| Approximate quantum error-correcting code (AQECC) |
Given an encoding, a decoder that yields the optimal entanglement fidelity can be obtained by solving a semi-definite program [1][2] (see also Ref. [3]).The Petz recovery map (a.k.a. the transpose map) [4][5], a quantum channel determined by the codespace and noise channel, recovers information perfectly for strictly correctable noise and yields an infidelity of recovery that is at most twice away from the infidelity of the best possible recovery [6]. The infidelity of a modified Petz recovery map can be bounded using relative entropies between uncorrupted and corrupted code states on countably infinite Hilbert spaces [7]. |
| Approximate secret-sharing code |
Decoding is analagous to reconstruction in a secret sharing scheme and is done in polynomial time. The only required operations are verification of quantum authentication, which is a pair of polynomial-time quantum algorithms that check if the fidelity of the received state is close to 1, and erasure correction for a stabilizer code, which involves solving a system of linear equations. |
| Bacon-Shor code |
While check operators are few-body, stabilizer weights scale with the number of qubits, and stabilizer expectation values are obtained by taking products of gauge-operator expectation values. It is thus not clear how to extract stabilizer values in a fault-tolerant manner [8][9]. |
| Balanced product code |
BP-OSD decoder [10]. |
| Binary Golay code |
Table lookup or algebraic algorithms such as Berlekamp-Welch [11].Both Golay codes have a trellis representation and can thus be decoded using trellis decoding [12][13]. |
| Binary code |
For few-bit codes (\(n\) is small), decoding can be based on a lookup table. For infinite code families, the size of such a table scales exponentially with \(n\), so approximate decoding algorithms scaling polynomially with \(n\) have to be used. The decoder determining the most likely error given a noise channel is called the maximum-likelihood decoder.Given a received string \(x\) and an error bound \(e\), a list decoder returns a list of all codewords that are at most \(e\) from \(x\) in Hamming distance. The number of codewords in a neighborhood of \(x\) has to be polynomial in \(n\) in order for this decoder to run in time polynomial in \(n\). |
| Binary quantum Goppa code |
Farran algorithm [14]. |
| Binary 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. |
| 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 [15]. 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 [16][15]. |
| 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 [17]. |
| Braunstein five-mode code |
Error correction can be done using linear-optical elements and feedback [18]. |
| Calderbank-Shor-Steane (CSS) stabilizer code |
Coherent decoders allow for measurement-free error correction [19]. One method is table/multi-control decoding [20], which scales exponentially with the number of ancillas used in syndrome measurement. Another method, the Ising-based decoder, utilizes the mapping of the effect of the noise to a statistical mechanical model [21][22] such that the decoding problem maps to preparation of the ground state of an Ising model. |
| Cat code |
Measurement in the Fock basis. For a \(2(S+1)\) cat code, a number measurement returns outcome \(2(S+1)k\), if \(k\) is even, then it corresponds to logical 0 state; if \(k\) is odd, then it corresponds to logical 1 state. |
| Chuang-Leung-Yamamoto 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. |
| Codeword stabilized (CWS) code |
There is no known efficient algorithm to decode non-additive (non-stabilizer) CWS codes. |
| Color code |
Projection decoder [23].Matching decoder gives low logical failure rate [24].Integer-program-based decoder [25].Restriction decoder [26].Cellular-automaton decoder for the \(XYZ\) color code [27]. |
| Convolutional code |
Decoders based on the Viterbi algorithm (trellis decoding) were developed first, which result in the most-likely codeword for the encoded bits [28]. Following, other trellis decoders such as the BCJR decoding algorithm [29] were developed later. |
| Dinur-Hsieh-Lin-Vidick (DHLV) code |
Linear-time decoder utilizing the small set-flip decoder [30] for \(Z\) errors and a reconstruction procedure for \(X\) errors [31]. |
| Eigenstate thermalization hypothesis (ETH) code |
An explicit universal recovery channel for the ETH code is given in [32]. |
