Here is a list of quantum codes that have been realized in devices and/or that have practical real-world relevance.

Name | Realization(s) |
---|---|

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 [1]. |

Analog-cluster-state code | Analog cluster states on a number of modes ranging from tens to millions [2–4] have been synthesized in photonic degrees of freedom.Required primitives for Gaussian gates have been realized [5]. |

Array-based LDPC (AB-LDPC) code | Certain AB-LDPC codes have been proposed to be used for DSL transmission [6]. |

BPSK c-q code | Linear-optical quantum receiver [7].Homodyne receiver [8].Kennedy receiver [8,9].Photon-number resolving detector [10].Communication over dephasing [11], time-varying phase-noise [12], and thermal-noise [13] channels.Adaptive decoder using displacements and photon detection [14].BPQM detector on a BPSK-modulated tree code [15]. |

Bacon-Shor code | Trapped-ion qubits: state preparation, logical measurement, and syndrome extraction (deferred to the end) for nine-qubit Bacon-Shor code demonstrated on a 13-qubit device by M. Cetina and C. Monroe groups [16]. |

Balanced code | Balanced length-eight code, known as a 6b/8b encoding, used for balancing direct current in a communications system [17] |

Binary BCH code | Satellite communication [18] |

Binary PSK (BPSK) code | Telephone-line modems throughout 1950s and 1960s: Bell 103 and 202, as well as international standards V.21 [19] and V.23 [20]. |

Binomial code | Microwave cavities coupled to superconducting circuits: state transfer between a binomial codeword to another system [21], error-correction protocol nearly reaching break-even [22], and a teleported CNOT gate [23]. 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 \(16\%\) [24] (see also [25]). See Ref. [26] for another experiment.Motional degree of freedom of a trapped ion: binomial state preparation for \(S=2\) realized by Tan group [27]. |

Bose–Chaudhuri–Hocquenghem (BCH) code | DVDs, disk drives, and two-dimensional bar codes [28]. |

Cat code | Parity-syndrome measurement tested [29] and implemented for the four-component (\(S=1\)) cat code [30] in a microwave cavity coupled to a superconducting circuit. The latter work [30] 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 [31]. |

Cluster-state code | Quantum compututation with cluster states has been realized in the polarizations of photons [32,33]. |

Coherent FSK (CFSK) c-q code | Time-resolving quantum receiver [34].Bondurant receiver [35].Bayesian inference [36]. |

Coherent-state c-q code | Continuous-variable quantum key distribution (CV-QKD) [37–39]. |

Constant-weight code | Radio-network frequency hopping [40]. |

Convolutional code | A type of convolutional code used in Real-time Application networks [41].Mobile and radio communications (3G networks) use convolutional codes concatenated with RS codes to obtain suitable performance [42].A convolutional code with rate 1/2 was used for deep-space and satellite communication [43] |

Covering code | Data compression both with or without compression [44].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 [45,46]. |

Cross-interleaved RS (CIRS) code | Compact discs (CDs); see [48][47; Sec. 5.6][49; Ch. 4]. |

Cycle LDPC code | Cycle LDPC codes have been proposed to be used for MIMO channels [50]. |

Cyclic redundancy check (CRC) code | CRC-16 and CRC-32 are used in data transmission, e.g., IEEE 802.16e, IEEE 802.3 [51] and TCP/IP communication [52; Sec. 2.3.3]. |

Delsarte-Goethals (DG) code | Space-time signaling [53].Compressed neighbor discovery in a network [54]. |

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 [55]. 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 [56], which is the same topological order as the \(G=D_4\) quantum double [57,58]. |

Dual-rail quantum code | The dual-rail code is ubiquitous in linear-optical quantum devices and is behind the KLM protocol, one of the first proposals for fault-tolerant computation. See reviews [59–61] for more details.Superconducting circuit devices: Gates have been demonstrated in the Schoelkopf group at Yale University [62]. Error detection has been demonstrated in 3D cavities in the Devoret group at Yale University [63] and Amazon Web Services [64] using transmon qubits, following earlier theoretical proposals [65,66]. Logical readout in 3D cavities has been demonstrated by Quantum Circuits Inc. [67].Photonic platforms: state preparation and measurement fidelity of \(99.98\%\) in the C telecom band by PsiQuantum [68]. |

