Here is a list of all classical and quantum codes that have been realized in experimental or real-world devices.
| Name | Realization(s) |
|---|---|
| 120-cell code | Improved proofs of the Bell-Kochen-Specker (BKS) theorem [1]. |
| 600-cell code | Improved proofs of the Bell-Kochen-Specker (BKS) theorem [2]. |
| Alamouti code | Wireless standards since: 3G, LTE, LTE-Advanced, and 5G.Wireless communication: IEEE 802.11n, IEEE 802.11ad, IEEE 802.11ay, etc. |
| Analog stabilizer code | One-sided device-independent QKD [3]. |
| Array-based LDPC (AB-LDPC) code | Certain AB-LDPC codes have been proposed to be used for DSL transmission [4]. |
| BPSK c-q code | Linear-optical quantum receiver [5].Homodyne receiver [6].Kennedy receiver [6,7].Photon-number resolving detector [8].Communication over dephasing [9], time-varying phase-noise [10], and thermal-noise [11] channels.Adaptive decoder using displacements and photon detection [12]. |
| Bacon-Shor code | Trapped-ion qubits: state preparation, logical measurement, and stabilizer measurement for nine-qubit Bacon-Shor code demonstrated on a 13-qubit device by M. Cetina and C. Monroe groups [13]. |
| Balanced code | Balanced length-eight code, known as a 6b/8b encoding, used for balancing direct current in a communications system [14] |
| Binary BCH code | Satellite communication [15] |
| Binary PSK (BPSK) code | Telephone-line modems throughout 1950s and 1960s: Bell 103 and 202, as well as international standards V.21 [16] and V.23 [17]. |
| Binomial code | Microwave cavities coupled to superconducting circuits: state transfer between a binomial codeword to another system [18], error-correction protocol nearly reaching break-even [19], and a teleported CNOT gate [20]. A realization of the "0-2-4" encoding is the first to go beyond break-even error-correction and yields a logical lifetime that exceeds the cavity lifetime by \(16\%\) [21] (see also [22]). |
| Bose–Chaudhuri–Hocquenghem (BCH) code | DVDs, disk drives, and two-dimensional bar codes [23]. |
| Bosonic c-q code | Coherent-state c-q encodings are used for continuous-variable quantum key distribution (CV-QKD) [24–26]. |
| Cat code | Parity-syndrome measurement tested [27] and implemented for the four-component (\(S=1\)) cat code [28] in a microwave cavity coupled to a superconducting circuit. The latter work [28] is the first to reach break-even error-correction, where the lifetime of a logical qubit is on par with the cavity lifetime, despite protection against dephasing not being implemented. A fault-tolerant version of parity measurement has also been realized [29]. |
| Classical Goppa code | Initial version of the McEliece public-key cryptosystem [30,31] and its variation by Niederreiter [32] where the generator matrix is replaced by the parity check matrix. Some of these were proven to be insecure since the public key exposes algebraic structure of code [33]. |
| Coherent FSK (CFSK) c-q code | Time-resolving quantum receiver [34].Bondurant receiver [35].Bayesian inference [36]. |
| Constant-weight code | Radio-network frequency hopping [37]. |
| Convolutional code | A type of convolutional code used in Real-time Application networks [38].Mobile and radio communications (3G networks) use convolutional codes concatenated with Reed-Solomon codes to obtain suitable performance [39].A convolutional code with rate 1/2 was used for deep-space and satellite communication [40] |
| Covering code | Data compression both with or without compression [41].Football-pool problem: finding the smallest number of bets on a set of matches needed to guarantee at least one bet has at most \(\rho\) errors [42,43]. |
| Cycle LDPC code | Cycle LDPC codes have been proposed to be used for MIMO channels [44]. |
| Dihedral \(G=D_m\) quantum-double code | Signatures of a phase equivalent to the \(G=D_4\) quantum double detected in a 27-qubit trapped-ion device by Quantinuum [45]. Preparation of ground states and braiding of anyons has also been performed. The phase was realized as a gauged \(G=\mathbb{Z}_3^2\) twisted quantum double [46], which is the same topological order as the \(G=D_4\) quantum double [47,48]. |