| Expander code |
Decoding can be done in \(O(n)\) runtime using a greedy algorithm. The algorithm consists of flipping a bit of the received word if it will result in a greater number of satisfied parity checks. This is repeated until a codeword is reached. |
| Expander lifted-product code |
Linear-time decoder [33]. |
| Fiber-bundle code |
Greedy algorithm can be used to efficiently decode \(X\) errors, but no known efficient decoding of \(Z\) errors yet [8]. |
| Fibonacci string-net code |
Clustering decoder (provides best known threshold for this code) [34].Fusion aware iterative minimum-weight perfect matching decoder. Note that ordinary MWPM decoders do not produce a threshold with this code [34]. |
| Finite-dimensional quantum error-correcting code |
The operation \(\cal{D}\) in the definition of this code is called the decoder. However, the term decoder can sometimes be used for the inverse of an encoder, which does not correct errors.Quantum machine-learning based decoders such as quantum convolutional neural networks [35] and quantum autoencoders [36]. |
| Folded RS code |
Guruswami–Rudra achieved list decoding capacity of radius \(1-R-\epsilon\) [37].Folded RS codes, concatenated with suitable inner codes, can be efficiently list-decoded up to the Zyablov bound [38]. |
| Fountain code |
Invert the fragment generator matrix resulting from the continuous encoding process. If exactly \(K\) packets are received, then the probability of decoding correctly is \(0.289\). Extra packets increase this probability exponentially. The decoding runtime is dominated by the matrix inversion step, which takes order \(O(n^3)\) time. |
| Frobenius code |
Adapted from the Berlekamp decoding algorithm for classical BCH codes. There exists a polynomial time quantum algorithm to correct errors of weight at most \(\tau\), where \(\delta=2\tau+1\) is the BCH distance of the code [39]. |
| Gabidulin code |
Fast decoder based on a transform-domain approach [40]. |
| Galois-field \(q\)-ary code |
For small \(n\), decoding can be based on a lookup table. For infinite code families, the size of such a table scales exponentially with \(n\), so approximate decoding algorithms scaling polynomially with \(n\) have to be used. The decoder determining the most likely error given a noise channel is called the maximum-likelihood decoder.Given a received string \(x\) and an error bound \(e\), a list decoder returns a list of all codewords that are at most \(e\) from \(x\). The number of codewords in a neighborhood of \(x\) has to be polynomial in \(n\) in order for this decoder to run in time polynomial in \(n\). |
| Galois-qudit code |
For few-qudit codes (\(n\) is small), decoding can be based on a lookup table. For infinite code families, the size of such a table scales exponentially with \(n\), so approximate decoding algorithms scaling polynomially with \(n\) have to be used. The decoder determining the most likely error given a noise channel is called the maximum-likelihood decoder. |
| Galois-qudit non-stabilizer code |
The decoding circuit involves the application of phase estimation. |
| Goppa code |
Algebraic decoding algorithms [41].If \( \text{deg} G(z) = 2t \) , then there exists a \(t\)-correcting algebraic decoding algorithm for \( \Gamma(L,G) \). |
| Gottesman-Kitaev-Preskill (GKP) code |
Syndrome measruement of displacement error 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 [42].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 [42]. |
| 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. |
| 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. [43] calls the QND Gate and which is defined as \(QND_c | x , y \rangle = |x + c y, y \rangle\). |
| Homological product code |
Union-find [44]. |
| Honeycomb code |
The ISG has a static subgroup for all time steps \(r\geq 3\) – that is, a subgroup which remains a subgroup of the ISG for all future times – given by so-called plaquette stabilizers. These are stabilizers consisting of products of check operators around homologically trivial paths. The syndrome bits correspond to the eigenvalues of the plaquette stabilizers. Because of the structure of the check operators, only one-third of all plaquettes are measured each round. The syndrome bits must therefore be represented by a lattice in spacetime, to reflect when and where the outcome was obtained. |
| Hypergraph product code |
