check matrix, followed by applying an iterative message passing alogirithm. [209]. |
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) [210] (a.k.a. closest lattice point problem) or by using other effective iterative schemes when, e.g., the lattice represents a concatenated GKP code [24,211–213]. While the decoder time scales exponentially with number of modes \(n\) generically, the time can be polynomial in \(n\) for certain codes [195].Babai's nearest plane algorithm [214] can be used for bounded-distance decoding [195].Combining AD noise with amplification yields displacement noise, the noise that GKP codes are designed to correct [215].ML decoder for correcting shift errors in GKP two-qubit gates [216]. |
Haah cubic code (CC) | Hard-decisions RG decoder [217]. |
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 [218] and neural-network [219] decoders. |
Heptagon holographic code | Optimal erasure decoder [220]. |
Hergert code | Since the equivalent \(\mathbb{Z_4}\) codes are extended cyclic codes, efficient encoding and decoding is possible. [134,221]. |
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 RS 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 [155]. |
Hexacode | Bounded-distance decoder requiring at most 34 real operations [208]. |
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 [222]. The bilayer architecture allows for logical entangling gates between logical surface-code patches.Soft output decoding [223]. |
High-dimensional expander (HDX) code | For 2D simplicial complexes, cycle code decoder admitting a polynomial-time decoding algorithm can be used [224]. |
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. [225] calls the QND Gate and which is defined as \(QND_c | x , y \rangle = |x + c y, y \rangle\). |
Homological code | Sweep decoder [226].Improved BP-OSD decoder [227].Renormalization group (RG) decoder [228]. |
Homological product code | Union-find [229]. |
Honeycomb (6.6.6) color code | Distance-three measurement schedule based on detector error models [230].Message-passing decoder [231].Adaptation of the restriction decoder [232].Neural-network decoder [233].Möbius matching decoder gives low logical failure rate [234] and has an open-source implementation called Chromöbius [4].AMBP4, a quaternary version [235] of the MBP decoder [236].MaxSAT-based decoder [237]. |
Honeycomb Floquet code | The ISG has a static subgroup for all time steps \(r\geq 3\) – that is, a subgroup which remains a subgroup of the ISG for all future times – given by so-called plaquette stabilizers. These are stabilizers consisting of products of check operators around homologically trivial paths. The syndrome bits correspond to the eigenvalues of the plaquette stabilizers. Because of the structure of the check operators, only one-third of all plaquettes are measured each round. The syndrome bits must therefore be represented by a lattice in spacetime, to reflect when and where the outcome was obtained. |
Hyperbolic Floquet code | Syndrome structure allows for MWPM decoding. |
Hyperbolic surface code | Hastings decoder [238]. |
Hypergraph product (HGP) code | ReShape decoder that uses minimum weight decoders for the classical codes used in the hypergraph construction [239].2D geometrically local syndrome extraction circuits with depth order \(O(\sqrt{n})\) using order \(O(n)\) ancilla qubits [240].Improved BP-OSD decoder [227].Erasure-correction can be implemented approximately with \(O(n^2)\) operations with quantum generalizations [241] of the peeling and pruned peeling decoders [242], 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 [243].Generalization [244] of Viderman's algorithm for expander codes [165]. |
Interleaved RS (IRS) code | Decoder that corrects up to \(1-\frac{2k+n}{3n}\) fraction of random errors [245].Decoder that corrects up to \(1-(\frac{k}{n})^{2/3}\) fraction of random errors [246]. |
Irregular repeat-accumulate (IRA) code | Linear-time decoder [247]. |
Justesen code | Generalized minimum distance decoding [248]. |
Kerdock code | Soft decision decoding involves extending the Fast Hadamard Transform decoding algorithm for the first-order RM code to Kerdock code [134].Complexity of soft decision decoding algorithm: \(4^m\) multiplications and \(m4^m\) additions [134,249]. |
Kitaev surface code | Using data from multiple syndrome measurements prior to decoding allows for correcting syndrome measurement errors. The surface code requires order \(O(d)\) extraction rounds in order to gain a reliable estimate. Syndrome measurements are distance-preserving because syndrome extraction circuits can be designed to avoid hook errors [59].Syndrome extraction circuits consist of CNOT gates and ancillary measurements since this is a stabilizer code [250]. Measurement schedules can be optimized using spacetime circuit codes to yield the 3CX surface code [251]. Schedules can also be optimized via ZX calculus [252,253]. Inspired by the honeycomb Floquet code, various weight-two measurement schemes have been designed [254–256].Expanding diamonds decoder correcting errors of some maximum fractal dimension [257]. 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) [59,258] (based on work by Edmonds on finding a matching in a graph [259,260]), 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 [59,259,261]. MWPM solves the MPE decoding problem exactly for independent \(X\) and \(Z\) noise. MPE decoding is \(NP\)-hard in general for the surface code [262].Bravyi-Suchara-Vargo (BSV) tensor network decoder [263] approximately solves the ML decoding problem under independent \(X,Z\) noise for the surface code and takes time of order \(O(n^2)\) [263]. ML decoding [59] is \(\#P\)-hard in general for the surface code [262].Union-find decoder [264] uses the union-find data structure [265–267], 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) [268]. Belief union find is a combination of belief-propagation and union-find [269]. Strictly local (as opposed to partially local) union find [270] has a worst-case runtime of order \(O(d^3)\) in the distance \(d\).Modified MWPM decoders: pipeline MWPM (accounting for correlations between events) [271,272]; modification tailored to asymmetric noise [273]; parity blossom MWPM and fusion blossom MWPM [274], 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) [268]; belief perfect matching (a combination of belief-propagation and MWPM) [269]; spanning tree matching (STM) and rapid-fire (RFire) decoders [275]; ordered decoding based on MWPM [276]. Combining, or harmonizing, various decoders can improve performance [277]. One such example is the Libra decoder [278], a combination of MWPM decoders combined with matching synthesis.Renormalization group (RG) [279–281]; see Ref. [282] for the planar surface code.Linear-time ML erasure decoder [17].Markov-chain Monte Carlo [283].Cellular automaton decoders [284–286]; see also [287].Neural network [288–290], reinforcement learning [291–293], and transformer-based [294,295] decoders.Lightweight low-latency look-up table (LILLIPUT) decoder for small surface codes [296].Decoders can be augmented with a pre-decoder [297,298], which can allow for some processing to be done inside the cryogenic environment of the quantum system [299].Sliding-window [300,301] and parallel-window [300] parallelizable decoders, designed to overcome the backlog problem, can be combined with many inner decoders, such as MWPM or union-find.Modifications of BP: generalized belief propagation (GBP) [302], based on a classical version [303]; AMBP4, a quaternary version [235] of the MBP decoder [236] of complexity \(O(n\log\log n)\); blockBP, a combination of BP and tensor-network decoders [304]; machine-learning inspired modifications [305]. See Ref. [306] for a review of BP decoders. The min-sum decoder, a simple variant of BP, cannot be used to attain the benefits of codes with distance greater than 9 [307].A color-code decoder can be used for the surface code [308].Progressive-Proximity Bit-Flipping (PPBF) decoder [309].Collision clustering decoder [310]. |
Ladder Floquet code | Period-four measurement sequence utilizing two-qubit measurements [63]. |
Lattice stabilizer code | Clustering decoder [217,284].Quantum neural-network (QNN) decoder [9]. |
Lattice-based code | Spherical decoder [311,312]. |
Lechner-Hauke-Zoller (LHZ) code | BP decoder [313]. |
Linear STC | Sphere decoder [314–316]. |
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 [317].Optimal symbol-by-symbol decoding rule [318].Information set decoding (ISD) [319], 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 [123].Soft-decision maximum-likelihood trellis-based decoder [320].Random linear codes over large fields are list-recoverable and list-decodable up to near-optimal rates [321].Extensions of algebraic-geometry decoders to linear codes [322,323]. |
Linear binary code | Decoding an arbitary linear binary code is \(NP\)-complete [324].Slepian's standard-array decoding [325].Recursive maximum likelihood decoding [326].Deep learning [327] and a transformer graph neural net (GNN) for soft decoding [328].Chase decoding, which uses channel measurement information [329]. |
Linear code with complementary dual (LCD) | The decoding problem reduces to finding the nearest codeword in \(C\) given a word in \(C^{\perp}\) [330]. |
Linearized RS code | Berlekamp-Welch-type decoder [331] and its sum-rank version [332]. |
Locally decodable code (LDC) | LDCs admit local decoders, i.e., decoders whose runtime scales polylogarithmically with \(n\). |
Loop toric code | Measurement-free local error correction circuit (LEC) using reinforcement learning [333]. |
Low-density parity-check (LDPC) code | Message-passing algorithm called belief propagation (BP) [334–336] (see also [337–339]).Soft-decision Sum-Product Algorithm (SPA) [334,337,340] and its simplification the Min-Sum Algorithm (MSA) [341].Linear programming [342–344].Iterative LDPC decoders can get stuck at stopping sets of their Tanner graphs [345], with decoder performance improving with the size of the smallest stopping set; see [346; Sec. 21.3.1] for more details. The smallest stopping set size can reach the minimum distance of the code [347].Ensembles of random LDPC codes under iterative decoders are subject to the concentration theorem [337,348]; see [346; Thm. 21.7.1] for the case of the BEC.Reinforcement learning [349]. |
Low-rank parity-check (LRPC) code | Efficient probabilistic decoder [350].Mixed decoder [351]. |
Luby transform (LT) code | Sum-Product Algorithm (SPA), often called a peeling decoder [352,353], similar to belief propagation [354]. |
MacKay-Neal LDPC (MN-LDPC) code | Free-energy minimization and a BP decoder [355]. |
Majorana box qubit | Qubit readout can be done by charge sensing [356–359]. |
Matrix-product code | Decoder up to half of the minimum distance for NSC codes [360]. |
Melas code | Algebraic decoder [361]. |
Modular-qudit code | For few-qudit codes (\(n\) is small), decoding can be based on a lookup table. For infinite code families, the size of such a table scales exponentially with \(n\), so approximate decoding algorithms scaling polynomially with \(n\) have to be used. The decoder determining the most likely error given a noise channel is called the maximum-likelihood (ML) decoder. |
Modular-qudit stabilizer code | Trellis decoder for prime-dimensional qudits, which builds a compact representation of the algebraic structure of the normalizer \(\mathsf{N(S)}\) [362]. |
Modular-qudit surface code | Renormalization group decoder [281,363]. |
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) [364] |
Multiplicity code | Multivariate multiplicity codes can be decoded up to half of the minimum distance in polynomial time [365,366].Univariate [367] and multivariate [365] multiplicity codes can be list-decoded up to the Johnson bound. Certain univariate code families achieve the list-decoding capacity for sufficiently large field characteristic [365,368]. |
NTRU-GKP code | Babai's nearest plane algorithm [214] can be used for bounded-distance decoding.An NTRU-based decoder against stochastic displacement noise is efficient because the decoding problem is equivalent to decrypting the NTRU cryptosystem with knowledge of the encoder. |
Newman-Moore code | Efficient decoder [369]. |
Niset-Andersen-Cerf code | Optical decoder using three beam splitters, electronic gain detectors, and two phase-insensitive amplifiers as described in Ref. [370]. |
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 [92]), which is based on teleportation [93,94]. This type of error correction avoids the complicated correction procedures typical in Fock-state codes, but requires a supply of clean codewords [91]. Performance of this method was analyzed in Ref. [371].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 [372,373] on an ancillary mode. See Section 4.B.1 of Ref. [91]. |
On-off keyed (OOK) c-q code | Dolinar receiver [374].Superconducting transition edge sensor (TES) photon-number resolving detector [375]. |
Orthogonal Spacetime Block Code (OSTBC) | Maximum-likelihood decoding can be achieved with only linear processing [376]. |
Oscillator-into-oscillator GKP code | Syndromes can be read off using ancilla modes, yielding partial information about noise in the logical modes that can then be used in an efficient ML decoding procedure [377]. |
PI qubit code | Schur-Weyl-transform based decoder [378]. 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 [379] and the quantum Schur transform. |
PPM c-q code | Conditional pulse nulling (CPN) receiver [380]. |
PSK c-q code | Multi-stage quantum receivers [381–386].Bayesian inference [109]. |
Pair-cat code | Lindbladian-based dissipative encoding utilizing two-mode two-photon absorption [387]. 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 | Hierarchical recovery model [388].Greedy decoder [388]. |
Permutation spherical code | Efficient maximum-likelihood decoder determining the Voronoi region of an error word. |
Plane-curve code | Generalization of the Peterson algorithm for BCH codes [150,389,390]. |
Polar c-q code | Quantum-limited successive-cancellation (SC) joint-detection receiver [391]. |
Polar code | Successive cancellation (SC) decoder [392].Successive cancellation list (SCL) decoder [393] and a modification utilizing sequence repetition (SR-List) [394].Soft cancellation (SCAN) decoder [395,396].Belief propagation (BP) decoder [397].Noisy quantum gate-vased decoder [398]. |
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 [134,249]. |
Quantum Goppa code | Farran algorithm [399]. |
Quantum Tamo-Barg (QTB) code | Polynomially efficient decoder for QTB codes against errors acting on a number of subsystems that can go up to half of their conjectured distance [400; Thm. 8]. The decoder is based on decoding RS codes, and its runtime is independent of the locality \(r\).Polynomially efficient decoder for FQTB codes against errors acting on a number of subsystems that can go up to half of their conjectured distance [400; Thm. 7]. The runtime depends on the locality \(r\). |
Quantum Tanner code | Linear-time potetial-based decoder similar to the small-set-flip decoder for quantum expander codes [135].Linear-time decoder [161].Logarithmic-time mismatch decomposition decoder [162]. |
Quantum convolutional code | ML decoder [401]. |
Quantum data-syndrome (QDS) code | Syndrome errors are decoded using redundant stabilizer measurements. |
Quantum error-correcting code (QECC) | The effect of an error is a mapping of the code subspace into another, potentially overlapping, subspace. To determine, or diagnose, the effect of the error in what is known as syndrome-based decoding, one can measure one or more operators called check operators, which resolve code and error spaces without collapsing the quantum information inside the spaces. The eigenvalues of check operators are called error syndromes. One round or cycle of quantum error correction proceeds by extracting syndromes and performing correcting operations to map the error space containing the logical information back into the codespace. For some codes, correcting operations are not necessary because one can instead track which error space contains the logical information. |