| Dual-rail quantum code | The dual-rail code is ubiquitous in linear optical quantum devices [49]. |
| Fibonacci string-net code | NMR: Implementation of braiding-based Hamamard gate on two qubits [50]. |
| Five-qubit perfect code | NMR: Implementation of perfect error correcting code on 5 spin subsystem of labeled crotonic acid for quantum network benchmarking [51]. Single-qubit logical gates [52].Superconducting qubits [53].Trapped-ion qubits: non-transversal CNOT gate between two logical qubits, including rounds of correction and fault-tolerant primitives such as flag qubits and pieceable fault tolerance, on a 12-qubit device by Quantinuum [54].Nitrogen-vacancy centers in diamond: fault-tolerant single-qubit Clifford operations [55]. |
| Floquet color code | Plaquette stabilizer measurement realized on the IBM Falcon superconducting-qubit device [56] |
| Fountain code | Designed for servers sending data to many recipients, such as during broadcasting or file distribution [57,58]. |
| Gabidulin code | Public-key cryptosystems [59,60].Digital watermarking. The Gabidulin code provides efficient correction against luminance tampering and image-slicing distortion due to the consistency of the rank against alterations such as column swapping [61]. |
| Generalized RS (GRS) code | Commonly used in mass storage systems such as CDs, DVDs, QR codes etc.Various cloud storage systems [62].Public-key cryptosystems generalizing those that used Goppa codes [30–32], some of which were proven to be insecure [33]. More recent works focus on methods to mask the algebraic structure using subcodes of GRS codes [59]. For example, a key-recovery attack was developed in Ref. [63] for a variant of masking method proposed in Ref. [64]. |
| Golay code | Extended Golay code used in the Voyager 1 and 2 spacecraft, transmitting hundreds of color pictures of Jupiter and Saturn in their 1979, 1980, and 1981 fly-bys [65].Extended Golay code used in American military standards for automatic link establishment in high frequency radio systems [66].Proofs of the quantum mechanical Kochen-Specker theorem [67]. |
| Gold code | Used in for synchronization purposes in telecommunication [68]GPS C/A for satellite navigation [69]. |
| Gray code | Three-dimensional imaging [70].Broadcasting and communication [71]. |
| Group GKP code | Cryptographic applications stemming from the monogamy of entanglement of group GKP code and error words [3]. |
| Hamming code | Commonly used when error rates are very low, for example, computer RAM or integrated circuits [72].Hamming-code based matrix embedding used in steganography [73,74]. |
| Heavy-hexagon code | Logical state preparation and flag-qubit error correction realized in superconducting-circuit devices (specifically, fixed-frequency transmon qubit architectures) by IBM for \(d=2\) [75,76] and \(d=3\) [77]. |
| Hessian polyhedron code | Quantum mechanical SIC-POVMs [78]. |
| Hexagonal GKP code | Microwave cavity coupled to superconducting circuits: reduced form of GKP error correction, where displacement error syndromes are measured to one bit of precision using an ancillary transmon [79]. |
| Honeycomb Floquet code | Plaquette stabilizer measurement realized on the IBM Falcon superconducting-qubit device [56] |
| Interleaved RS (IRS) code | The cross-interleaved RS (CIRC), an IRS code using two shortened RS codes and two forms of interleaving, was used for compact discs (CDs) [80] (see Ref. [81], Sec. 5.6 and Ref. [82], Ch. 4). |
| Irregular LDPC code | Satellite communication after concatenating with a modulation scheme [83]. |
| Irregular repeat-accumulate (IRA) code | LDPC codes are used for digital satellite video broadcasting per the DVB-S2 standard [84,85] utilize IRA code features and are subject to ongoing litigation; see Ref. [86].Apple and Broadcom Wi-Fi devices utilize IRA encoding and decoding code features and are subject to ongoing litigation; see Ref. [86]. |