ReShape decoder that uses minimum weight decoders for the classical codes used in the hypergraph construction [45]. |
| Justesen code |
Generalized minimum distance decoding [46]. |
| Kitaev surface code |
Maximum-likelihood (ML) [21].Minimum weight perfect-matching (MWPM) [21][47] (based on work by Edmonds on finding a matching in a graph [48][49]) and pipeline MWPM [50][51], a modification accounting for correlations between events.Renormalization group (RG) [52][53][54].Markov-chain Monte Carlo [55].Tensor network [56].Cellular automaton [57].Machine learning [58][59][60].Union-find [61]. |
| Luby transform (LT) code |
Sum-product algorithm, often called a peeling decoder [62][63], similar to belief propagation [64]. |
| Modular-qudit code |
For few-qudit codes (\(n\) is small), decoding can be based on a lookup table. For infinite code families, the size of such a table scales exponentially with \(n\), so approximate decoding algorithms scaling polynomially with \(n\) have to be used. The decoder determining the most likely error given a noise channel is called the maximum-likelihood decoder. |
| Modular-qudit stabilizer code |
The structure of stabilizer codes allows for syndrome-based decoding, where errors are corrected based on the results of stabilizer measurements (syndromes).Trellis decoder for prime-dimensional qudits, which builds a compact representation of the algebraic structure of the normalizer \(\mathsf{N(S)}\) [65]. |
| 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) [66] |
| Multi-mode GKP code |
The MLD decoder for Gaussian displacement errors is realized by evaluating a lattice theta function, and in general the decision can be approximated by either solving (approximating) the closest vector problem (CVP) or by using other effective iterative schemes when e.g. the lattice represents a concatenated GKP code [67][68][69][70]. |
| Niset-Andersen-Cerf code |
Optical decoder using three beam splitters, electronic gain detectors, and two phase-insensitive amplifiers as described in Ref. [71]. |
| Number-phase code |
Measurement done in the logical \(X\), or dual, basis has zero uncertainty in the case of ideal number phase codes. Approximate number-phase codes are characterized by vanishing phase uncertainty.Number measurement can be done by using the CROT gate \(\mathrm{e}^{(2\pi \mathrm{i} / NM) \hat n \otimes \hat n}\). See Section 4.B.1 of Ref. [72]. |
| Orthogonal Spacetime Block Code (OSTBC) |
Maximum-likelihood decoding can be achieved with only linear processing [73]. |
| Pastawski-Yoshida-Harlow-Preskill (HaPPY) code |
Greedy algorithm for decoding specified in Ref. [74]. |
| Permutation-invariant code |
For a family of codes, using projection, probability amplitude rebalancing, and gate teleportation can be done in \(O(N^2)\) [75]. |
| Polar code |
Successive cancellation decoder [76].Successive cancellation list (SCL) decoder [77] and a modification utilizing sequence repetition (SR-List) [78]. |
| Quantum Lego code |
The decoder is created by creating a decoding quantum circuit with dangling legs replaced with input/output wires, and tensors converted to unitary gates. Maximum likelihood decoding can be used when the tensors are stabilizer codes.Tensor-network decoder when the tensor network is contractible via stabilizer isometries [79].Tensor-network-based decoder when the encoding unitary is known [80]. |
| Quantum Tanner code |
Linear-time decoder similar to the small-set-flip decoder for quantum expander codes [30].Linear-time decoder [33]. |
| Quantum cyclic code |
Adapted from the Berlekamp decoding algorithm for classical BCH codes [81]. |
| Quantum expander code |
Small set-flip linear-time decoder, which corrects \(\Omega(n^{1/2})\) adversarial errors [82]. |
| Quantum low-density parity-check (QLDPC) code |
Non-binary decoding algorithm for CSS-type QLDPC codes [83].BP-OSD decoder adds a post-processing step based on ordered statistics decoding (OSD) to the belief propogation (BP) decoder [10].Soft-syndrome iterative belief propagation for CSS-type QLDPC codes, utilizing the entire continuous signal obtained in the physical implementation of the stabilizer measurement (as opposed to discretizing the signal into a syndrome bit) [84].Message-passing decoder utilizing stabilizer inactivation (MP-SI) for CSS-type QLDPC qubit codes [85]. |
| 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 [86]. |
| Qubit code |