Quantum expander code | Small set-flip linear-time decoder, which corrects order \(\Omega(n^{1/2})\) adversarial errors [402].Log-time decoder [403].Constant-time decoder [404].2D geometrically local syndrome extraction circuits acting on a patch of \(N\) physical qubits have to be of depth of order \(\Omega(n/\sqrt{N})\) or deeper. More generally, there is a tradeoff between the depth \(D\) and width \(W\) of a syndrome extraction circuit, namely, \(D \geq n/\sqrt{W}\) [240]. |
Quantum locally recoverable code (QLRC) | Codes constructed with the help of AEL distance amplification [405,406] admit efficient decoders [400]. |
Quantum low-density parity-check (QLDPC) code | Iterative error estimation based on the MIN-SUM and SUM-PRODUCT algorithms [96].Quantum belief propagation (BP) decoder [407–409] is a quantum version of the classical BP decoder, but performance suffers due to degeneracy [410]. Various post-processing algorithms have been proposed (see below and also Refs. [411,412]).BP-OSD decoder, scaling as \(O(n^3)\), adds a post-processing step based on ordered statistics decoding (OSD) to the belief propogation (BP) decoder [68]. For an open-source implementation, see [413,414].Neural BP decoder [415,416] and GNN decoders [417,418] for qubit codes.Partially and fully decoupled BP decoders, which use the decoupling representation, yield improvements against depolarizing noise [419].Message-passing decoder utilizing stabilizer inactivation (MP-SI a.k.a. BP-SI) for CSS-type QLDPC qubit codes [420].BP localized statistics decoding (BP-LSD) that exploits error clustering [421].Syndrome-based linear programming (SB-LP) algorithm can be applied as a post-processing step after syndrome-based min-sum (SM-MS) decoding [422].BP guided decimation (BPGD) decoder [423].Ambiguity clustering (AC) decoder, in which measurement data is divided into clusters and decoded independently [424].Non-binary decoding algorithm for CSS-type QLDPC codes [425].2D geometrically local syndrome extraction circuits with bounded depth using order \(O(n^2)\) ancilla qubits [240]. For CSS codes, syndrome extraction can be implemented in constant depth [426].Soft (i.e., analog) syndrome iterative BP for CSS-type QLDPC codes, utilizing the continuous signal obtained in the physical implementation of the stabilizer measurement (as opposed to discretizing the signal into a syndrome bit) [427].The MWPM decoder for surface codes may be generalizable to QLDPC codes [428].Extensions of the union-find decoder for qubit QLDPC codes [429,430].Sliding-window decoding [431].Closed-branch decoder [432].BP with guided decimation guessing (GDG) sliding-window decoder for CSS qubit codes [433].Performing \(d\) syndrome extraction rounds obtains an effective distance of \(d\) for a qubit QLDPC code [434].Fault-tolerant constant-depth encoder and unencoder [435].BP plus ordered Tanner forest (BP+OTF) almost-linear time decoder [436]. |
Quantum parity code (QPC) | Teleportation-based QEC [437]. |
Quantum polar code | Quantum successive-cancellation list decoder (SCL-E) for quantum polar codes that do not need entanglement assistance [438]. |
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 [439,440]. An automaton by Gacs yields a decoder for a 1D lattice [441].Machine learning algorithm to implement continuous error-correction for the three-qubit quantum repetition code [442]. |
Quantum spherical code (QSC) | Lindbladian scheme stabilizing all points in the constellation and protecting from the AD operator \(E_{0}^{\otimes n}\) [443]. |
Quantum turbo code | Turbo decoder [444; Sec. V].Modified decoder yield improvement over the memoryless depolarizing channel [445]. |
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 [446]. |
Qubit CSS code | Coherent decoders allow for measurement-free error correction [447]. One method is table/multi-control decoding [448], which scales exponentially with the number of ancillas used in syndrome measurement. A fault-tolerant measurement-free scheme for low-distance CSS codes is formulated in Ref. [449]. Another method, the Ising-based decoder, utilizes the mapping of the effect of the noise to a statistical mechanical model [59,60] such that the decoding problem maps to preparation of the ground state of an Ising model. See [450; Table 3.2] for a Rosetta stone comparing statistical mechanical models, CSS codes, and chain complexes.Decoders based on neural networks [451]. |
Qubit c-q code | BP with quantum messages (BPQM) decoder [306,452,453]. |
Qubit code | Incorporating faulty syndrome measurements can be done using the phenomenological noise model, which simulates errors during syndrome extraction by flipping some of the bits of the measured syndrome bitstring. In the more involved circuit-level noise model, every component of the syndrome extraction circuit can be faulty.Hook errors are syndrome measurement circuit faults that cause more than one data-qubit error [59]. Hook errors occur at specific places in a syndrome extraction circuit and can sometimes be removed by re-ordering the gates of the circuit. If not, the use of flag qubits (see [454]) 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 size of the circuit extracting the syndrome depends on the weight of its corresponding stabilizer generator. Syndrome extraction circuits can be simulated efficiently using dedicated software (e.g., STIM [455]) and there are many general schemes for generating them [456] (see also [82]).MPE decoding, i.e., the process of finding the most likely error, is \(NP\)-complete in general [457,458]. 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 [459].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 [460], slowing down the computation.Splitting decoders [461].Trellis decoder, which builds a compact representation of the algebraic structure of the normalizer \(\mathsf{N(S)}\) [462].Quantum extension of GRAND decoder [463].Deep neural-network probabilistic decoder [464].Generalized belief propagation (GBP) [302] based on a classical version [303].Integer optimization decoder [465].Autonomous Lindbladian based decoders for codes encoding a single logical qubit [466].For codes encoding a single logical qubit, logical information can be extracted by single-qubit operations and classical communication [467].Correlated decoding can improve performance of Clifford and non-Clifford entangling gates [468].Detector graphs [455,469] and detector error models [230] can be used to design syndrome extraction circuits and logical measurements.Fault-tolerant constant-depth unencoder transforming logical states into physical states using single-qubit measurements [435]. |
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 [25,26] (see also Ref. [27]). 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 [80]. |
Random code | Ball-collision decoding [470].Information set decoding (ISD) [471] and Finiasz and Sendrier (FS-ISD) decoding [472]. |
Rank-metric code | Polynomial-reconstruction Berlekamp-Welch based decoder [473].Berlekamp-Massey based decoder [474]. |
Raptor (RAPid TORnado) code | Raptor codes can be decoded using inactivation decoding [475], 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) [59,258] (based on work by Edmonds on finding a matching in a graph [259,260]). |
Reed-Muller (RM) code | Reed decoder with \(r+1\)-step majority decoding corrects \(\frac{1}{2}(2^{m-r}-1)\) errors [476] (see also Ch. 13 of Ref. [21]).Sequential code-reduction decoding [477].Matrix factorization can be used to decode an RM\((n,n-3)\) code [478]; see [479]. |
Reed-Solomon (RS) code | Decoding general RS codes is \(NP\)-hard [480].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 \(O(n^2)\) [72,73].Gorenstein-Peterson-Zierler decoder with runtime of order \(O(n^3)\) [69,89] (see exposition in Ref. [71]).Berlekamp-Welch decoder with runtime of order \(O(n^3)\) [481] (see exposition in Ref. [482]), assuming that \(t \geq (n+k)/2\).Gao decoder using extended Euclidean algorithm [483].Fast-Fourier-transform decoder with runtime of order \(O(n \text{polylog}n)\) [484].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 asymptotically in \(n\) [485]. Roth and Ruckenstein proposed a modified key equation that allows for correction of more than \(\left\lfloor (n-k)/2 \right\rfloor\) errors [486]. The Guruswami-Sudan algorithm improved the Sudan algorithm to \(1-\sqrt{R}\) [23], meaning that RS codes achieve list-decoding capacity; see Ref. [487] for bounds. It was later shown that generic RS codes achieve list-decoding capacity [488]. A modification of the Guruswami-Sudan algorithm by Koetter and Vardy is used for soft-decision decoding [199] (see also Ref. [489]). Subcodes of RS codes whose evaluation points lie in a subfield can be decoded up to the \(1-R\) [198].The ubiquity of RS codes has yielded off-the-shelf VLSI intergrated-circuit decoding hardware [490] (see also Ref. [491], Ch. 5 and 10). |
Regenerating code (RGC) | If the recovered symbols are exactly equal to the erased symbols, the repair is called an exact repair.If the recovered symbols are not exactly equal to the erased symbols but still preserve the code properties, the repair is called a functional repair. |
Regular binary Tanner code | Parallel decoding algorithm corrects a fraction \(\delta_0^2/48\) of errors for Tanner codes [163]. A modification of said algorithm improves the fraction to \(\delta_0^2/4\) with no extra cost to complexity [492].Soft-decision linear-time decoder correcting errors almost up to half of the Blokh-Zyablov bound [493]. |
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 [439,440]. An automaton by Gacs yields a decoder for a 1D lattice [441]. |
Rotated surface code | Only certain syndrome extraction schedules are distance-preserving [494].Local neural-network using 3D convolutions, combined with a separate global decoder [495].Iterative CNOT decoder [496]. |
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 \(n\) [497] (see [498; Sec. 29.7.2] for a description). |
Singleton-bound approaching AQECC | Quantum list decodable [192]. |
Spacetime circuit code | Efficient decoders can be constructed for some circuits [499]. |
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 in which it is located upon reception. |
Spin cat code | Measurement-free error correction protocol [500]. |
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 [501].Decoder [502] based on Knill error correction (a.k.a. telecorrection [92]), which is based on teleportation [93,94].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 [501].Reinforcement learning decoder that uses only one ancilla qubit [503]. It has been extended to utilize previously measured syndrome information [504]. |
Square-octagon (4.8.8) color code | Fault-tolerant syndrome extraction circuits [505].Matching decoder [506,507].Integer-program (IP) decoder [505].Two-copy surface-code decoder [508]. |
Stabilizer code | The structure of stabilizer codes allows for straightforward syndrome-based decoding because the stabilizer generators serve as the code's check operators, and their eigenvalues serve as the error syndromes. The error correction process involves measuring the stabilizer generators and applying correcting Pauli-type operators based on the measurement outcomes. |
String-net code | Fusing non-Abelian anyons cannot be done in one step [509].Syndrome measurement circuits analyzed in Ref. [510].Clustering decoder [168]. |
Subspace code | List decoding up to the Singleton bound [198]. |
Subsystem CSS code | Steane-type decoder utilizing data from the underlying classical codes [511]. |
Subsystem color code | Clustering decoder [512].Erasure decoder [513].Gauge-fixing decoders [513,514]. |
Subsystem hypergraph product (SHP) code | Efficient decoder [201]. |
Subsystem modular-qudit CSS code | Steane-type decoder utilizing data from the underlying classical codes [511]. |
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 [202; Appendix].Decoder for certain geometrically local subsystem codes from hypergraphs [203]. |
Surface-17 code | Lookup table [494]. |
Ta-Shma zigzag code | Unique and list decoders [515]. |
Tamo-Barg code | Polynomial evaluation over \(r\) points [516]. |
Tanner code | Min-sum and sum-product iterative decoders for binary Tanner codes [517,518]; see also [78,519]. These decoders can be improved using a probabilistic message-passing schedule [520].Any code can be put into normal form without significantly altering the underlying graph or the decoding complexity [521]. For an algebraic viewpoint on decoding, see [522].Iterative decoding is optimal for Tanner graphs that are free of cycles [518]. However, codes that admit cycle-free representations have bounds on their distances [523,524]; see [346,525]. |
Tensor-network code | The decoder is created by creating a decoding quantum circuit with dangling legs replaced with input/output wires, and tensors converted to unitary gates. Maximum likelihood decoding can be used when the tensors are stabilizer codes.Tensor-network decoder when the tensor network is contractible via stabilizer isometries [526]. Independent logical qubits can be decoded in parallel [527].Tensor-network-based decoder when the encoding unitary is known [528]. |
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 [123] 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 [529]. |
Ternary Golay code | Decoder for the extended ternary Golay code using the tetracode [205]. |
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 [242]. This decoder either terminates when it has removed a given erasure pattern or when it is stuck in a stopping set. |
Torus-layer spherical code (TLSC) | Efficiently decodable [530]. |
Triangular surface code | The decoding uses a single decoding graph since the triangle code is not a CSS code. Nodes of the graph are located at each stabilizer (center of the triangle graph) and have red or blue edges, where red associates with \(X\) errors and blue with \(Z\) errors. To take into account any errors from measuring the error syndrome, a three-dimensional stack of the decoding graphs is laid on top of the code with vertical edges connecting to qubits between layers [531]. |
Turbo code | Turbo decoder [532], an instance of BP decoding [533].Maximum A Posteriori (MAP) decoder [534] and a soft output derivative [535]. The use of soft outputs can improve code performance [536].List decoding [537].VLSI intergrated-circuit decoding hardware [538]. |
Twisted XZZX toric code | Fault-tolerant syndrome extraction circuits using flag qubits [539].AMBP4, a quaternary version [235] of the MBP decoder [236]. |
Two-component cat code | All-optical decoder [540] based on Knill error correction (a.k.a. telecorrection [92]), which is based on teleportation [93,94]. |
Union stabilizer (USt) code | Error-detection algorithm [103–105]. |
Varshamov-Tenengolts (VT) code | Decoder based on checksums \(\sum_{i=1}^n i~x_i^{\prime}\) of corrupted codewords \(x_i^{\prime}\) [541,542]. |
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 \(\sigma_x (a+a^\dagger)\) [543]. This results in an \(X\)-dependent shift of the readout cavity resonance which can be measured.Star-code autonomous correction scheme [544]. |
Wasilewski-Banaszek code | Destructive measurement with photon number measurements on each mode. |
X-cube Floquet code | Period-six measurement sequence utilizing two-qubit measurements [545]. |
X-cube model code | Parallelized matching decoder [98]. |
XYZ color code | Efficient ML decoder at infinite bias [546].Cellular-automaton decoder [546]. |
XYZ ruby Floquet code | Period-three and period-six measurement sequences utilizing two-qubit measurements [547]. |
XYZ\(^2\) hexagonal stabilizer code | Maximum-likelihood decoding using the EWD decoder [548]. |