| Justesen code | Generating small-bias sample spaces, i.e., probability distributions that parity functions cannot typically distinguish from the uniform distribution [87]. |
| Kitaev chain code | Photonic systems: braiding of defects has been simulated in a device that has a different notion of locality than a bona-fide fermionic system [88].Superconducting circuits: initialization [89], braiding [90] and detection [90,91] of defects has been simulated in devices that have a different notion of locality than a bona-fide fermionic system. |
| Kitaev surface code | One cycle of syndrome readout on 19-qubit planar and 24-qubit toric codes realized in two-dimensional Rydberg atomic arrays [92]. Signatures of corresponding topological phase of matter detected in superconducting circuits [93] and two-dimensional Rydberg atomic arrays [94]. Ground state of the toric code has been implemented with and without twists, and the non-Abelian braiding behavior of the twists, which realize Ising anyons, has been demonstrated [95]. |
| Lloyd-Slotine nine-mode code | Optical network by the Furusawa group [96]. |
| Locally recoverable code (LRC) | An \((18,14,7)\) LRC code has beed used in the Windows Azure cloud storage system [97]; see also [98; 31.3.1.2]. |
| Low-rank parity-check (LRPC) code | Cryptosystem [99] that is a rank-metric analogue of NTRU [100] and MDPC [101] cryptosystems.Post-quantum cryptography [102]. |
| Matching code | Braiding of defects has been demonstrated for a five-qubit version of code [103]. |
| Maximum distance separable (MDS) code | The McEliece Public Key Cryptosystem [104] uses algebraic coding theory to secure communications against eavesdropping attack, in which private keys are generator matrices of linear codes, i.e., \(G\). Public Keys shared to outside world are scrambled and permutated versions of \(G\), i.e., \(G^\prime=SGP\). Data to be encrypted, \(u\), is multiplied by public key and added intentional errors \(e\), i.e., \(x=uG^\prime+e\). Upon receiving encrypted data, private key owner can apply inverse permutation \(P^{-1}\) to \(x\), decode the scrambled message given the presence of \(e\) errors, and finally unscramble to obtain \(u\). Security parameters of the system are \(n\) and \(e\), hence MDS codes, such as GRS codes may provide the same security level for shorter key sizes, and the private key owner can decode arguably fast enough using known decoding algorithms.Automatic repeat request (ARQ) data transmission protocols ([82], Ch. 7). |
| Maximum-rank distance (MRD) code | Useful for error and erasure correction in network coding [105,106]. |
| Monitored random-circuit code | Measurement induced quantum phases have been realized in a trapped-ion processor [107]. |
| NTRU-GKP code | Public-key quantum communication protocol [108]. |
| Niset-Andersen-Cerf code | Realized in Ref. [109] in an optical system with 3 beam-splitters. The fidelity peaked around \(0.6\) for deterministic approach, and around \(0.77\) for the probabilistic approach (with a 25% chance of error). |
| On-off keyed (OOK) c-q code | Proof-of-principle experiments using Dolinar [110] and TES receivers [111]. |
| One-hot code | The bi-quinary code, a combination of one-hot 1-in-2 and 1-in-5 one-hot codes to encode decimal digits, was used in several early computers ([112], Ch. 27).Marking the state of a finite automaton [113].Used in machine-learning based classification tasks because one-hot encodings, as opposed to integer encodings, do not presume an order [114]. |
| PPM c-q code | Conditional pulse nulling (CPN) receiver [115]. |
| PSK c-q code | Unambiguous state discrimination using displacement-based receiver for 4-PSK [116].Multi-stage quantum receivers [117–120].Bayesian inference [121].Time resolving quantum receiver opertaing in the telecom C band [122].Displacements and photon detection [123].Adaptive decoder using linear-optical elements and photon detection [12]. |
| Pair-cat code | Microwave cavities coupled to superconducting circuits by the Wang group [124]. |
| Phase-shift keying (PSK) code | Telephone-line modems: 1967 Milgo 4400/48 and international standard V.27 used 8-PSK [125]. |