For few-qubit 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. |
| Qubit stabilizer code |
The structure of stabilizer codes allows for syndrome-based decoding, where errors are corrected based on the results of stabilizer measurements (syndromes). Finding an optimal decoder is \(\#P\)-hard [87].Trellis decoder, which builds a compact representation of the algebraic structure of the normalizer \(\mathsf{N(S)}\) [88]. |
| 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 [1][2] (see also Ref. [3]). 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 [15]. |
| Ramanujan-complex product code |
For 2D simplicial complexes, cycle code decoder admitting a polynomial-time decoding algorithm can be used [89]. |
| Random code |
Ball-collision decoding [90].Finiasz and Sendrier (FS-ISD) decoding [91]. |
| Rank-metric code |
Polynomial-reconstruction Berlekamp-Welch based decoder [92].Berlekamp-Massey based decoder [93]. |
| Raptor (RAPid TORnado) code |
Raptor codes can be decoded using inactivation decoding [94], a combination of belief-propogation and Gaussian elimination decoding. |
| Reed-Muller (RM) code |
Reed decoder with \(r+1\)-step majority decoding corrects \(\frac{1}{2}(2^{m-r}-1)\) errors [95] (see also Ch. 13 of Ref. [96]).Sequential code-reduction decoding [97].First-order (\(r=1\)) RM codes admit specialized decoders [98]. |
| Reed-Solomon (RS) code |
Berlekamp-Massey decoder [99][100].Gorenstein-Peterson-Zierler decoder [101].Berlekamp-Welch decoder [11], assuming that \(t \geq (n+k)/2\).Gao decoder using extended Euclidean algorithm [102].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 evaulation of \(Q(\alpha_i,y_i)\) and homogenous linear equation(interpolation). Once this is done, list roots of \(y\) that agree at \(\geq t\) points. The Sudan list decoding algorithm corrects up to \(1-\sqrt{2R}\) proportion of errors [103]. The Sudan–Guruswami algorithm improves that to \(1-\sqrt{R}\) [104]. |
| Single parity-check 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. |
| Skew-cyclic code |
Only given for skew-BCH codes, adapted froom standard BCH codes. |
| String-net code |
Syndrome measurement circuits analyzed in Ref. [105]. |
| Subsystem qubit stabilizer code |
Syndrome measurements are obtained by first measuring gauge operators of the code and taking their products, which give the stabilizer measurement outcomes. |
| Surface-17 code |
Lookup table [106]. |
| Tanner code |
Parallel decoding algorithm corrects a fraction \(\delta_0^2/48\) of errors for Tanner codes [107]. A modification of said algorithm improves the fraction to \(\delta_0^2/4\) with no extra cost to complexity [108]. |
| Three qutrit code |
The quantum information (the secret) can be recovered from a unitary transformation acting on only two qutrits, \( U_{ij} \otimes I \), where \(U_{ij}\) acts on qutrits \(i,j\) and \(I\) is the identity on the remaining qutrit. By the cyclic structure of the codewords, this unitary transformation performs a permutation that recovers the information and stores it in one of the two qutrits involved in recovery. |
| Tornado code |
Linear-time decoder. |
| Translationally-invariant stabilizer code |
Clustering decoder [109][110]. |
| Two-dimensional hyperbolic surface code |
Due to the symmetries of hyperbolic surface codes, optimal measurement schedules of the stabilizers can be found [111]. |
| 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 \(\sigma_x (a+a^\dagger)\) [112]. This results in an \(X\)-dependent shift of the readout cavity resonance which can be measured. |
| Wasilewski-Banaszek code |
Destructive measurement with photon number measurements on each mode. |
| XYZ\(^2\) hexagonal stabilizer code |
Maximum-likelihood decoding using the EWD decoder [113]. |
| XZZX surface code |
Minimum-weight perfect matching decoder, which can be used for \(X\) and \(Z\) noise. For \(Y\) noise, a variant of the matching decoder could be used like it is used for the XY code in Ref. [114]. |
| Zetterberg code |
Kallquist first described an algebraic decoding theorem [115]. A faster version was later provided in Ref. [116] and further modified in Ref. [117]. |
| \([[5,1,3]]\) perfect code |
Combined dynamical decoupling and error correction protocol on individually-controlled qubits with always-on Ising couplings [118].Symmetric decoder correcting all weight-one Pauli errors. The resulting logical error channel after coherent noise has been explicitly derived [119]. |