XZZX surface code | MWPM decoder, which can be used for \(X\) and \(Z\) noise. For \(Y\) noise, a variant of the matching decoder could be used like it is used for the XY code in Ref. [549]. Decoding complexity scales as order \(O(n^3)\) because the code is non-CSS [235][549; Supplement]. |
Yoked surface code | Soft information from the inner surface codes can be utilized via a message passing algorithm [125].Yokes can be measured using lattice surgery [550]. |
Zetterberg code | Kallquist first described an algebraic decoding theorem [551]. A faster version was later provided in Ref. [552] and further modified in Ref. [553]. |
\(((9,12,3))\) qubit code | Fault-tolerant scheme that converts the required POVM into 10 binary measurements whose redundancy is guaranteed by a classical code [82]. |
\([2^m,m+1,2^{m-1}]\) First-order RM code | First-order RM codes admit specialized decoders, such as the Fast Hadamard Transform decoder [554]. |
\([2^m-1,m,2^{m-1}]\) simplex code | Serial orthogonal decoder [555,556]Quantum decoder [557]. |
\([[12,2,4]]\) carbon code | Syndrome extraction circuit based on Knill error correction (a.k.a. telecorrection [92]), but using only two code blocks instead of three [558; Fig. 5]. |
\([[144,12,12]]\) gross code | The GDG sliding-window decoder [433], with a realization achieving a worst-case decoding latency of 3ms per window.AC decoder is faster than ordinary BP-OSD with no reduction of fidelity [424]. |
\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code | Latin rectangle encoder [559].Efficient decoder [560]. |
\([[5,1,3]]_{\mathbb{R}}\) Braunstein five-mode code | Error correction can be done using linear-optical elements and feedback [561]. |
\([[6,4,2]]\) error-detecting code | Efficient decoder for the many-hypercube code [562]. |
\([[7,1,3]]\) Steane code | Shor error correction fidelity calculation [563,564]. |
\([[9,1,3]]\) Shor code | Bit- and phase-flip circuits utilize CNOT and Hadamard gates ([565], Fig. 10.6). |
\([[9,1,3]]_{\mathbb{R}}\) Lloyd-Slotine code | Syndromes are real-valued, and decoding is done by a continuous version of majority voting (a.k.a. triple modular redundancy). |
\(\chi^{(2)}\) code | Linear optics and \(\chi^{(2)}\) interactions. |
\(q\)-ary simplex code | Permutation decoder [566] and MacDonald [567] codes. |
References
- [1]
- N. Delfosse, “Decoding color codes by projection onto surface codes”, Physical Review A 89, (2014) arXiv:1308.6207 DOI
- [2]
- A. M. Stephens, “Efficient fault-tolerant decoding of topological color codes”, (2014) arXiv:1402.3037
- [3]
- S.-H. Lee, A. Li, and S. D. Bartlett, “Color code decoder with improved scaling for correcting circuit-level noise”, (2024) arXiv:2404.07482
- [4]
- C. Gidney and C. Jones, “New circuits and an open source decoder for the color code”, (2023) arXiv:2312.08813
- [5]
- O. Higgott and N. P. Breuckmann, “Subsystem Codes with High Thresholds by Gauge Fixing and Reduced Qubit Overhead”, Physical Review X 11, (2021) arXiv:2010.09626 DOI
- [6]
- A. A. Kovalev et al., “Numerical and analytical bounds on threshold error rates for hypergraph-product codes”, Physical Review A 97, (2018) arXiv:1804.01950 DOI
- [7]
- G. Duclos-Cianci and D. Poulin, “A renormalization group decoding algorithm for topological quantum codes”, (2010) arXiv:1006.1362
- [8]
- C. T. Chubb and S. T. Flammia, “Statistical mechanical models for quantum codes with correlated noise”, Annales de l’Institut Henri Poincaré D 8, 269 (2021) arXiv:1809.10704 DOI
- [9]
- W. Zhong, O. Shtanko, and R. Movassagh, “Advantage of Quantum Neural Networks as Quantum Information Decoders”, (2024) arXiv:2401.06300
- [10]
- A. Lavasani and S. Vijay, “The Stability of Gapped Quantum Matter and Error-Correction with Adiabatic Noise”, (2024) arXiv:2402.14906
- [11]
- F. Pastawski et al., “Limitations of Passive Protection of Quantum Information”, (2009) arXiv:0911.3843
- [12]
- A. Kay, “Capabilities of a Perturbed Toric Code as a Quantum Memory”, Physical Review Letters 107, (2011) arXiv:1107.3940 DOI
- [13]
- A. B. Aloshious and P. K. Sarvepalli, “Projecting three-dimensional color codes onto three-dimensional toric codes”, Physical Review A 98, (2018) arXiv:1606.00960 DOI
- [14]
- T. R. Scruby and K. Nemoto, “Local Probabilistic Decoding of a Quantum Code”, Quantum 7, 1093 (2023) arXiv:2212.06985 DOI
- [15]
- C. Piveteau, C. T. Chubb, and J. M. Renes, “Tensor Network Decoding Beyond 2D”, (2023) arXiv:2310.10722
- [16]
- A. Kulkarni and P. K. Sarvepalli, “Decoding the three-dimensional toric codes and welded codes on cubic lattices”, Physical Review A 100, (2019) arXiv:1808.03092 DOI
- [17]
- N. Delfosse and G. Zémor, “Linear-time maximum likelihood decoding of surface codes over the quantum erasure channel”, Physical Review Research 2, (2020) arXiv:1703.01517 DOI
- [18]
- S. Miao et al., “A Joint Code and Belief Propagation Decoder Design for Quantum LDPC Codes”, (2024) arXiv:2401.06874
- [19]
- S. X. Cui, C. Galindo, and D. Romero, “Abelian Group Quantum Error Correction in Kitaev’s Model”, (2024) arXiv:2404.08552
- [20]
- H. Helgert, “Decoding of alternant codes (Corresp.)”, IEEE Transactions on Information Theory 23, 513 (1977) DOI
- [21]
- F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
- [22]
- V. Guruswami and M. Sudan, “Improved decoding of Reed-Solomon and algebraic-geometry codes”, IEEE Transactions on Information Theory 45, 1757 (1999) DOI
- [23]
- 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
- [24]
- C. Vuillot et al., “Quantum error correction with the toric Gottesman-Kitaev-Preskill code”, Physical Review A 99, (2019) arXiv:1810.00047 DOI
- [25]
- K. Audenaert and B. De Moor, “Optimizing completely positive maps using semidefinite programming”, Physical Review A 65, (2002) arXiv:quant-ph/0109155 DOI
- [26]
- M. Reimpell and R. F. Werner, “Iterative Optimization of Quantum Error Correcting Codes”, Physical Review Letters 94, (2005) arXiv:quant-ph/0307138 DOI
- [27]
- A. S. Fletcher, “Channel-Adapted Quantum Error Correction”, (2007) arXiv:0706.3400
- [28]
- A. S. Fletcher, P. W. Shor, and M. Z. Win, “Structured near-optimal channel-adapted quantum error correction”, Physical Review A 77, (2008) arXiv:0708.3658 DOI
- [29]
- G. Balló and P. Gurin, “Robustness of channel-adapted quantum error correction”, Physical Review A 80, (2009) arXiv:0905.3838 DOI
- [30]
- P. Hayden et al., “A Decoupling Approach to the Quantum Capacity”, Open Systems & Information Dynamics 15, 7 (2008) arXiv:quant-ph/0702005 DOI
- [31]
- F. Dupuis, “The decoupling approach to quantum information theory”, (2010) arXiv:1004.1641
- [32]
- F. Dupuis et al., “One-Shot Decoupling”, Communications in Mathematical Physics 328, 251 (2014) arXiv:1012.6044 DOI
- [33]
- I. Cong, S. Choi, and M. D. Lukin, “Quantum convolutional neural networks”, Nature Physics 15, 1273 (2019) arXiv:1810.03787 DOI
- [34]
- D. F. Locher, L. Cardarelli, and M. Müller, “Quantum Error Correction with Quantum Autoencoders”, Quantum 7, 942 (2023) arXiv:2202.00555 DOI
- [35]
- D. W. Leung et al., “Approximate quantum error correction can lead to better codes”, Physical Review A 56, 2567 (1997) DOI
- [36]
- M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2012) DOI
- [37]
- C. Cafaro and P. van Loock, “Approximate quantum error correction for generalized amplitude-damping errors”, Physical Review A 89, (2014) arXiv:1308.4582 DOI
- [38]
- S. Dutta, D. Biswas, and P. Mandayam, “Noise-adapted qudit codes for amplitude-damping noise”, (2024) arXiv:2406.02444
- [39]
- D. Petz, “Sufficient subalgebras and the relative entropy of states of a von Neumann algebra”, Communications in Mathematical Physics 105, 123 (1986) DOI
- [40]
- D. PETZ, “SUFFICIENCY OF CHANNELS OVER VON NEUMANN ALGEBRAS”, The Quarterly Journal of Mathematics 39, 97 (1988) DOI
- [41]
- H. Kwon and M. S. Kim, “Fluctuation Theorems for a Quantum Channel”, Physical Review X 9, (2019) arXiv:1810.03150 DOI
- [42]
- H. Barnum and E. Knill, “Reversing quantum dynamics with near-optimal quantum and classical fidelity”, (2000) arXiv:quant-ph/0004088
- [43]
- G. Zheng et al., “Near-Optimal Performance of Quantum Error Correction Codes”, Physical Review Letters 132, (2024) arXiv:2401.02022 DOI
- [44]
- H. K. Ng and P. Mandayam, “Simple approach to approximate quantum error correction based on the transpose channel”, Physical Review A 81, (2010) arXiv:0909.0931 DOI
- [45]
- O. Fawzi and R. Renner, “Quantum Conditional Mutual Information and Approximate Markov Chains”, Communications in Mathematical Physics 340, 575 (2015) arXiv:1410.0664 DOI
- [46]
- M. Junge et al., “Universal Recovery Maps and Approximate Sufficiency of Quantum Relative Entropy”, Annales Henri Poincaré 19, 2955 (2018) arXiv:1509.07127 DOI
- [47]
- S. T. Flammia et al., “Limits on the storage of quantum information in a volume of space”, Quantum 1, 4 (2017) arXiv:1610.06169 DOI
- [48]
- Y. Hu and Y. Zou, “Petz map recovery for long-range entangled quantum many-body states”, (2024) arXiv:2408.00857
- [49]
- T. Utsumi and Y. Nakata, “Explicit decoders using quantum singular value transformation”, (2024) arXiv:2405.06051
- [50]
- B. Yoshida and A. Kitaev, “Efficient decoding for the Hayden-Preskill protocol”, (2017) arXiv:1710.03363
- [51]
- M. Blaum and R. M. Roth, “New array codes for multiple phased burst correction”, IEEE Transactions on Information Theory 39, 66 (1993) DOI
- [52]
- K. Tsujino et al., “Quantum Receiver beyond the Standard Quantum Limit of Coherent Optical Communication”, Physical Review Letters 106, (2011) arXiv:1103.5592 DOI
- [53]
- Kennedy, Robert S. "A near-optimum receiver for the binary coherent state quantum channel." Quarterly Progress Report 108 (1973): 219-225.
- [54]
- 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) arXiv:0706.1038 DOI
- [55]
- M. T. DiMario and F. E. Becerra, “Robust Measurement for the Discrimination of Binary Coherent States”, Physical Review Letters 121, (2018) arXiv:1807.05199 DOI
- [56]
- D. Sych and G. Leuchs, “Practical Receiver for Optimal Discrimination of Binary Coherent Signals”, Physical Review Letters 117, (2016) arXiv:1404.5033 DOI
- [57]
- Z. W. E. Evans and A. M. Stephens, “Message passing in fault-tolerant quantum error correction”, Physical Review A 78, (2008) arXiv:0806.2188 DOI
- [58]
- G. Escobar-Arrieta and M. Gutiérrez, “Improved performance of the Bacon-Shor code with Steane’s syndrome extraction method”, (2024) arXiv:2403.01659
- [59]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [60]
- A. T. Schmitz, “Thermal Stability of Dynamical Phase Transitions in Higher Dimensional Stabilizer Codes”, (2020) arXiv:2002.11733
- [61]
- H. Bombin, “Topological subsystem codes”, Physical Review A 81, (2010) arXiv:0908.4246 DOI
- [62]
- M. B. Hastings, J. Haah, and R. O’Donnell, “Fiber bundle codes: breaking the n \({}^{\text{1/2}}\) polylog( n ) barrier for Quantum LDPC codes”, Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (2021) arXiv:2009.03921 DOI
- [63]
- M. B. Hastings and J. Haah, “Dynamically Generated Logical Qubits”, Quantum 5, 564 (2021) arXiv:2107.02194 DOI
- [64]
- G. Sarma and H. Mabuchi, “Gauge subsystems, separability and robustness in autonomous quantum memories”, New Journal of Physics 15, 035014 (2013) arXiv:1212.3564 DOI
- [65]
- D. Knuth, “Efficient balanced codes”, IEEE Transactions on Information Theory 32, 51 (1986) DOI
- [66]
- S. Al-Bassam and B. Bose, “On balanced codes”, IEEE Transactions on Information Theory 36, 406 (1990) DOI
- [67]
- K. A. Schouhamer Immink and J. H. Weber, “Very Efficient Balanced Codes”, IEEE Journal on Selected Areas in Communications 28, 188 (2010) DOI
- [68]
- P. Panteleev and G. Kalachev, “Degenerate Quantum LDPC Codes With Good Finite Length Performance”, Quantum 5, 585 (2021) arXiv:1904.02703 DOI
- [69]
- W. Peterson, “Encoding and error-correction procedures for the Bose-Chaudhuri codes”, IEEE Transactions on Information Theory 6, 459 (1960) DOI
- [70]
- 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.
- [71]
- R.E. Blahut, Theory and practice of error-control codes, Addison-Wesley 1983.
- [72]
- J. Massey, “Shift-register synthesis and BCH decoding”, IEEE Transactions on Information Theory 15, 122 (1969) DOI
- [73]
- E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, 1968
- [74]
- H. Burton, “Inversionless decoding of binary BCH codes”, IEEE Transactions on Information Theory 17, 464 (1971) DOI
- [75]
- W. W. Peterson and E. J. Weldon, Error-correcting codes. MIT press 1972.
- [76]
- R. Gallager, Information Theory and Reliable Communication (Springer Vienna, 1972) DOI
- [77]
- Y. Sugiyama et al., “A method for solving key equation for decoding goppa codes”, Information and Control 27, 87 (1975) DOI
- [78]
- R. McEliece, The Theory of Information and Coding (Cambridge University Press, 2002) DOI
- [79]
- Chen, Y. H., Truong, T. K., Chang, Y., Lee, C. D., & Chen, S. H. (2007). Algebraic decoding of quadratic residue codes using Berlekamp-Massey algorithm. Journal of information science and engineering, 23(1), 127-145.
- [80]
- V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018) arXiv:1708.05010 DOI
- [81]
- M. H. Michael et al., “New Class of Quantum Error-Correcting Codes for a Bosonic Mode”, Physical Review X 6, (2016) arXiv:1602.00008 DOI
- [82]
- Y. Ouyang, “Robust projective measurements through measuring code-inspired observables”, (2024) arXiv:2402.04093
- [83]
- S. Bravyi et al., “High-threshold and low-overhead fault-tolerant quantum memory”, Nature 627, 778 (2024) arXiv:2308.07915 DOI
- [84]
- A. G. Fowler, A. M. Stephens, and P. Groszkowski, “High-threshold universal quantum computation on the surface code”, Physical Review A 80, (2009) arXiv:0803.0272 DOI
- [85]
- N. Berthusen et al., “Toward a 2D Local Implementation of Quantum LDPC Codes”, (2024) arXiv:2404.17676
- [86]
- M. H. Shaw and B. M. Terhal, “Lowering Connectivity Requirements For Bivariate Bicycle Codes Using Morphing Circuits”, (2024) arXiv:2407.16336
- [87]
- N. Sourlas, “Spin-glass models as error-correcting codes”, Nature 339, 693 (1989) DOI
- [88]
- E. Berlekamp, “Nonbinary BCH decoding (Abstr.)”, IEEE Transactions on Information Theory 14, 242 (1968) DOI
- [89]
- 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
- [90]
- S. Kwon, S. Watabe, and J.-S. Tsai, “Autonomous quantum error correction in a four-photon Kerr parametric oscillator”, npj Quantum Information 8, (2022) arXiv:2203.09234 DOI
- [91]
- A. L. Grimsmo, J. Combes, and B. Q. Baragiola, “Quantum Computing with Rotation-Symmetric Bosonic Codes”, Physical Review X 10, (2020) arXiv:1901.08071 DOI
- [92]
- C. M. Dawson, H. L. Haselgrove, and M. A. Nielsen, “Noise thresholds for optical cluster-state quantum computation”, Physical Review A 73, (2006) arXiv:quant-ph/0601066 DOI
- [93]
- E. Knill, “Quantum computing with realistically noisy devices”, Nature 434, 39 (2005) arXiv:quant-ph/0410199 DOI
- [94]
- E. Knill, “Scalable Quantum Computation in the Presence of Large Detected-Error Rates”, (2004) arXiv:quant-ph/0312190
- [95]
- J. Nelson et al., “Fault-Tolerant Quantum Memory using Low-Depth Random Circuit Codes”, (2023) arXiv:2311.17985
- [96]
- T. Camara, H. Ollivier, and J.-P. Tillich, “Constructions and performance of classes of quantum LDPC codes”, (2005) arXiv:quant-ph/0502086
- [97]
- J. Zhao, Y.-C. Wu, and G.-P. Guo, “Quantum memory error correction computation based on Chamon model”, (2023) arXiv:2303.05267
- [98]
- B. J. Brown and D. J. Williamson, “Parallelized quantum error correction with fracton topological codes”, Physical Review Research 2, (2020) arXiv:1901.08061 DOI
- [99]
- T. C. Bohdanowicz et al., “Good approximate quantum LDPC codes from spacetime circuit Hamiltonians”, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (2019) arXiv:1811.00277 DOI
- [100]
- N. Patterson, “The algebraic decoding of Goppa codes”, IEEE Transactions on Information Theory 21, 203 (1975) DOI
- [101]
- Daniel J. Bernstein, "Understanding binary-Goppa decoding." Cryptology ePrint Archive (2022).