Error-corrected sensing code | A single physical qubit entangled with an NV spin was used to measure an incoming signal in a way that bit-flip errors on the qubit were correctable [69]. |

Error-correcting output code (ECOC) | Multiclass problems in machine learning, relevant to facial recognition [70,71], text recognition [72], or digit classification [73]. |

Fibonacci string-net code | NMR: Implementation of braiding-based Hamamard gate on two qubits [74].Superconducting qubits: state preparation, fusion, and braiding [75,76]. The latter work utilized DSNP and sampled the string-net wavefunction to estimate the underlying chromatic polynomial. |

Five-qubit perfect code | NMR: Implementation of perfect error correcting code on 5 spin subsystem of labeled crotonic acid for quantum network benchmarking [77]. Single-qubit logical gates [78]. Magic-state distillation using 7-qubit device [79].Superconducting qubits [80].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 [81]. Real-time magic-state distillation [82].Nitrogen-vacancy centers in diamond: fault-tolerant single-qubit Clifford operations using two ancillas [83]. The fault-tolerant circuit yields better fidelity than the non-fault-tolerant circuit. |

Floquet color code | Plaquette stabilizer measurement realized on the IBM Falcon superconducting-qubit device [84] |

Fountain code | Designed for servers sending data to many recipients, such as during broadcasting or file distribution [85,86].DNA storage [87]. |

Frameproof (FP) code | FP codes are utilized in digital fingerprinting and watermarking [88]. |

Gabidulin code | Public-key cryptosystems [89,90].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 [91]. |

Generalized RS (GRS) code | Commonly used in mass storage systems such as CDs, DVDs, QR codes etc.Various cloud storage systems [92].A variation of the McEliece public-key cryptosystem [93,94] by Niederreiter [95] replaced the generator matrix by the parity check matrix of a GRS code. This was proven to be insecure since the public key exposes the algebraic structure of code [96]. More recent works focus on methods to mask the algebraic structure using subcodes of GRS codes [89]. For example, a key-recovery attack was developed in Ref. [97] for a variant of masking method proposed in Ref. [98]. |

Generalized Shor code | The \([[m^2,1,m]]\) codes for \(m\leq 7\) have been realized in trapped-ion quantum devices [99]. |

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 [100].Extended Golay code used in American military standards for automatic link establishment in high frequency radio systems [101].Proofs of the quantum mechanical Kochen-Specker theorem [102]. |

Gold code | Used in for synchronization purposes in telecommunication [103]GPS C/A for satellite navigation [104]. |

Goppa code | The McEliece public-key cryptosystem [93,94]. The protocol relies on the assumptions that Goppa-code generator matrices are hard to distinguish from random linear codes. However, there is an algorithm distinguishing between the two code classes in a time subexponential in \(n\) [105]. |

Gray code | Three-dimensional imaging [106].Broadcasting and communication [107]. |

Group GKP code | Cryptographic applications stemming from the monogamy of entanglement of group GKP code and error words [1]. |

Heavy-hexagon code | Superconducting qubits: Logical state preparation and flag-qubit error correction realized in superconducting-circuit devices (specifically, fixed-frequency transmon qubit architectures) by IBM for \(d=2\) [108,109] and \(d=3\) [110]. Simultaneous syndrome extraction and logical Bell-state preparation for both the embedded surface and Bacon-Shor codes of distance \(\leq 4\) on an IBM 133-qubit device [111]. |

Hessian polyhedron code | Quantum mechanical SIC-POVMs [112]. |

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 [113]. |

Honeycomb Floquet code | Plaquette stabilizer measurement realized on the IBM Falcon superconducting-qubit device [84] |

Irregular LDPC code | Satellite communication after concatenating with a modulation scheme [114]. |

Irregular repeat-accumulate (IRA) code | LDPC codes are used for digital satellite video broadcasting per the DVB-S2 standard [115,116] utilize IRA code features and are subject to ongoing litigation; see Ref. [117].Apple and Broadcom Wi-Fi devices utilize IRA encoding and decoding code features and are subject to ongoing litigation; see Ref. [117]. |

Justesen code | Generating small-bias sample spaces, i.e., probability distributions that parity functions cannot typically distinguish from the uniform distribution [118]. |

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 [119].Superconducting circuits: initialization [120], braiding [121] and detection [121,122] of defects has been simulated in devices that have a different notion of locality than a bona-fide fermionic system. |