| Polar code | Code control channels for the 5G NR (New Radio) interfaces [126]. |
| Pulse-position modulation (PPM) code | Greek hydraulic semaphore system [127,128].Telegraph time-division multiplexing.Radio-control, fiber-optic communications, and deep-space communications. |
| Quadrature PSK (QPSK) code | Japanese and North American digital cellular and personal systems [129].Telephone-line modems: 1962 Bell 201 and international standard V.24 [130]. |
| Quadrature-amplitude modulation (QAM) code | Optical communication (e.g., Ref. [131]).Telephone-line modems: 1971 Codex 9600C and international standard V.29 used 16-QAM [132]. |
| Quantum Tanner code | Used to obtain explicit lower bounds in the sum-of-squares game [133].States that, on average, achieve small violations of check operators for quantum Tanner codes require a circuit of non-constant depth to make. They are used in the proof [134] of the No low-energy trivial states (NLTS) conjecture [135]. |
| Quantum lattice code | Quantum lattice states are featured in the proof of hardness of LWE [136; pg. 12]. |
| Quantum parity code (QPC) | The \([[m^2,1,m]]\) codes for \(m\leq 7\) have been realized in trapped-ion quantum devices [137]. |
| Quantum repetition code | NMR: 3-qubit phase-flip code [138,139], with up to two rounds of error correction in liquid-state NMR [140].Superconducting circuits: 3-qubit phase-flip and bit-flip code by Schoelkopf group [141]; 3-qubit bit-flip code [142]; 3-qubit phase-flip code up to 3 cycles of error correction [143]; IBM 15-qubit device [144]; IBM Rochester device using 43-qubit code [145]; Google system performing up to 8 error-correction cycles on 5 and 9 qubits [146]; Google Quantum AI Sycamore utilizing up to 11 physical qubits and running 50 correction rounds [147]; Google Quantum AI Sycamore utilizing up to 25 qubits for comparison of logical error scaling with a quantum code [148] (see also [149]).Continuous error correction protocols have been implemented on a 3-qubit superconducting qubit device [150].Semiconductor spin-qubit devices: 3-qubit devices at RIKEN [151] and Delft [152].Nitrogen-vacancy centers in diamond: 3-qubit phase-flip code [153,154] (see also Ref. [155]).Trapped-ion device: 3-qubit phase-flip algorithm implemented in 3 cycles on high fidelity gate operations [156]. Both phase- and bit-flip codes for 31 qubits and their stabilizer measurements have been realized by Quantinuum [157]. |
| Quasi-cyclic LDPC (QC-LDPC) code | 5G NR cellular communication for the traffic channel [158,159].Wireless communication: WiMAX (IEEE 802.16e) [160–162], WiFi 4 (IEEE 802.11n) [163], and WPAN (IEEE 802.15.3c) [164]. |
| Qubit CSS code | Fully homomorphic encryption [165].Entanglement purification protocols related to quantum key distribution (QKD) [166].Cryptographic applications stemming from the monogamy of entanglement of CSS code and error words [167]. |
| Qubit c-q code | Quantum enhancement was demonstrated using a polarization-based non-error-correcting c-q encodings [168]. |
| Random code | Distributed storage systems [169].Classical and quantum cryptography based on the learning-with-errors problem, which is related to decoding a random linear code [136]. |
| Rank-metric code | Identity-Based Encryption [170].Digital watermarking [171].Network coding and streaming media broadcasting [172]. |
| Rank-modulation Gray code (RMGC) | Electronic devices where charges can either increase in an individual cell or decrease in a block of adjacent cells, e.g., flash memories [173]. |
| Raptor (RAPid TORnado) code | Two versions of Raptor codes have been standardized by IETF: R10 and the more recent RaptorQ. RaptorQ is used in mobile multimedia broadcasts as specified in ETSI technical specifications. It is also used in the mobile Next Gen TV standard.Raptor codes are useful in scenarios where erasure (i.e. weak signal or noisy channel) is common, such as in military or disaster scenarios. |
| Reed-Muller (RM) code | Deep-space communication [174,175]. For example, the \((32, 6, 16)\) RM code was used for the Mariner 9 spacecraft. |