- [102]
- P. Beelen et al., “On Rational Interpolation-Based List-Decoding and List-Decoding Binary Goppa Codes”, IEEE Transactions on Information Theory 59, 3269 (2013) DOI
- [103]
- Y. Li, I. Dumer, and L. P. Pryadko, “Clustered Error Correction of Codeword-Stabilized Quantum Codes”, Physical Review Letters 104, (2010) arXiv:0907.2038 DOI
- [104]
- Y. Li et al., “Clustered bounded-distance decoding of codeword-stabilized quantum codes”, 2010 IEEE International Symposium on Information Theory (2010) DOI
- [105]
- Li, Yunfan. Codeword Stabilized Quantum Codes and Their Error Correction. Diss. UC Riverside, 2010.
- [106]
- Y. Li et al., “Structured error recovery for code-word-stabilized quantum codes”, Physical Review A 81, (2010) arXiv:0912.3245 DOI
- [107]
- R. S. Bondurant, “Near-quantum optimum receivers for the phase-quadrature coherent-state channel”, Optics Letters 18, 1896 (1993) DOI
- [108]
- M. V. Jabir et al., “Experimental demonstration of the near-quantum optimal receiver”, OSA Continuum 3, 3324 (2020) DOI
- [109]
- I. A. Burenkov, O. V. Tikhonova, and S. V. Polyakov, “Quantum receiver for large alphabet communication”, Optica 5, 227 (2018) arXiv:1802.08287 DOI
- [110]
- I. A. Burenkov et al., “Time-Resolving Quantum Measurement Enables Energy-Efficient, Large-Alphabet Communication”, PRX Quantum 1, (2020) DOI
- [111]
- M. V. Jabir et al., “Energy and bandwidth efficiency optimization of quantum-enabled optical communication channels”, npj Quantum Information 8, (2022) DOI
- [112]
- H. Yuen, R. Kennedy, and M. Lax, “Optimum testing of multiple hypotheses in quantum detection theory”, IEEE Transactions on Information Theory 21, 125 (1975) DOI
- [113]
- I. A. Burenkov, M. V. Jabir, and S. V. Polyakov, “Practical quantum-enhanced receivers for classical communication”, AVS Quantum Science 3, (2021) DOI
- [114]
- S. Guha, “Structured Optical Receivers to Attain Superadditive Capacity and the Holevo Limit”, Physical Review Letters 106, (2011) arXiv:1101.1550 DOI
- [115]
- J. S. Sidhu et al., “Linear optics and photodetection achieve near-optimal unambiguous coherent state discrimination”, Quantum 7, 1025 (2023) arXiv:2109.00008 DOI
- [116]
- Y. C. Eldar and G. D. Forney Jr, “On Quantum Detection and the Square-Root Measurement”, (2000) arXiv:quant-ph/0005132
- [117]
- Y. C. Eldar, A. Megretski, and G. C. Verghese, “Optimal Detection of Symmetric Mixed Quantum States”, (2002) arXiv:quant-ph/0211111
- [118]
- M. Rosati, “Performance of Coherent Frequency-Shift Keying for Classical Communication on Quantum Channels”, 2021 IEEE International Symposium on Information Theory (ISIT) (2021) arXiv:2203.09822 DOI
- [119]
- M. Li et al., “2D Compass Codes”, Physical Review X 9, (2019) arXiv:1809.01193 DOI
- [120]
- B. Pato, T. Tansuwannont, and K. R. Brown, “Concatenated Steane code with single-flag syndrome checks”, (2024) arXiv:2403.09978
- [121]
- A. Jain et al., “Improved quantum error correction with randomized compiling”, Physical Review Research 5, (2023) arXiv:2303.06846 DOI
- [122]
- L. Berent et al., “Analog Information Decoding of Bosonic Quantum Low-Density Parity-Check Codes”, PRX Quantum 5, (2024) arXiv:2311.01328 DOI
- [123]
- G. Forney, “Generalized minimum distance decoding”, IEEE Transactions on Information Theory 12, 125 (1966) DOI
- [124]
- B. Rahn, A. C. Doherty, and H. Mabuchi, “Exact performance of concatenated quantum codes”, Physical Review A 66, (2002) arXiv:quant-ph/0206061 DOI
- [125]
- D. Poulin, “Optimal and efficient decoding of concatenated quantum block codes”, Physical Review A 74, (2006) arXiv:quant-ph/0606126 DOI
- [126]
- A. Viterbi, “Error bounds for convolutional codes and an asymptotically optimum decoding algorithm”, IEEE Transactions on Information Theory 13, 260 (1967) DOI
- [127]
- L. Bahl et al., “Optimal decoding of linear codes for minimizing symbol error rate (Corresp.)”, IEEE Transactions on Information Theory 20, 284 (1974) DOI
- [128]
- P.-S. Hsin, R. Kobayashi, and G. Zhu, “Non-Abelian Self-Correcting Quantum Memory”, (2024) arXiv:2405.11719
- [129]
- J. Meggitt, “Error correcting codes and their implementation for data transmission systems”, IEEE Transactions on Information Theory 7, 234 (1961) DOI
- [130]
- E. Prange, “The use of information sets in decoding cyclic codes”, IEEE Transactions on Information Theory 8, 5 (1962) DOI
- [131]
- J. Macwilliams, “Permutation Decoding of Systematic Codes”, Bell System Technical Journal 43, 485 (1964) DOI
- [132]
- S. Dutta and P. P. Kurur, “Quantum Cyclic Code”, (2010) arXiv:1007.1697
- [133]
- W. An, M. Medard, and K. R. Duffy, “CRC Codes as Error Correction Codes”, ICC 2021 - IEEE International Conference on Communications (2021) arXiv:2104.13663 DOI
- [134]
- A. R. Hammons et al., “The Z/sub 4/-linearity of Kerdock, Preparata, Goethals, and related codes”, IEEE Transactions on Information Theory 40, 301 (1994) DOI
- [135]
- S. Gu, C. A. Pattison, and E. Tang, “An efficient decoder for a linear distance quantum LDPC code”, (2022) arXiv:2206.06557
- [136]
- I. Dinur et al., “Good Quantum LDPC Codes with Linear Time Decoders”, (2022) arXiv:2206.07750
- [137]
- S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239
- [138]
- P. J. Nadkarni and S. S. Garani, “Quantum error correction architecture for qudit stabilizer codes”, Physical Review A 103, (2021) DOI
- [139]
- I. B. Djordjevic, “Photonic entanglement-assisted quantum low-density parity-check encoders and decoders”, Optics Letters 35, 1464 (2010) DOI
- [140]
- M.-H. Hsieh, W.-T. Yen, and L.-Y. Hsu, “High Performance Entanglement-Assisted Quantum LDPC Codes Need Little Entanglement”, IEEE Transactions on Information Theory 57, 1761 (2011) arXiv:0906.5532 DOI
- [141]
- M. M. Wilde, “Quantum Coding with Entanglement”, (2008) arXiv:0806.4214
- [142]
- M. M. Wilde and D. B. Uskov, “Linear-optical hyperentanglement-assisted quantum error-correcting code”, Physical Review A 79, (2009) arXiv:0807.4906 DOI
- [143]
- M. Blaum et al., “EVENODD: an efficient scheme for tolerating double disk failures in RAID architectures”, IEEE Transactions on Computers 44, 192 (1995) DOI
- [144]
- N. Bao and N. Cheng, “Eigenstate thermalization hypothesis and approximate quantum error correction”, Journal of High Energy Physics 2019, (2019) arXiv:1906.03669 DOI
- [145]
- Nilsson, Nils J. "Learning machines." (1965).
- [146]
- T. Windeatt and R. Ghaderi, “Coding and decoding strategies for multi-class learning problems”, Information Fusion 4, 11 (2003) DOI
- [147]
- O. Pujol, S. Escalera, and P. Radeva, “An incremental node embedding technique for error correcting output codes”, Pattern Recognition 41, 713 (2008) DOI
- [148]
- Allwein, Erin L., Robert E. Schapire, and Yoram Singer. "Reducing multiclass to binary: A unifying approach for margin classifiers." Journal of machine learning research 1.Dec (2000): 113-141.
- [149]
- A. Passerini, M. Pontil, and P. Frasconi, “New Results on Error Correcting Output Codes of Kernel Machines”, IEEE Transactions on Neural Networks 15, 45 (2004) DOI
- [150]
- J. Justesen et al., “Construction and decoding of a class of algebraic geometry codes”, IEEE Transactions on Information Theory 35, 811 (1989) DOI
- [151]
- 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
- [152]
- D. Ehrhard, “Decoding Algebraic-Geometric Codes by solving a key equation”, Lecture Notes in Mathematics 18 (1992) DOI
- [153]
- R. Pellikaan, “On a decoding algorithm for codes on maximal curves”, IEEE Transactions on Information Theory 35, 1228 (1989) DOI
- [154]
- S. Vladut, “On the decoding of algebraic-geometric codes over F/sub q/ for q<or=16”, IEEE Transactions on Information Theory 36, 1461 (1990) DOI
- [155]
- 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
- [156]
- S. Sakata, “Extension of the Berlekamp-Massey algorithm to N dimensions”, Information and Computation 84, 207 (1990) DOI
- [157]
- S. Sakata, “Decoding binary 2-D cyclic codes by the 2-D Berlekamp-Massey algorithm”, IEEE Transactions on Information Theory 37, 1200 (1991) DOI
- [158]
- 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
- [159]
- D. Ehrhard, “Achieving the designed error capacity in decoding algebraic-geometric codes”, IEEE Transactions on Information Theory 39, 743 (1993) DOI
- [160]
- M. A. Shokrollahi and H. Wasserman, “List decoding of algebraic-geometric codes”, IEEE Transactions on Information Theory 45, 432 (1999) DOI
- [161]
- A. Leverrier and G. Zémor, “Efficient decoding up to a constant fraction of the code length for asymptotically good quantum codes”, (2022) arXiv:2206.07571
- [162]
- A. Leverrier and G. Zémor, “Decoding quantum Tanner codes”, (2022) arXiv:2208.05537
- [163]
- M. Sipser and D. A. Spielman, “Expander codes”, IEEE Transactions on Information Theory 42, 1710 (1996) DOI
- [164]
- J. Feldman et al., “LP Decoding Corrects a Constant Fraction of Errors”, IEEE Transactions on Information Theory 53, 82 (2007) DOI
- [165]
- M. Viderman, “Linear-time decoding of regular expander codes”, ACM Transactions on Computation Theory 5, 1 (2013) DOI
- [166]
- G. M. Nixon and B. J. Brown, “Correcting Spanning Errors With a Fractal Code”, IEEE Transactions on Information Theory 67, 4504 (2021) arXiv:2002.11738 DOI
- [167]
- 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) arXiv:1506.03815 DOI
- [168]
- G. Dauphinais and D. Poulin, “Fault-Tolerant Quantum Error Correction for non-Abelian Anyons”, Communications in Mathematical Physics 355, 519 (2017) arXiv:1607.02159 DOI
- [169]
- A. Schotte et al., “Quantum error correction thresholds for the universal Fibonacci Turaev-Viro code”, (2021) arXiv:2012.04610
- [170]
- A. Schotte, L. Burgelman, and G. Zhu, “Fault-tolerant error correction for a universal non-Abelian topological quantum computer at finite temperature”, (2022) arXiv:2301.00054
- [171]
- K. R. Duffy, J. Li, and M. Medard, “Capacity-Achieving Guessing Random Additive Noise Decoding”, IEEE Transactions on Information Theory 65, 4023 (2019) arXiv:1802.07010 DOI
- [172]
- K. R. Duffy, J. Li, and M. Medard, “Guessing noise, not code-words”, 2018 IEEE International Symposium on Information Theory (ISIT) (2018) DOI
- [173]
- J. P. Paz and W. H. Zurek, “Continuous error correction”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 355 (1998) DOI
- [174]
- J. P. Barnes and W. S. Warren, “Automatic Quantum Error Correction”, Physical Review Letters 85, 856 (2000) arXiv:quant-ph/9912104 DOI
- [175]
- C. Ahn, A. C. Doherty, and A. J. Landahl, “Continuous quantum error correction via quantum feedback control”, Physical Review A 65, (2002) arXiv:quant-ph/0110111 DOI
- [176]
- M. Sarovar and G. J. Milburn, “Continuous quantum error correction by cooling”, Physical Review A 72, (2005) arXiv:quant-ph/0501038 DOI
- [177]
- R. van Handel and H. Mabuchi, “Optimal error tracking via quantum coding and continuous syndrome measurement”, (2005) arXiv:quant-ph/0511221
- [178]
- J. F. Poyatos, J. I. Cirac, and P. Zoller, “Quantum Reservoir Engineering with Laser Cooled Trapped Ions”, Physical Review Letters 77, 4728 (1996) DOI
- [179]
- O. Oreshkov, “Continuous-time quantum error correction”, (2013) arXiv:1311.2485
- [180]
- J.-M. Lihm, K. Noh, and U. R. Fischer, “Implementation-independent sufficient condition of the Knill-Laflamme type for the autonomous protection of logical qudits by strong engineered dissipation”, Physical Review A 98, (2018) arXiv:1711.02999 DOI
- [181]
- O. Oreshkov and T. A. Brun, “Continuous quantum error correction for non-Markovian decoherence”, Physical Review A 76, (2007) arXiv:0705.2342 DOI
- [182]
- A. V. Antipov, E. O. Kiktenko, and A. K. Fedorov, “Realizing a class of stabilizer quantum error correction codes using a single ancilla and circular connectivity”, Physical Review A 107, (2023) arXiv:2207.13356 DOI
- [183]
- 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) arXiv:1509.01239 DOI
- [184]
- C. Liu, “Exact performance of the five-qubit code with coherent errors”, (2022) arXiv:2203.01706
- [185]
- M. Davydova, N. Tantivasadakarn, and S. Balasubramanian, “Floquet Codes without Parent Subsystem Codes”, PRX Quantum 4, (2023) arXiv:2210.02468 DOI
- [186]
- V. Guruswami and A. Rudra, “Explicit Codes Achieving List Decoding Capacity: Error-correction with Optimal Redundancy”, (2007) arXiv:cs/0511072
- [187]
- Atri Rudra. List Decoding and Property Testing of Error Correcting Codes. PhD thesis, University of Washington, 8 2007.