Kitaev surface code | Signatures of corresponding topological phase of matter detected in superconducting circuits [123] and two-dimensional Rydberg atomic arrays [124]. |

Linearized RS code | Network coding [125].Code-based cryptography [126,127]. |

Locally recoverable code (LRC) | An \((18,14,7)\) LRC has beed used in the Windows Azure cloud storage system [128]; see also [129; 31.3.1.2].Facebook f4 BLOB cloud storage system [130] |

Long-range enhanced surface code (LRESC) | Preparation of GHZ state of four logical qubits with beyond break-even fidelity in a \([[25,4,3]]\) LRESC [131]. |

Loop toric code | Trapped ions: single-shot QEC realized using the \([[33,1,4]]\) loop toric code on the Quantinuum H2 device [132]. |

Low-rank parity-check (LRPC) code | Cryptosystem [133] that is a rank-metric analogue of NTRU [134] and MDPC [135] cryptosystems.Post-quantum cryptography [136]. |

Matching code | Braiding of defects has been demonstrated for a five-qubit version of code [137]. |

Maximally recoverable (MR) code | RAID storage [138]. |

Maximum distance separable (MDS) code | Automatic repeat request (ARQ) data transmission protocols ([49], Ch. 7). |

Maximum-rank distance (MRD) code | Useful for error and erasure correction in network coding [139,140]. |

Modular-qudit cluster-state code | Quantum compututation with cluster states has been realized using photons in the time and frequency domains [141]. |

Monitored random-circuit code | Measurement induced quantum phases have been realized in a trapped-ion processor [142]. |

Multi-edge LDPC code | Quantum key distribution [143–145]. |

NTRU-GKP code | Public-key NTRU-based quantum communication protocol [146]. |

Niset-Andersen-Cerf code | Realized in Ref. [147] 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 [148] and TES receivers [149]. |

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 ([150], Ch. 27).Marking the state of a finite automaton [151]. |

PPM c-q code | Conditional pulse nulling (CPN) receiver [152]. |

PSK c-q code | Unambiguous state discrimination using displacement-based receiver for 4-PSK [153].Multi-stage quantum receivers [154–157].Bayesian inference [158].Time resolving quantum receiver opertaing in the telecom C band [159].Displacements and photon detection [160].Adaptive decoder using linear-optical elements and photon detection [14]. |

Pair-cat code | Microwave cavities coupled to superconducting circuits by the Wang group [161]. |

Phase-shift keying (PSK) code | Telephone-line modems: 1967 Milgo 4400/48 and international standard V.27 used 8-PSK [162]. |

Polar code | Code control channels for the 5G NR (New Radio) interfaces [163]. |

Pulse-position modulation (PPM) code | Greek hydraulic semaphore system [164,165].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 [166].Telephone-line modems: 1962 Bell 201 and international standard V.24 [167]. |

Quadrature-amplitude modulation (QAM) code | Optical communication (e.g., Ref. [168]).Telephone-line modems: 1971 Codex 9600C and international standard V.29 used 16-QAM [169]. |

Quantum Tanner code | Used to obtain explicit lower bounds in the sum-of-squares game [170].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 [171] of the NLTS conjecture [172]. |

Quantum divisible code | Triply even codes can yield secure multi-party quantum computation [173]. |

Quantum parity code (QPC) | The \([[m^2,1,m]]\) codes for \(m\leq 7\) have been realized in trapped-ion quantum devices [99].QPCs have been discussed independently in the context of superconducting circuits [174; Eq. (1)][175; Eqs. (8-10)], and aspects of such designs have been realized in experiments [176]. |