| Reed-Solomon (RS) code | RS Product Code (RSPC) was used in DVDs (see Ref. [82], Ch. 4).DSL technologies and their variants against impluse noise [176].Cryptographic primitives based on the hardness of decoding RS codes for more than \(1-\sqrt{k/n}+\epsilon\) errors. This is equivalent to the polynomial reconstruction problem [177].RS codes as outer codes concatenated with convolutional codes are used indirectly in space exploration programs such as Voyager and Galileo. RS codes were part of a temetry channel coding standard issued by the Consultative Committee for Space Data Systems (see Ref. [82], Ch. 3).The ubiquity of RS codes has yielded off-the-shelf VLSI intergrated-circuit decoding hardware [178] (see also Ref. [82], Ch. 5 and 10).Automatic repeat request (ARQ) data transmission protocols (see Ref. [82], Ch. 7).Slow-frequency-hop spread-spectrum transmission (see Ref. [82], Chs. 8-9).Coded sharding designs in blockchains to increase efficiency [179].Private Information Retrieval [180].Used in QR-Codes to retrieve damaged barcodes [181].Wireless communication systems such as 3G, DVB, and WiMAX [182].Correcting pooled testing results for SARS-CoV-2 [183]. |
| Regular binary Tanner code | First hardware implementation was done using a semi-systolic decoding architecture [184]. |
| Repetition code | Repetition codes, in conjunction with other codes, were used in magnetic disks [185].Communication protocol such as FlexRay [186] is using repetition code' |
| Residue AG code | Improvements over the McEliece public-key cryptosystem to linear AG codes on curves of arbitrary genus [187]. Only the subfield subcode proposal remains resilient to attacks [188; Sec. 15.7.5.3].Algebraic secret-sharing schemes [189]. |
| Single parity-check (SPC) code | Can be realized on almost every communication device. SPCs are some of the earliest error-correcting codes ([112], Ch. 27). |
| Skew-cyclic code | Not directly implemented, but BCH codes form a subclass, and are used in DVD, solid state drive storage, etc. |
| Spacetime block code (STBC) | High data-rate wireless communication, e.g., WiMAX (IEEE 802.16m) [190–192]. |
| Square-lattice GKP code | Motional degree of freedom of a trapped ion: square-lattice GKP encoding realized with the help of post-selection [193,194], followed by realization of reduced form of GKP error correction, where displacement error syndromes are measured to one bit of precision using an ion electronic state [195].Microwave cavity coupled to superconducting circuits: reduced form of square-lattice GKP error correction, where displacement error syndromes are measured to one bit of precision using an ancillary transmon [79]. Subsequent paper [22] uses reinforcement learning for error-correction cycle design and is the first to go beyond break-even error-correction, with the lifetime of a logical qubit exceeding the cavity lifetime by about a factor of two (see also [21]).Single-qubit \(Z\)-gate has been demonstrated in the single-photon subspace of an infinite-mode space [196], in which time and frequency become bosonic conjugate variables of a single effective bosonic mode.In signal processing, GKP state position-state wavefunctions are related to Dirac combs [197]. |
| Surface-17 code | Implemented at ETH Zurich by the Wallraff group [198] and on the Zuchongzhi 2.1 superconducting quantum processor [199]. Both experimental error rates are above the pseudothreshold for this code relative to a single qubit. See Physics viewpoint for a summary [200]. |
| Tensor-product code | Construction can be used in magnetic recording by taking the tensor product of a Reed-Solomon code and a parity-check code [201]. |
| Ternary Golay code | Code used in football pools with at least one good bet [43,202]. In fact, the code was originally constructed by Juhani Virtakallio and published in the Finnish football pool magazine Veikkaaja [43,203].Proofs of the quantum mechanical Kochen-Specker theorem [67]. |