- [188]
- 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
- [189]
- V. Guruswami, “Linear-Algebraic List Decoding of Folded Reed-Solomon Codes”, 2011 IEEE 26th Annual Conference on Computational Complexity (2011) arXiv:1106.0436 DOI
- [190]
- S. Bhandari et al., “Ideal-Theoretic Explanation of Capacity-Achieving Decoding”, (2021) arXiv:2103.07930 DOI
- [191]
- V. Guruswami and A. Rudra, “Better Binary List Decodable Codes Via Multilevel Concatenation”, IEEE Transactions on Information Theory 55, 19 (2009) DOI
- [192]
- T. Bergamaschi, L. Golowich, and S. Gunn, “Approaching the Quantum Singleton Bound with Approximate Error Correction”, (2022) arXiv:2212.09935
- [193]
- S. Dutta and P. P. Kurur, “Quantum Cyclic Code of length dividing \(p^{t}+1\)”, (2011) arXiv:1011.5814
- [194]
- N. C. Menicucci, “Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States”, Physical Review Letters 112, (2014) arXiv:1310.7596 DOI
- [195]
- M. Lin, C. Chamberland, and K. Noh, “Closest Lattice Point Decoding for Multimode Gottesman-Kitaev-Preskill Codes”, PRX Quantum 4, (2023) arXiv:2303.04702 DOI
- [196]
- Y. Ouyang, “Permutation-invariant quantum coding for quantum deletion channels”, 2021 IEEE International Symposium on Information Theory (ISIT) (2021) arXiv:2102.02494 DOI
- [197]
- D. Silva and F. R. Kschischang, “Fast encoding and decoding of Gabidulin codes”, 2009 IEEE International Symposium on Information Theory (2009) arXiv:0901.2483 DOI
- [198]
- V. Guruswami and C. Xing, “List decoding reed-solomon, algebraic-geometric, and gabidulin subcodes up to the singleton bound”, Proceedings of the forty-fifth annual ACM symposium on Theory of Computing (2013) DOI
- [199]
- R. Koetter and A. Vardy, “Algebraic soft-decision decoding of reed-solomon codes”, IEEE Transactions on Information Theory 49, 2809 (2003) DOI
- [200]
- Berman, A., Dor, A., Shany, Y., Shapir, I., and Doubchak, A. (2023). U.S. Patent No. 11,855,658. Washington, DC: U.S. Patent and Trademark Office.
- [201]
- P. K. Sarvepalli, A. Klappenecker, and M. Rotteler, “New decoding algorithms for a class of subsystem codes and generalized shor codes”, 2009 IEEE International Symposium on Information Theory (2009) DOI
- [202]
- M. Suchara, S. Bravyi, and B. Terhal, “Constructions and noise threshold of topological subsystem codes”, Journal of Physics A: Mathematical and Theoretical 44, 155301 (2011) arXiv:1012.0425 DOI
- [203]
- V. V. Gayatri and P. K. Sarvepalli, “Decoding Algorithms for Hypergraph Subsystem Codes and Generalized Subsystem Surface Codes”, (2018) arXiv:1805.12542
- [204]
- J.-M. Goethals, “On the Golay perfect binary code”, Journal of Combinatorial Theory, Series A 11, 178 (1971) DOI
- [205]
- V. Pless, “Decoding the Golay codes”, IEEE Transactions on Information Theory 32, 561 (1986) DOI
- [206]
- A. J. VITERBI, “Error Bounds for Convolutional Codes and an Asymptotically Optimum Decoding Algorithm”, The Foundations of the Digital Wireless World 41 (2009) DOI
- [207]
- B. Honary and G. Markarian, “New simple encoder and trellis decoder for Golay codes”, Electronics Letters 29, 2170 (1993) DOI
- [208]
- 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
- [209]
- 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
- [210]
- E. Agrell et al., “Closest point search in lattices”, IEEE Transactions on Information Theory 48, 2201 (2002) DOI
- [211]
- K. Noh and C. Chamberland, “Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code”, Physical Review A 101, (2020) arXiv:1908.03579 DOI
- [212]
- J. Conrad, J. Eisert, and F. Arzani, “Gottesman-Kitaev-Preskill codes: A lattice perspective”, Quantum 6, 648 (2022) arXiv:2109.14645 DOI
- [213]
- N. Raveendran et al., “Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound”, Quantum 6, 767 (2022) arXiv:2111.07029 DOI
- [214]
- L. Babai, “On Lovász’ lattice reduction and the nearest lattice point problem”, Combinatorica 6, 1 (1986) DOI
- [215]
- K. Noh, V. V. Albert, and L. Jiang, “Quantum Capacity Bounds of Gaussian Thermal Loss Channels and Achievable Rates With Gottesman-Kitaev-Preskill Codes”, IEEE Transactions on Information Theory 65, 2563 (2019) arXiv:1801.07271 DOI
- [216]
- K. Noh, C. Chamberland, and F. G. S. L. Brandão, “Low-Overhead Fault-Tolerant Quantum Error Correction with the Surface-GKP Code”, PRX Quantum 3, (2022) arXiv:2103.06994 DOI
- [217]
- S. Bravyi and J. Haah, “Quantum Self-Correction in the 3D Cubic Code Model”, Physical Review Letters 111, (2013) arXiv:1112.3252 DOI
- [218]
- D. Bhoumik et al., “Efficient Syndrome Decoder for Heavy Hexagonal QECC via Machine Learning”, ACM Transactions on Quantum Computing 5, 1 (2024) arXiv:2210.09730 DOI
- [219]
- B. Hall, S. Gicev, and M. Usman, “Artificial neural network syndrome decoding on IBM quantum processors”, Physical Review Research 6, (2024) arXiv:2311.15146 DOI
- [220]
- R. J. Harris et al., “Calderbank-Shor-Steane holographic quantum error-correcting codes”, Physical Review A 98, (2018) arXiv:1806.06472 DOI
- [221]
- T. Helleseth and P. V. Kumar, “The algebraic decoding of the Z/sub 4/-linear Goethals code”, IEEE Transactions on Information Theory 41, 2040 (1995) DOI
- [222]
- F. Annexstein and M. Baumslag, “A unified approach to off-line permutation routing on parallel networks”, Proceedings of the second annual ACM symposium on Parallel algorithms and architectures - SPAA ’90 (1990) DOI
- [223]
- N. Meister, C. A. Pattison, and J. Preskill, “Efficient soft-output decoders for the surface code”, (2024) arXiv:2405.07433
- [224]
- S. Evra, T. Kaufman, and G. Zémor, “Decodable quantum LDPC codes beyond the \(\sqrt{n}\) distance barrier using high dimensional expanders”, (2020) arXiv:2004.07935
- [225]
- P. Hayden et al., “Spacetime replication of continuous variable quantum information”, New Journal of Physics 18, 083043 (2016) arXiv:1601.02544 DOI
- [226]
- 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
- [227]
- O. Higgott and N. P. Breuckmann, “Improved Single-Shot Decoding of Higher-Dimensional Hypergraph-Product Codes”, PRX Quantum 4, (2023) arXiv:2206.03122 DOI
- [228]
- K. Duivenvoorden, N. P. Breuckmann, and B. M. Terhal, “Renormalization Group Decoder for a Four-Dimensional Toric Code”, IEEE Transactions on Information Theory 65, 2545 (2019) arXiv:1708.09286 DOI
- [229]
- N. Delfosse and M. B. Hastings, “Union-Find Decoders For Homological Product Codes”, Quantum 5, 406 (2021) arXiv:2009.14226 DOI
- [230]
- P.-J. H. S. Derks et al., “Designing fault-tolerant circuits using detector error models”, (2024) arXiv:2407.13826
- [231]
- P. Sarvepalli and R. Raussendorf, “Efficient decoding of topological color codes”, Physical Review A 85, (2012) arXiv:1111.0831 DOI
- [232]
- C. Chamberland et al., “Triangular color codes on trivalent graphs with flag qubits”, New Journal of Physics 22, 023019 (2020) arXiv:1911.00355 DOI
- [233]
- N. Maskara, A. Kubica, and T. Jochym-O’Connor, “Advantages of versatile neural-network decoding for topological codes”, Physical Review A 99, (2019) arXiv:1802.08680 DOI
- [234]
- K. Sahay and B. J. Brown, “Decoder for the Triangular Color Code by Matching on a Möbius Strip”, PRX Quantum 3, (2022) arXiv:2108.11395 DOI
- [235]
- K.-Y. Kuo and C.-Y. Lai, “Comparison of 2D topological codes and their decoding performances”, 2022 IEEE International Symposium on Information Theory (ISIT) (2022) arXiv:2202.06612 DOI
- [236]
- K.-Y. Kuo and C.-Y. Lai, “Exploiting degeneracy in belief propagation decoding of quantum codes”, npj Quantum Information 8, (2022) arXiv:2104.13659 DOI
- [237]
- L. Berent et al., “Decoding quantum color codes with MaxSAT”, (2023) arXiv:2303.14237
- [238]
- M. B. Hastings, “Decoding in Hyperbolic Spaces: LDPC Codes With Linear Rate and Efficient Error Correction”, (2013) arXiv:1312.2546
- [239]
- A. O. Quintavalle and E. T. Campbell, “ReShape: a decoder for hypergraph product codes”, (2022) arXiv:2105.02370
- [240]
- N. Delfosse, M. E. Beverland, and M. A. Tremblay, “Bounds on stabilizer measurement circuits and obstructions to local implementations of quantum LDPC codes”, (2021) arXiv:2109.14599
- [241]
- N. Connolly et al., “Fast erasure decoder for hypergraph product codes”, (2024) arXiv:2208.01002
- [242]
- M. G. Luby et al., “Efficient erasure correcting codes”, IEEE Transactions on Information Theory 47, 569 (2001) DOI
- [243]
- A. G. Manes and J. Claes, “Distance-preserving stabilizer measurements in hypergraph product codes”, (2023) arXiv:2308.15520
- [244]
- A. Krishna, I. L. Navon, and M. Wootters, “Viderman’s algorithm for quantum LDPC codes”, (2023) arXiv:2310.07868
- [245]
- D. Bleichenbacher, A. Kiayias, and M. Yung, “Decoding interleaved Reed–Solomon codes over noisy channels”, Theoretical Computer Science 379, 348 (2007) DOI
- [246]
- D. Coppersmith and M. Sudan, “Reconstructing curves in three (and higher) dimensional space from noisy data”, Proceedings of the thirty-fifth annual ACM symposium on Theory of computing (2003) DOI
- [247]
- Jin, Hui, Aamod Khandekar, and Robert McEliece. "Irregular repeat-accumulate codes." Proc. 2nd Int. Symp. Turbo codes and related topics. 2000.
- [248]
- J. Justesen, “Class of constructive asymptotically good algebraic codes”, IEEE Transactions on Information Theory 18, 652 (1972) DOI
- [249]
- A. R. Hammons Jr. et al., “The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes”, (2002) arXiv:math/0207208
- [250]
- A. G. Fowler et al., “Surface codes: Towards practical large-scale quantum computation”, Physical Review A 86, (2012) arXiv:1208.0928 DOI
- [251]
- M. McEwen, D. Bacon, and C. Gidney, “Relaxing Hardware Requirements for Surface Code Circuits using Time-dynamics”, Quantum 7, 1172 (2023) arXiv:2302.02192 DOI
- [252]
- B. Coecke and R. Duncan, “Interacting Quantum Observables”, Automata, Languages and Programming 298 DOI
- [253]
- B. Coecke and R. Duncan, “Interacting quantum observables: categorical algebra and diagrammatics”, New Journal of Physics 13, 043016 (2011) arXiv:0906.4725 DOI
- [254]
- R. Chao et al., “Optimization of the surface code design for Majorana-based qubits”, Quantum 4, 352 (2020) arXiv:2007.00307 DOI
- [255]
- C. Gidney, “A Pair Measurement Surface Code on Pentagons”, Quantum 7, 1156 (2023) arXiv:2206.12780 DOI
- [256]
- L. Grans-Samuelsson et al., “Improved Pairwise Measurement-Based Surface Code”, Quantum 8, 1429 (2024) arXiv:2310.12981 DOI
- [257]
- Andrew Landahl, private communication, 2023
- [258]
- A. G. Fowler, “Minimum weight perfect matching of fault-tolerant topological quantum error correction in average \(O(1)\) parallel time”, (2014) arXiv:1307.1740
- [259]
- J. Edmonds, “Paths, Trees, and Flowers”, Canadian Journal of Mathematics 17, 449 (1965) DOI
- [260]
- 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
- [261]
- F. Barahona et al., “Morphology of ground states of two-dimensional frustration model”, Journal of Physics A: Mathematical and General 15, 673 (1982) DOI
- [262]
- A. Fischer and A. Miyake, “Hardness results for decoding the surface code with Pauli noise”, (2024) arXiv:2309.10331
- [263]
- S. Bravyi, M. Suchara, and A. Vargo, “Efficient algorithms for maximum likelihood decoding in the surface code”, Physical Review A 90, (2014) arXiv:1405.4883 DOI
- [264]
- N. Delfosse and N. H. Nickerson, “Almost-linear time decoding algorithm for topological codes”, Quantum 5, 595 (2021) arXiv:1709.06218 DOI
- [265]
- B. A. Galler and M. J. Fisher, “An improved equivalence algorithm”, Communications of the ACM 7, 301 (1964) DOI
- [266]
- J. E. Hopcroft and J. D. Ullman, “Set Merging Algorithms”, SIAM Journal on Computing 2, 294 (1973) DOI
- [267]
- R. E. Tarjan and J. van Leeuwen, “Worst-case Analysis of Set Union Algorithms”, Journal of the ACM 31, 245 (1984) DOI
- [268]
- C. A. Pattison et al., “Improved quantum error correction using soft information”, (2021) arXiv:2107.13589
- [269]
- O. Higgott et al., “Improved decoding of circuit noise and fragile boundaries of tailored surface codes”, (2023) arXiv:2203.04948
- [270]
- T. Chan and S. C. Benjamin, “Actis: A Strictly Local Union–Find Decoder”, Quantum 7, 1183 (2023) arXiv:2305.18534 DOI
- [271]
- A. G. Fowler, “Optimal complexity correction of correlated errors in the surface code”, (2013) arXiv:1310.0863
- [272]
- A. Paler and A. G. Fowler, “Pipelined correlated minimum weight perfect matching of the surface code”, Quantum 7, 1205 (2023) arXiv:2205.09828 DOI
- [273]
- X. Xu et al., “High-Threshold Code for Modular Hardware With Asymmetric Noise”, Physical Review Applied 12, (2019) arXiv:1812.01505 DOI
- [274]
- Y. Wu and L. Zhong, “Fusion Blossom: Fast MWPM Decoders for QEC”, (2023) arXiv:2305.08307
- [275]
- D. Forlivesi, L. Valentini, and M. Chiani, “Spanning Tree Matching Decoder for Quantum Surface Codes”, (2024) arXiv:2405.01151
- [276]
- K. Sahay et al., “Error correction of transversal controlled-NOT gates for scalable surface code computation”, (2024) arXiv:2408.01393
- [277]
- N. Shutty, M. Newman, and B. Villalonga, “Efficient near-optimal decoding of the surface code through ensembling”, (2024) arXiv:2401.12434
- [278]
- C. Jones, “Improved accuracy for decoding surface codes with matching synthesis”, (2024) arXiv:2408.12135
- [279]
- G. Duclos-Cianci and D. Poulin, “Fast Decoders for Topological Quantum Codes”, Physical Review Letters 104, (2010) arXiv:0911.0581 DOI
- [280]
- G. Duclos-Cianci and D. Poulin, “Fault-Tolerant Renormalization Group Decoder for Abelian Topological Codes”, (2013) arXiv:1304.6100
- [281]
- 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) arXiv:1411.3028 DOI
- [282]
- J. Wootton, “A Simple Decoder for Topological Codes”, Entropy 17, 1946 (2015) arXiv:1310.2393 DOI
- [283]
- A. Hutter, J. R. Wootton, and D. Loss, “Efficient Markov chain Monte Carlo algorithm for the surface code”, Physical Review A 89, (2014) arXiv:1302.2669 DOI
- [284]
- J. W. Harrington, Analysis of Quantum Error-Correcting Codes: Symplectic Lattice Codes and Toric Codes, California Institute of Technology, 2004 DOI
- [285]
- M. Herold et al., “Cellular-automaton decoders for topological quantum memories”, npj Quantum Information 1, (2015) arXiv:1406.2338 DOI
- [286]
- M. Herold et al., “Cellular automaton decoders of topological quantum memories in the fault tolerant setting”, New Journal of Physics 19, 063012 (2017) arXiv:1511.05579 DOI
- [287]
- C.-E. Bardyn and T. Karzig, “Exponential lifetime improvement in topological quantum memories”, Physical Review B 94, (2016) arXiv:1512.04528 DOI
- [288]
- G. Torlai and R. G. Melko, “Neural Decoder for Topological Codes”, Physical Review Letters 119, (2017) arXiv:1610.04238 DOI
- [289]
- C. Chamberland and P. Ronagh, “Deep neural decoders for near term fault-tolerant experiments”, Quantum Science and Technology 3, 044002 (2018) arXiv:1802.06441 DOI
- [290]
- Y. Ueno et al., “NEO-QEC: Neural Network Enhanced Online Superconducting Decoder for Surface Codes”, (2022) arXiv:2208.05758
- [291]
- R. Sweke et al., “Reinforcement learning decoders for fault-tolerant quantum computation”, Machine Learning: Science and Technology 2, 025005 (2020) arXiv:1810.07207 DOI
- [292]
- E. S. Matekole et al., “Decoding surface codes with deep reinforcement learning and probabilistic policy reuse”, (2022) arXiv:2212.11890
- [293]
- K. Meinerz, C.-Y. Park, and S. Trebst, “Scalable Neural Decoder for Topological Surface Codes”, Physical Review Letters 128, (2022) arXiv:2101.07285 DOI
- [294]
- J. Bausch et al., “Learning to Decode the Surface Code with a Recurrent, Transformer-Based Neural Network”, (2023) arXiv:2310.05900
- [295]
- H. Wang et al., “Transformer-QEC: Quantum Error Correction Code Decoding with Transferable Transformers”, (2023) arXiv:2311.16082
- [296]
- P. Das, A. Locharla, and C. Jones, “LILLIPUT: A Lightweight Low-Latency Lookup-Table Based Decoder for Near-term Quantum Error Correction”, (2021) arXiv:2108.06569
- [297]
- N. Delfosse, “Hierarchical decoding to reduce hardware requirements for quantum computing”, (2020) arXiv:2001.11427
- [298]
- S. C. Smith, B. J. Brown, and S. D. Bartlett, “Local Predecoder to Reduce the Bandwidth and Latency of Quantum Error Correction”, Physical Review Applied 19, (2023) arXiv:2208.04660 DOI
- [299]
- G. S. Ravi et al., “Better Than Worst-Case Decoding for Quantum Error Correction”, (2022) arXiv:2208.08547
- [300]
- X. Tan et al., “Scalable surface code decoders with parallelization in time”, (2022) arXiv:2209.09219
- [301]
- L. Skoric et al., “Parallel window decoding enables scalable fault tolerant quantum computation”, Nature Communications 14, (2023) arXiv:2209.08552 DOI
- [302]
- J. Old and M. Rispler, “Generalized Belief Propagation Algorithms for Decoding of Surface Codes”, Quantum 7, 1037 (2023) arXiv:2212.03214 DOI
- [303]
- J. S. Yedidia, W. T. Freeman, and Y. Weiss, Generalized belief propagation, in NIPS, Vol. 13 (2000) pp. 689–695.
- [304]
- A. Kaufmann and I. Arad, “A blockBP decoder for the surface code”, (2024) arXiv:2402.04834
- [305]
- J. Chen et al., “Improved Belief Propagation Decoding Algorithms for Surface Codes”, (2024) arXiv:2407.11523
- [306]
- H. D. Pfister et al., “Belief Propagation for Classical and Quantum Systems: Overview and Recent Results”, IEEE BITS the Information Theory Magazine 2, 20 (2022) DOI
- [307]
- J. du Crest, M. Mhalla, and V. Savin, “A blindness property of the Min-Sum decoding for the toric code”, (2024) arXiv:2406.14968
- [308]
- A. Benhemou et al., “Minimising surface-code failures using a color-code decoder”, (2024) arXiv:2306.16476
- [309]
- M. Pacenti et al., “Progressive-Proximity Bit-Flipping for Decoding Surface Codes”, (2024) arXiv:2402.15924
- [310]
- B. Barber et al., “A real-time, scalable, fast and highly resource efficient decoder for a quantum computer”, (2023) arXiv:2309.05558
- [311]
- U. Fincke and M. Pohst, “Improved methods for calculating vectors of short length in a lattice, including a complexity analysis”, Mathematics of Computation 44, 463 (1985) DOI
- [312]
- C. P. Schnorr and M. Euchner, “Lattice basis reduction: Improved practical algorithms and solving subset sum problems”, Mathematical Programming 66, 181 (1994) DOI
- [313]
- F. Pastawski and J. Preskill, “Error correction for encoded quantum annealing”, Physical Review A 93, (2016) arXiv:1511.00004 DOI
- [314]
- O. Damen, A. Chkeif, and J.-C. Belfiore, “Lattice code decoder for space-time codes”, IEEE Communications Letters 4, 161 (2000) DOI
- [315]
- B. Hassibi and H. Vikalo, “On the sphere-decoding algorithm I. Expected complexity”, IEEE Transactions on Signal Processing 53, 2806 (2005) DOI
- [316]
- E. Viterbo and J. Bouros, “A universal lattice code decoder for fading channels”, IEEE Transactions on Information Theory 45, 1639 (1999) DOI
- [317]
- I. Dumer, “Maximum likelihood decoding with reduced complexity”, Proceedings of IEEE International Symposium on Information Theory DOI
- [318]
- C. Hartmann and L. Rudolph, “An optimum symbol-by-symbol decoding rule for linear codes”, IEEE Transactions on Information Theory 22, 514 (1976) DOI
- [319]
- C. Peters, “Information-Set Decoding for Linear Codes over F q”, Post-Quantum Cryptography 81 (2010) DOI
- [320]
- J. Wolf, “Efficient maximum likelihood decoding of linear block codes using a trellis”, IEEE Transactions on Information Theory 24, 76 (1978) DOI
- [321]
- A. Rudra and M. Wootters, “Average-radius list-recovery of random linear codes: it really ties the room together”, (2017) arXiv:1704.02420
- [322]
- 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.
- [323]
- R. Pellikaan, “On decoding by error location and dependent sets of error positions”, Discrete Mathematics 106–107, 369 (1992) DOI
- [324]
- 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
- [325]
- D. Slepian, “Some Further Theory of Group Codes”, Bell System Technical Journal 39, 1219 (1960) DOI
- [326]
- Y. S. Han et al., “Maximum-likelihood Soft-decision Decoding for Binary Linear Block Codes Based on Their Supercodes”, (2014) arXiv:1408.1310
- [327]
- E. Nachmani, Y. Beery, and D. Burshtein, “Learning to Decode Linear Codes Using Deep Learning”, (2016) arXiv:1607.04793
- [328]
- Y. Choukroun and L. Wolf, “Error Correction Code Transformer”, (2022) arXiv:2203.14966
- [329]
- D. Chase, “Class of algorithms for decoding block codes with channel measurement information”, IEEE Transactions on Information Theory 18, 170 (1972) DOI
- [330]
- J. L. Massey, “Linear codes with complementary duals”, Discrete Mathematics 106–107, 337 (1992) DOI
- [331]
- S. Liu, F. Manganiello, and F. R. Kschischang, “Construction and decoding of generalized skew-evaluation codes”, 2015 IEEE 14th Canadian Workshop on Information Theory (CWIT) (2015) DOI
- [332]
- U. Martinez-Penas and F. R. Kschischang, “Reliable and Secure Multishot Network Coding Using Linearized Reed-Solomon Codes”, IEEE Transactions on Information Theory 65, 4785 (2019) arXiv:1805.03789 DOI
- [333]
- M. Park et al., “Enhancing Quantum Memory Lifetime with Measurement-Free Local Error Correction and Reinforcement Learning”, (2024) arXiv:2408.09524
- [334]
- R. Gallager, Low-density parity check codes. 1963. PhD thesis, MIT Cambridge, MA.
- [335]
- Pearl, J. (2022). Reverend Bayes on inference engines: A distributed hierarchical approach. In Probabilistic and Causal Inference: The Works of Judea Pearl (pp. 129-138).
- [336]
- Probabilistic Reasoning in Intelligent Systems (Elsevier, 1988) DOI
- [337]
- T. J. Richardson and R. L. Urbanke, “The capacity of low-density parity-check codes under message-passing decoding”, IEEE Transactions on Information Theory 47, 599 (2001) DOI
- [338]
- S. Lin and D. J. Costello, Error Control Coding, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 2004.
- [339]
- K. Shimizu et al., “A parallel LSI architecture for LDPC decoder improving message-passing schedule”, 2006 IEEE International Symposium on Circuits and Systems DOI
- [340]
- F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm”, IEEE Transactions on Information Theory 47, 498 (2001) DOI
- [341]
- J. Chen et al., “Reduced-Complexity Decoding of LDPC Codes”, IEEE Transactions on Communications 53, 1288 (2005) DOI
- [342]
- J. Feldman. Decoding Error-Correcting Codes via Linear Programming. PhD thesis, Massachusetts Institute of Technology, 2003.
- [343]
- J. Feldman, M. J. Wainwright, and D. R. Karger, “Using Linear Programming to Decode Binary Linear Codes”, IEEE Transactions on Information Theory 51, 954 (2005) DOI
- [344]
- J. Feldman, “LP Decoding”, Encyclopedia of Algorithms 1177 (2016) DOI
- [345]
- Changyan Di et al., “Finite-length analysis of low-density parity-check codes on the binary erasure channel”, IEEE Transactions on Information Theory 48, 1570 (2002) DOI
- [346]
- C. A. Kelley, "Codes over Graphs." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [347]
- M. Schwartz and A. Vardy, “On the stopping distance and the stopping redundancy of codes”, IEEE Transactions on Information Theory 52, 922 (2006) DOI
- [348]
- T. J. Richardson, M. A. Shokrollahi, and R. L. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes”, IEEE Transactions on Information Theory 47, 619 (2001) DOI
- [349]
- S. Habib, A. Beemer, and J. Kliewer, “RELDEC: Reinforcement Learning-Based Decoding of Moderate Length LDPC Codes”, (2023) arXiv:2112.13934
- [350]
- 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).
- [351]
- P. Gaborit et al., “RankSign: An Efficient Signature Algorithm Based on the Rank Metric”, Post-Quantum Cryptography 88 (2014) DOI
- [352]
- T. Richardson and R. Urbanke, Modern Coding Theory (Cambridge University Press, 2008) DOI
- [353]
- David J. C. MacKay. 2002. Information Theory, Inference & Learning Algorithms. Cambridge University Press, USA
- [354]
- J. Pearl, “Reverend Bayes on Inference Engines: A Distributed Hierarchical Approach”, Probabilistic and Causal Inference 129 (2022) DOI
- [355]
- D. J. C. MacKay and R. M. Neal, “Good codes based on very sparse matrices”, Cryptography and Coding 100 (1995) DOI
- [356]
- D. Aasen et al., “Milestones Toward Majorana-Based Quantum Computing”, Physical Review X 6, (2016) arXiv:1511.05153 DOI
- [357]
- S. Plugge et al., “Majorana box qubits”, New Journal of Physics 19, 012001 (2017) arXiv:1609.01697 DOI
- [358]
- T. Karzig et al., “Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes”, Physical Review B 95, (2017) arXiv:1610.05289 DOI
- [359]
- J. F. Steiner and F. von Oppen, “Readout of Majorana qubits”, Physical Review Research 2, (2020) arXiv:2004.02124 DOI
- [360]
- 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
- [361]
- A. Alahmadi et al., “On the lifted Melas code”, Cryptography and Communications 8, 7 (2015) DOI
- [362]
- E. Sabo, A. B. Aloshious, and K. R. Brown, “Trellis Decoding For Qudit Stabilizer Codes And Its Application To Qubit Topological Codes”, (2022) arXiv:2106.08251
- [363]
- H. Anwar et al., “Fast decoders for qudit topological codes”, New Journal of Physics 16, 063038 (2014) arXiv:1311.4895 DOI
- [364]
- M. J. Gullans and D. A. Huse, “Dynamical Purification Phase Transition Induced by Quantum Measurements”, Physical Review X 10, (2020) arXiv:1905.05195 DOI
- [365]
- S. Kopparty, Theory of Computing 11, 149 (2015) DOI
- [366]
- S. Bhandari et al., “Decoding multivariate multiplicity codes on product sets”, Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (2021) arXiv:2012.01530 DOI
- [367]
- Rasmus R. Nielsen. List decoding of linear block codes. PhD thesis, Technical University of Denmark, 2001
- [368]
- V. Guruswami and C. Wang, “Optimal rate list decoding via derivative codes”, (2011) arXiv:1106.3951
- [369]
- D. R. Chowdhury et al., “Design of CAECC - cellular automata based error correcting code”, IEEE Transactions on Computers 43, 759 (1994) DOI
- [370]
- J. Niset, U. L. Andersen, and N. J. Cerf, “Experimentally Feasible Quantum Erasure-Correcting Code for Continuous Variables”, Physical Review Letters 101, (2008) arXiv:0710.4858 DOI
- [371]
- T. Hillmann et al., “Performance of Teleportation-Based Error-Correction Circuits for Bosonic Codes with Noisy Measurements”, PRX Quantum 3, (2022) arXiv:2108.01009 DOI
- [372]
- Carl W. Helstrom. Quantum Detection and Estimation Theory. Elsevier, 1976.