| Two-component cat code | Lindbladian-based [204,205] and Hamiltonian-based 'Kerr-cat' [206] encodings have been achieved in superconducting circuit devices by the Devoret group; Ref. [205] also demonstrated a displacement-based gate. The Lindbladian-based scheme has further achieved a suppression of bit-flip errors that is exponential in the average photon number [207] up to a bit-flip time of 1ms. A bit-flip time of 1s has been achieved in a similar system in the classical bit regime [208].T4C code realized in a superconducting circuit device by the Wang group [209]. |
| Very small logical qubit (VSLQ) code | Star-code autonomous correction scheme realized using superconducting circuits [210]. |
| Weight-two code | Two-in-five, also known as the two-out-of-five code, was used in the United States Postal Service's POSTNET barcode system as well as the Postal Alpha-numeric Encoding Technique (PLANET).Two-in-five code forms the numerical part of the Code 39 barcode encoding.Two-in-five code was used on early IBM computers [203,211]. |
| XZZX surface code | Distance-five 25-qubit code implemented on a superconducting quantum processor by Google Quantum AI [148]. This code outperformed the average of several instances of the smaller distance-three 9-qubit \(XZZX\) variant of the surface-17 code realized on the same device, both in terms of logical error probability over 25 cycles and in terms of logical error per cycle. This increase in error-correcting capabilities while using more physical qubits supports the notion of an error threshold. Braiding of defects has been demonstrated for the distance-five code [212]. Leakage errors have been handled in a separate work in a distance-three code [149]. |
| Zero-pi qubit code | A related superconducting circuit has been realized by the Houck group [213]. |
| Zetterberg code | Code used to provide better protection of data transmission with its double error correcting capacity [214]. |
| \(A_2\) hexagonal lattice code | Wireless communication [215,216]. |
| \([[2m,2m-2,2]]\) error-detecting code | Trapped-ion devices: 12-qubit device by Quantinuum [217]. |
| \([[4,2,2]]\) CSS code | \([[4,1,2]]\) subcode implemented using four-qubit graph state of photons [218]. Trapped-ion device by IonQ [219].Logical state preparation and flag-qubit error correction realized in superconducting-circuit devices by IBM [75,76].The subcode \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [220] and \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) [147], treated as a planar surface code, has been realized in superconducting-circuit devices.Logical gates between two copies of the subcode \(\{|\overline{10}\rangle,|\overline{11}\rangle\}\), interpreted as lattice surgery between planar surface codes, realized in superconducting circuits [221].Logical gates for the \(\{|\overline{00}\rangle,|\overline{11}\rangle\}\) subcode, treated as a planar code, realized in superconducting circuits [222]. |
| \([[7,1,3]]\) Steane code | Trapped-ion qubits: seven-qubit device in Blatt group [223], ten-qubit QCCD device by Quantinuum [224] (see APS Physics Synopsys [225]). Fault-tolerant universal two-qubit gate set by Monz group [226]. Logical CNOT gate between two logical qubits, including rounds of correction and fault-tolerant primitives such as flag qubits and pieceable fault tolerance, on a 20-qubit device by Quantinuum [54]; logical fidelity interval of the combined preparation-CNOT-measurement procedure was higher than that of the unencoded physical qubits.Rydberg atom arrays: Lukin group [92]. |
| \([[9,1,3]]\) Shor code | Trapped-ion qubits: state preparation with 98.8(1)% and 98.5(1)% fidelity for state \(|\overline{0}\rangle\) and \(|\overline{1}\rangle\), respectively, by N. Linke group [137]. Variants of the code to handle coherent noise studied and realized by K. Brown and C. Monroe groups [227].Optical systems: quantum teleportation of information implemented by J.-W. Pan group on maximally entangled pair of one physical and one logical qubit with fidelity rate of up to 78.6% [228]. All-photonic quantum repeater architecture tested on the same code [229]. |