- [373]
- A. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Edizioni della Normale, 2011) DOI
- [374]
- 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.
- [375]
- 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
- [376]
- 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
- [377]
- Y. Xu et al., “Qubit-Oscillator Concatenated Codes: Decoding Formalism and Code Comparison”, PRX Quantum 4, (2023) arXiv:2209.04573 DOI
- [378]
- Y. Ouyang and G. K. Brennen, “Finite-round quantum error correction on symmetric quantum sensors”, (2024) arXiv:2212.06285
- [379]
- X. Wang and P. Zanardi, “Simulation of many-body interactions by conditional geometric phases”, Physical Review A 65, (2002) arXiv:quant-ph/0111017 DOI
- [380]
- 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)
- [381]
- M. Takeoka et al., “Implementation of projective measurements with linear optics and continuous photon counting”, Physical Review A 71, (2005) arXiv:quant-ph/0410133 DOI
- [382]
- F. E. Becerra et al., “M-ary-state phase-shift-keying discrimination below the homodyne limit”, Physical Review A 84, (2011) DOI
- [383]
- 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) arXiv:0905.2496 DOI
- [384]
- S. Izumi et al., “Displacement receiver for phase-shift-keyed coherent states”, Physical Review A 86, (2012) arXiv:1208.1815 DOI
- [385]
- S. Izumi et al., “Quantum receivers with squeezing and photon-number-resolving detectors forM-ary coherent state discrimination”, Physical Review A 87, (2013) arXiv:1302.2691 DOI
- [386]
- 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) arXiv:1304.7316 DOI
- [387]
- 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
- [388]
- F. Pastawski et al., “Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence”, Journal of High Energy Physics 2015, (2015) arXiv:1503.06237 DOI
- [389]
- A. N. Skorobogatov and S. G. Vladut, “On the decoding of algebraic-geometric codes”, IEEE Transactions on Information Theory 36, 1051 (1990) DOI
- [390]
- V. Yu. Krachkovskii, "Decoding of codes on algebraic curves," (in Russian), Conference Odessa, 1988.
- [391]
- M. M. Wilde and S. Guha, “Polar Codes for Classical-Quantum Channels”, IEEE Transactions on Information Theory 59, 1175 (2013) arXiv:1109.2591 DOI
- [392]
- 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
- [393]
- I. Tal and A. Vardy, “List Decoding of Polar Codes”, IEEE Transactions on Information Theory 61, 2213 (2015) DOI
- [394]
- Y. Ren et al., “A Sequence Repetition Node-Based Successive Cancellation List Decoder for 5G Polar Codes: Algorithm and Implementation”, IEEE Transactions on Signal Processing 70, 5592 (2022) arXiv:2205.08857 DOI
- [395]
- 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
- [396]
- U. U. Fayyaz and J. R. Barry, “Polar codes for partial response channels”, 2013 IEEE International Conference on Communications (ICC) (2013) DOI
- [397]
- E. Arkan, “A performance comparison of polar codes and Reed-Muller codes”, IEEE Communications Letters 12, 447 (2008) DOI
- [398]
- S. Kasi et al., “Decoding Polar Codes via Noisy Quantum Gates: Quantum Circuits and Insights”, (2022) arXiv:2210.10854
- [399]
- J. I. Farran, “Decoding Algebraic Geometry codes by a key equation”, (1999) arXiv:math/9910151
- [400]
- L. Golowich and V. Guruswami, “Quantum Locally Recoverable Codes”, (2023) arXiv:2311.08653
- [401]
- H. Ollivier and J.-P. Tillich, “Description of a Quantum Convolutional Code”, Physical Review Letters 91, (2003) arXiv:quant-ph/0304189 DOI
- [402]
- A. Leverrier, J.-P. Tillich, and G. Zemor, “Quantum Expander Codes”, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (2015) arXiv:1504.00822 DOI
- [403]
- 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) arXiv:1808.03821 DOI
- [404]
- A. Grospellier. Constant time decoding of quantum expander codes and application to fault-tolerant quantum computation. PhD thesis, Inria Paris (2019).
- [405]
- N. Alon, J. Edmonds, and M. Luby, “Linear time erasure codes with nearly optimal recovery”, Proceedings of IEEE 36th Annual Foundations of Computer Science DOI
- [406]
- N. Alon and M. Luby, “A linear time erasure-resilient code with nearly optimal recovery”, IEEE Transactions on Information Theory 42, 1732 (1996) DOI
- [407]
- M. B. Hastings, “Quantum belief propagation: An algorithm for thermal quantum systems”, Physical Review B 76, (2007) arXiv:0706.4094 DOI
- [408]
- M. S. Leifer and D. Poulin, “Quantum Graphical Models and Belief Propagation”, Annals of Physics 323, 1899 (2008) arXiv:0708.1337 DOI
- [409]
- D. Poulin and Y. Chung, “On the iterative decoding of sparse quantum codes”, (2008) arXiv:0801.1241
- [410]
- N. Raveendran and B. Vasić, “Trapping Sets of Quantum LDPC Codes”, Quantum 5, 562 (2021) arXiv:2012.15297 DOI
- [411]
- J. Kim, H. Jung, and J. Ha, “Low-Complexity Decoding Algorithm Utilizing Degeneracy for Quantum LDPC Codes”, MILCOM 2023 - 2023 IEEE Military Communications Conference (MILCOM) (2023) DOI
- [412]
- T.-H. Huang and Y.-L. Ueng, “A Binary BP Decoding Using Posterior Adjustment for Quantum LDPC Codes”, ICASSP 2024 - 2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (2024) DOI
- [413]
- J. Roffe et al., “Decoding across the quantum low-density parity-check code landscape”, Physical Review Research 2, (2020) arXiv:2005.07016 DOI
- [414]
- Roffe, Joschka. "LDPC: Python tools for low density parity check codes." (2022).
- [415]
- Y.-H. Liu and D. Poulin, “Neural Belief-Propagation Decoders for Quantum Error-Correcting Codes”, Physical Review Letters 122, (2019) arXiv:1811.07835 DOI
- [416]
- S. Miao et al., “Neural Belief Propagation Decoding of Quantum LDPC Codes Using Overcomplete Check Matrices”, (2023) arXiv:2212.10245
- [417]
- M. Lange et al., “Data-driven decoding of quantum error correcting codes using graph neural networks”, (2023) arXiv:2307.01241
- [418]
- A. Gong, S. Cammerer, and J. M. Renes, “Graph Neural Networks for Enhanced Decoding of Quantum LDPC Codes”, (2023) arXiv:2310.17758
- [419]
- Z. Yi et al., “Improved belief propagation decoding algorithm based on decoupling representation of Pauli operators for quantum LDPC codes”, (2023) arXiv:2305.17505
- [420]
- J. du Crest, M. Mhalla, and V. Savin, “Stabilizer Inactivation for Message-Passing Decoding of Quantum LDPC Codes”, (2023) arXiv:2205.06125
- [421]
- T. Hillmann et al., “Localized statistics decoding: A parallel decoding algorithm for quantum low-density parity-check codes”, (2024) arXiv:2406.18655
- [422]
- S. Javed et al., “Low-Complexity Linear Programming Based Decoding of Quantum LDPC codes”, (2024) arXiv:2311.18488
- [423]
- H. Yao et al., “Belief Propagation Decoding of Quantum LDPC Codes with Guided Decimation”, (2024) arXiv:2312.10950
- [424]
- S. Wolanski and B. Barber, “Ambiguity Clustering: an accurate and efficient decoder for qLDPC codes”, (2024) arXiv:2406.14527
- [425]
- Z. Babar et al., “Fifteen Years of Quantum LDPC Coding and Improved Decoding Strategies”, IEEE Access 3, 2492 (2015) DOI
- [426]
- M. A. Tremblay, N. Delfosse, and M. E. Beverland, “Constant-Overhead Quantum Error Correction with Thin Planar Connectivity”, Physical Review Letters 129, (2022) arXiv:2109.14609 DOI
- [427]
- N. Raveendran et al., “Soft Syndrome Decoding of Quantum LDPC Codes for Joint Correction of Data and Syndrome Errors”, (2022) arXiv:2205.02341
- [428]
- B. J. Brown, “Conservation Laws and Quantum Error Correction: Toward a Generalized Matching Decoder”, IEEE BITS the Information Theory Magazine 2, 5 (2022) arXiv:2207.06428 DOI
- [429]
- L. Berent, L. Burgholzer, and R. Wille, “Software Tools for Decoding Quantum Low-Density Parity-Check Codes”, Proceedings of the 28th Asia and South Pacific Design Automation Conference (2023) arXiv:2209.01180 DOI
- [430]
- M. C. Löbl et al., “Breadth-first graph traversal union-find decoder”, (2024) arXiv:2407.15988
- [431]
- S. Huang and S. Puri, “Improved Noisy Syndrome Decoding of Quantum LDPC Codes with Sliding Window”, (2023) arXiv:2311.03307
- [432]
- A. deMarti iOlius and J. E. Martinez, “The closed-branch decoder for quantum LDPC codes”, (2024) arXiv:2402.01532
- [433]
- A. Gong, S. Cammerer, and J. M. Renes, “Toward Low-latency Iterative Decoding of QLDPC Codes Under Circuit-Level Noise”, (2024) arXiv:2403.18901
- [434]
- D. Gottesman, “Fault-Tolerant Quantum Computation with Constant Overhead”, (2014) arXiv:1310.2984
- [435]
- Y. Shi, A. Patil, and S. Guha, “Stabilizer Entanglement Distillation and Efficient Fault-Tolerant Encoder”, (2024) arXiv:2408.06299
- [436]
- A. deMarti iOlius et al., “An almost-linear time decoding algorithm for quantum LDPC codes under circuit-level noise”, (2024) arXiv:2409.01440
- [437]
- S. Muralidharan et al., “Ultrafast and Fault-Tolerant Quantum Communication across Long Distances”, Physical Review Letters 112, (2014) arXiv:1310.5291 DOI
- [438]
- A. Gong and J. M. Renes, “Improved Logical Error Rate via List Decoding of Quantum Polar Codes”, (2023) arXiv:2304.04743
- [439]
- A. L. Toom, “Nonergodic Multidimensional System of Automata”, Probl. Peredachi Inf., 10:3 (1974), 70–79; Problems Inform. Transmission, 10:3 (1974), 239–246
- [440]
- L. F. Gray, “Toom’s Stability Theorem in Continuous Time”, Perplexing Problems in Probability 331 (1999) DOI
- [441]
- P. Gács, Journal of Statistical Physics 103, 45 (2001) DOI
- [442]
- I. Convy et al., “Machine learning for continuous quantum error correction on superconducting qubits”, New Journal of Physics 24, 063019 (2022) arXiv:2110.10378 DOI
- [443]
- S. P. Jain et al., “Quantum spherical codes”, Nature Physics (2024) arXiv:2302.11593 DOI
- [444]
- D. Poulin, J.-P. Tillich, and H. Ollivier, “Quantum serial turbo-codes”, (2009) arXiv:0712.2888
- [445]
- M. M. Wilde, M.-H. Hsieh, and Z. Babar, “Entanglement-Assisted Quantum Turbo Codes”, IEEE Transactions on Information Theory 60, 1203 (2014) arXiv:1010.1256 DOI
- [446]
- S. Bravyi et al., “Adaptive constant-depth circuits for manipulating non-abelian anyons”, (2022) arXiv:2205.01933
- [447]
- T. Inada et al., “Measurement-Free Ultrafast Quantum Error Correction by Using Multi-Controlled Gates in Higher-Dimensional State Space”, (2021) arXiv:2109.00086
- [448]
- G. A. Paz-Silva, G. K. Brennen, and J. Twamley, “Fault Tolerance with Noisy and Slow Measurements and Preparation”, Physical Review Letters 105, (2010) arXiv:1002.1536 DOI
- [449]
- S. Heußen, D. F. Locher, and M. Müller, “Measurement-free fault-tolerant quantum error correction in near-term devices”, (2023) arXiv:2307.13296
- [450]
- K. Fujii, “Quantum Computation with Topological Codes: from qubit to topological fault-tolerance”, (2015) arXiv:1504.01444
- [451]
- Y. Choukroun and L. Wolf, “Deep Quantum Error Correction”, (2023) arXiv:2301.11930
- [452]
- S. Brandsen, A. Mandal, and H. D. Pfister, “Belief Propagation with Quantum Messages for Symmetric Classical-Quantum Channels”, (2022) arXiv:2207.04984
- [453]
- N. Rengaswamy et al., “Belief propagation with quantum messages for quantum-enhanced classical communications”, npj Quantum Information 7, (2021) arXiv:2003.04356 DOI
- [454]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [455]
- C. Gidney, “Stim: a fast stabilizer circuit simulator”, Quantum 5, 497 (2021) arXiv:2103.02202 DOI
- [456]
- G. Zhang and Y. Li, “Time-efficient logical operations on quantum LDPC codes”, (2024) arXiv:2408.01339
- [457]
- M.-H. Hsieh and F. Le Gall, “NP-hardness of decoding quantum error-correction codes”, Physical Review A 83, (2011) arXiv:1009.1319 DOI
- [458]
- 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.
- [459]
- P. Iyer and D. Poulin, “Hardness of decoding quantum stabilizer codes”, (2013) arXiv:1310.3235
- [460]
- B. M. Terhal, “Quantum error correction for quantum memories”, Reviews of Modern Physics 87, 307 (2015) arXiv:1302.3428 DOI
- [461]
- N. Delfosse et al., “Splitting decoders for correcting hypergraph faults”, (2023) arXiv:2309.15354
- [462]
- H. Ollivier and J.-P. Tillich, “Trellises for stabilizer codes: Definition and uses”, Physical Review A 74, (2006) arXiv:quant-ph/0512041 DOI
- [463]
- D. Cruz, F. A. Monteiro, and B. C. Coutinho, “Quantum Error Correction Via Noise Guessing Decoding”, IEEE Access 11, 119446 (2023) arXiv:2208.02744 DOI
- [464]
- S. Krastanov and L. Jiang, “Deep Neural Network Probabilistic Decoder for Stabilizer Codes”, Scientific Reports 7, (2017) arXiv:1705.09334 DOI
- [465]
- R. J. Harris et al., “Decoding holographic codes with an integer optimization decoder”, Physical Review A 102, (2020) arXiv:2008.10206 DOI
- [466]
- O. Shtanko et al., “Bounds on Autonomous Quantum Error Correction”, (2023) arXiv:2308.16233
- [467]
- K. Shiraishi, H. Yamasaki, and M. Murao, “Efficient decoding of stabilizer code by single-qubit local operations and classical communication”, (2023) arXiv:2308.14054
- [468]
- M. Cain et al., “Correlated decoding of logical algorithms with transversal gates”, (2024) arXiv:2403.03272
- [469]
- O. Higgott and C. Gidney, “Sparse Blossom: correcting a million errors per core second with minimum-weight matching”, (2023) arXiv:2303.15933
- [470]
- D. J. Bernstein, T. Lange, and C. Peters, “Smaller Decoding Exponents: Ball-Collision Decoding”, Advances in Cryptology – CRYPTO 2011 743 (2011) DOI
- [471]
- 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
- [472]
- M. Finiasz and N. Sendrier, “Security Bounds for the Design of Code-Based Cryptosystems”, Advances in Cryptology – ASIACRYPT 2009 88 (2009) DOI
- [473]
- P. Loidreau, “A Welch–Berlekamp Like Algorithm for Decoding Gabidulin Codes”, Coding and Cryptography 36 (2006) DOI
- [474]
- 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
- [475]
- F. Lazaro, G. Liva, and G. Bauch, “Inactivation Decoding of LT and Raptor Codes: Analysis and Code Design”, IEEE Transactions on Communications 1 (2017) arXiv:1706.05814 DOI
- [476]
- 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
- [477]
- L. Rudolph and C. Hartmann, “Decoding by sequential code reduction”, IEEE Transactions on Information Theory 19, 549 (1973) DOI
- [478]
- G. Seroussi and A. Lempel, “Factorization of Symmetric Matrices and Trace-Orthogonal Bases in Finite Fields”, SIAM Journal on Computing 9, 758 (1980) DOI
- [479]
- E. T. Campbell and M. Howard, “Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost”, Physical Review A 95, (2017) arXiv:1606.01904 DOI
- [480]
- V. Guruswami and A. Vardy, “Maximum-likelihood decoding of Reed-Solomon Codes is NP-hard”, (2004) arXiv:cs/0405005
- [481]
- E. R. Berlekamp and L. Welch, Error Correction of Algebraic Block Codes. U.S. Patent, Number 4,633,470 1986.
- [482]
- P. Gemmell and M. Sudan, “Highly resilient correctors for polynomials”, Information Processing Letters 43, 169 (1992) DOI
- [483]
- S. Gao, “A New Algorithm for Decoding Reed-Solomon Codes”, Communications, Information and Network Security 55 (2003) DOI
- [484]
- 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
- [485]
- M. Sudan, “Decoding of Reed Solomon Codes beyond the Error-Correction Bound”, Journal of Complexity 13, 180 (1997) DOI
- [486]
- 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
- [487]
- V. Guruswami and A. Rudra, “Limits to List Decoding Reed–Solomon Codes”, IEEE Transactions on Information Theory 52, 3642 (2006) DOI
- [488]
- J. Brakensiek, S. Gopi, and V. Makam, “Generic Reed-Solomon Codes Achieve List-decoding Capacity”, (2024) arXiv:2206.05256
- [489]
- A. Vardy and Y. Be’ery, “Bit-level soft-decision decoding of Reed-Solomon codes”, IEEE Transactions on Communications 39, 440 (1991) DOI
- [490]
- D. V. Sarwate and N. R. Shanbhag, “High-speed architectures for Reed-Solomon decoders”, IEEE Transactions on Very Large Scale Integration (VLSI) Systems 9, 641 (2001) DOI
- [491]
- S. B. Wicker and V. K. Bhargava, Reed-Solomon Codes and Their Applications (IEEE, 1999) DOI
- [492]
- G. Zemor, “On expander codes”, IEEE Transactions on Information Theory 47, 835 (2001) DOI
- [493]
- A. Barg and G. Zemor, “Concatenated Codes: Serial and Parallel”, IEEE Transactions on Information Theory 51, 1625 (2005) DOI
- [494]
- Y. Tomita and K. M. Svore, “Low-distance surface codes under realistic quantum noise”, Physical Review A 90, (2014) arXiv:1404.3747 DOI
- [495]
- C. Chamberland et al., “Techniques for combining fast local decoders with global decoders under circuit-level noise”, Quantum Science and Technology 8, 045011 (2023) arXiv:2208.01178 DOI
- [496]
- K. H. Wan et al., “An iterative transversal CNOT decoder”, (2024) arXiv:2407.20976
- [497]
- R. Silverman and M. Balser, “Coding for constant-data-rate systems”, Transactions of the IRE Professional Group on Information Theory 4, 50 (1954) DOI
- [498]
- A. Lapidoth, A Foundation in Digital Communication (Cambridge University Press, 2017) DOI
- [499]
- N. Delfosse and A. Paetznick, “Spacetime codes of Clifford circuits”, (2023) arXiv:2304.05943
- [500]
- S. Omanakuttan et al., “Fault-Tolerant Quantum Computation Using Large Spin-Cat Codes”, PRX Quantum 5, (2024) arXiv:2401.04271 DOI
- [501]
- A. L. Grimsmo and S. Puri, “Quantum Error Correction with the Gottesman-Kitaev-Preskill Code”, PRX Quantum 2, (2021) arXiv:2106.12989 DOI
- [502]
- B. W. Walshe et al., “Continuous-variable gate teleportation and bosonic-code error correction”, Physical Review A 102, (2020) arXiv:2008.12791 DOI
- [503]
- V. V. Sivak et al., “Real-time quantum error correction beyond break-even”, Nature 616, 50 (2023) arXiv:2211.09116 DOI
- [504]
- M. Puviani et al., “Boosting the Gottesman-Kitaev-Preskill quantum error correction with non-Markovian feedback”, (2023) arXiv:2312.07391
- [505]
- A. J. Landahl, J. T. Anderson, and P. R. Rice, “Fault-tolerant quantum computing with color codes”, (2011) arXiv:1108.5738
- [506]
- D. S. Wang et al., “Graphical algorithms and threshold error rates for the 2d colour code”, (2009) arXiv:0907.1708
- [507]
- A. J. Landahl and C. Ryan-Anderson, “Quantum computing by color-code lattice surgery”, (2014) arXiv:1407.5103
- [508]
- Duclos-Cianci, Guillaume, Héctor Bombın, and David Poulin. "Fast decoding algorithm for subspace and subsystem color codes and local equivalence of topological phases." Personal communication (2011).
- [509]
- D. Beckman et al., “Measurability of Wilson loop operators”, Physical Review D 65, (2002) arXiv:hep-th/0110205 DOI
- [510]
- N. E. Bonesteel and D. P. DiVincenzo, “Quantum circuits for measuring Levin-Wen operators”, Physical Review B 86, (2012) arXiv:1206.6048 DOI
- [511]
- M. L. Liu, N. Tantivasadakarn, and V. V. Albert, “Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat's Lemma”, Quantum 8, 1403 (2024) arXiv:2311.18003 DOI
- [512]
- B. J. Brown, N. H. Nickerson, and D. E. Browne, “Fault-tolerant error correction with the gauge color code”, Nature Communications 7, (2016) arXiv:1503.08217 DOI
- [513]
- H. M. Solanki and P. Kiran Sarvepalli, “Correcting Erasures with Topological Subsystem Color Codes”, 2020 IEEE Information Theory Workshop (ITW) (2021) DOI
- [514]
- H. M. Solanki and P. K. Sarvepalli, “Decoding Topological Subsystem Color Codes Over the Erasure Channel using Gauge Fixing”, (2022) arXiv:2111.14594
- [515]
- F. G. Jeronimo et al., “Unique Decoding of Explicit \(ε\)-balanced Codes Near the Gilbert-Varshamov Bound”, (2020) arXiv:2011.05500
- [516]
- I. Tamo and A. Barg, “A Family of Optimal Locally Recoverable Codes”, IEEE Transactions on Information Theory 60, 4661 (2014) arXiv:1311.3284 DOI
- [517]
- N. Wiberg, H. Loeliger, and R. Kotter, “Codes and iterative decoding on general graphs”, European Transactions on Telecommunications 6, 513 (1995) DOI
- [518]
- Niclas Wiberg, Codes and decoding on general graphs. 1996. PhD thesis, Linköping University, Linköping, Sweden
- [519]
- Brendan J. Frey. Graphical models for machine learning and digital communication. MIT press, 1998.
- [520]
- Yongyi Mao and A. H. Banihashemi, “Decoding low-density parity-check codes with probabilistic schedule”, 2001 IEEE Pacific Rim Conference on Communications, Computers and Signal Processing (IEEE Cat. No.01CH37233) DOI
- [521]
- G. D. Forney, “Codes on graphs: normal realizations”, 2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060) DOI
- [522]
- E. Soljanin and E. Offer, “LDPC codes: a group algebra formulation”, Electronic Notes in Discrete Mathematics 6, 148 (2001) DOI
- [523]
- T. Etzion, A. Trachtenberg, and A. Vardy, “Which codes have cycle-free Tanner graphs?”, IEEE Transactions on Information Theory 45, 2173 (1999) DOI
- [524]
- R. Tanner, “A recursive approach to low complexity codes”, IEEE Transactions on Information Theory 27, 533 (1981) DOI
- [525]
- D. K. Kythe and P. K. Kythe, “Algebraic and Stochastic Coding Theory”, (2017) DOI
- [526]
- T. Farrelly et al., “Tensor-Network Codes”, Physical Review Letters 127, (2021) arXiv:2009.10329 DOI
- [527]
- T. Farrelly et al., “Parallel decoding of multiple logical qubits in tensor-network codes”, (2020) arXiv:2012.07317
- [528]
- A. J. Ferris and D. Poulin, “Tensor Networks and Quantum Error Correction”, Physical Review Letters 113, (2014) arXiv:1312.4578 DOI
- [529]
- P. Chaichanavong and P. H. Siegel, “Tensor-product parity code for magnetic recording”, IEEE Transactions on Magnetics 42, 350 (2006) DOI
- [530]
- C. Torezzan, S. I. R. Costa, and V. A. Vaishampayan, “Spherical codes on torus layers”, 2009 IEEE International Symposium on Information Theory (2009) DOI
- [531]
- T. J. Yoder and I. H. Kim, “The surface code with a twist”, Quantum 1, 2 (2017) arXiv:1612.04795 DOI
- [532]
- C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon limit error-correcting coding and decoding: Turbo-codes. 1”, Proceedings of ICC ’93 - IEEE International Conference on Communications DOI
- [533]
- R. J. McEliece, D. J. C. MacKay, and Jung-Fu Cheng, “Turbo decoding as an instance of Pearl’s “belief propagation” algorithm”, IEEE Journal on Selected Areas in Communications 16, 140 (1998) DOI
- [534]
- H. R. Sadjadpour, “<title>Maximum a posteriori decoding algorithms for turbo codes</title>”, SPIE Proceedings (2000) DOI
- [535]
- J. D. Kene and K. D. Kulat, “Soft Output Decoding Algorithm for Turbo Codes Implementation in Mobile Wi-Max Environment”, Procedia Technology 6, 666 (2012) DOI
- [536]
- B. Sklar, “A primer on turbo code concepts”, IEEE Communications Magazine 35, 94 (1997) DOI
- [537]
- K. R. Narayanan and G. L. Stuber, “List decoding of turbo codes”, IEEE Transactions on Communications 46, 754 (1998) DOI
- [538]
- G. Masera et al., “VLSI architectures for turbo codes”, IEEE Transactions on Very Large Scale Integration (VLSI) Systems 7, 369 (1999) DOI
- [539]
- Q. Xu et al., “Tailored XZZX codes for biased noise”, (2022) arXiv:2203.16486
- [540]
- J. Hastrup and U. L. Andersen, “All-optical cat-code quantum error correction”, (2021) arXiv:2108.12225
- [541]
- V. I. Levenshtein, Binary codes capable of correcting deletions, insertions and reversals (translated to English), Soviet Physics Dokl., 10(8), 707-710 (1966).
- [542]
- 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).
- [543]
- 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
- [544]
- Z. Li et al., “Autonomous error correction of a single logical qubit using two transmons”, (2023) arXiv:2302.06707
- [545]
- Z. Zhang, D. Aasen, and S. Vijay, “The X-Cube Floquet Code”, (2022) arXiv:2211.05784
- [546]
- J. F. S. Miguel, D. J. Williamson, and B. J. Brown, “A cellular automaton decoder for a noise-bias tailored color code”, Quantum 7, 940 (2023) arXiv:2203.16534 DOI
- [547]
- J. C. M. de la Fuente et al., “The XYZ ruby code: Making a case for a three-colored graphical calculus for quantum error correction in spacetime”, (2024) arXiv:2407.08566
- [548]
- K. Hammar et al., “Error-rate-agnostic decoding of topological stabilizer codes”, Physical Review A 105, (2022) arXiv:2112.01977 DOI
- [549]
- D. K. Tuckett et al., “Fault-Tolerant Thresholds for the Surface Code in Excess of 5% Under Biased Noise”, Physical Review Letters 124, (2020) arXiv:1907.02554 DOI
- [550]
- C. Gidney et al., “Yoked surface codes”, (2023) arXiv:2312.04522
- [551]
- 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
- [552]
- S. M. Dodunekov and J. E. M. Nilsson, “Algebraic decoding of the Zetterberg codes”, IEEE Transactions on Information Theory 38, 1570 (1992) DOI
- [553]
- M.-H. Jing et al., “A Result on Zetterberg Codes”, IEEE Communications Letters 14, 662 (2010) DOI
- [554]
- E.C. Posner, Combinatorial Structures in Planetary Reconnaissance in Error Correcting Codes, ed. H.B. Mann, Wiley, NY 1968.
- [555]
- R. R. Green, "A serial orthogonal decoder," JPL Space Programs Summary, vol. 37–39-IV, pp. 247–253, 1966.
- [556]
- 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
- [557]
- 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
- [558]
- M. P. da Silva et al., “Demonstration of logical qubits and repeated error correction with better-than-physical error rates”, (2024) arXiv:2404.02280
- [559]
- A. M. Steane, “Fast fault-tolerant filtering of quantum codewords”, (2004) arXiv:quant-ph/0202036
- [560]
- H. Yamasaki and M. Koashi, “Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation”, Nature Physics 20, 247 (2024) arXiv:2207.08826 DOI
- [561]
- S. L. Braunstein, “Quantum error correction for communication with linear optics”, Nature 394, 47 (1998) DOI
- [562]
- H. Goto, “Many-hypercube codes: High-rate quantum error-correcting codes for high-performance fault-tolerant quantum computing”, (2024) arXiv:2403.16054
- [563]
- Y. S. Weinstein, “Fidelity of an encoded [7,1,3] logical zero”, Physical Review A 84, (2011) arXiv:1101.1950 DOI
- [564]
- S. D. Buchbinder, C. L. Huang, and Y. S. Weinstein, “Encoding an Arbitrary State in a [7,1,3] Quantum Error Correction Code”, (2011) arXiv:1109.1714
- [565]
- M. Nakahara, “Quantum Computing”, (2008) DOI
- [566]
- W. Fish et al., “Partial permutation decoding for simplex codes”, Advances in Mathematics of Communications 6, 505 (2012) DOI
- [567]
- J. D. Key and P. Seneviratne, “Partial permutation decoding for MacDonald codes”, Applicable Algebra in Engineering, Communication and Computing 27, 399 (2016) DOI