Quantum repetition code | NMR: 3-qubit phase-flip code [177,178], with up to two rounds of error correction in liquid-state NMR [179].Trapped ions: 3-qubit bit-flip code by Wineland group [180], and 3-qubit phase-flip algorithm implemented in 3 cycles on high fidelity gate operations [181]. Both phase- and bit-flip codes for 31 qubits and their stabilizer measurements have been realized by Quantinuum [182]. Multiple rounds of Steane error correction [183].Superconducting circuits: 3-qubit phase-flip and bit-flip code by Schoelkopf group [184,185]; 3-qubit bit-flip code [186]; 3-qubit phase-flip code up to 3 cycles of error correction [187]; IBM 15-qubit device [188]; IBM Rochester device using 43-qubit code [189]; Google system performing up to 8 error-correction cycles on 5 and 9 qubits [190]; Google Quantum AI Sycamore utilizing up to 11 physical qubits and running 50 correction rounds [191]; Google Quantum AI Sycamore utilizing up to 25 qubits for comparison of logical error scaling with a quantum code [192] (see also [193]). Google Quantum AI follow-up experiment on codes up to (classical) distance 29, demonstrating exponential suppression to an error floor of \(10^{-10}\) [194].Continuous error correction protocols have been implemented on a 3-qubit superconducting qubit device [195].Semiconductor spin-qubit devices: 3-qubit devices at RIKEN [196] and Delft [197].Nitrogen-vacancy centers in diamond: 3-qubit phase-flip code [198,199] (see also Ref. [200]). |

Quasi-cyclic LDPC (QC-LDPC) code | 5G NR cellular communication for the traffic channel [201,202].Wireless communication: WiMAX (IEEE 802.16e) [203–205], WiFi 4 (IEEE 802.11n) [206], and WPAN (IEEE 802.15.3c) [207]. |

Qubit CSS code | Fully homomorphic encryption [208].Cryptographic applications stemming from the monogamy of entanglement of CSS code and error words [209]. |

Qubit c-q code | Quantum enhancement was demonstrated using a polarization-based non-error-correcting c-q encodings [210]. |

Random code | Distributed storage systems [211].Classical and quantum cryptography based on the learning-with-errors problem, which is related to decoding a random linear code [212].Random codes can be used to realize secure computation [213]. |

Rank-metric code | Identity-Based Encryption [214].Digital watermarking [215].Network coding and streaming media broadcasting [216]. |

Rank-modulation code | Electronic devices where charges can either increase in an individual cell or decrease in a block of adjacent cells, e.g., flash memories [217]. |

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 [218,219]. |

Reed-Solomon (RS) code | RS Product Code (RSPC) was used in DVDs (see Ref. [49], Ch. 4).DSL technologies and their variants against impluse noise [220].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 [221].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 telemetry channel coding standard issued by the Consultative Committee for Space Data Systems (see Ref. [49], Ch. 3).Automatic repeat request (ARQ) data transmission protocols (see Ref. [49], Ch. 7).Slow-frequency-hop spread-spectrum transmission (see Ref. [49], Chs. 8-9).RS codes over \(q=2^m\) are used in RAID 6 [222,223]; see [48].Coded sharding designs in blockchains to increase efficiency [224].Used in QR-Codes to retrieve damaged barcodes [225].Wireless communication systems such as 3G, DVB, and WiMAX [226].Correcting pooled testing results for SARS-CoV-2 [227].DNA storage [228]. |

Regular binary Tanner code | First hardware implementation was done using a semi-systolic decoding architecture [229]. |

Repetition code | Repetition codes, in conjunction with other codes, were used in magnetic disks [230].Communication protocols such as FlexRay [231].' |

Residue AG code | Improvements over the McEliece public-key cryptosystem to linear AG codes on curves of arbitrary genus [232]. Only the subfield subcode proposal remains resilient to attacks [233; Sec. 15.7.5.3].Algebraic geometric secret-sharing schemes [234]. |

Single parity-check (SPC) code | Can be realized on almost every communication device. SPCs are some of the earliest error-correcting codes ([150], 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) [235–237]. |

Square-lattice GKP code | Motional degree of freedom of a trapped ion: square-lattice GKP encoding realized with the help of post-selection by Home group [238,239], 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 [240]. State preparation also realized by Tan group [27]. Universal gate set, including a two-qubit entangling gate, realized by Tan group [241].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 [113]. Subsequent paper by Devoret group [25] 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 [24]). See Ref. [26] for another experiment. A feed-forward-free, i.e., fully autonomous protocol has also been implemented by Nord Quantique [242].GKP states and homodyne measurements have been realized in propagating telecom light by the Furusawa group [243].Single-qubit \(Z\)-gate has been demonstrated [244] in the single-photon subspace of an infinite-mode space [245], in which time and frequency become bosonic conjugate variables of a single effective bosonic mode. In this context, GKP position-state wavefunctions are called Dirac combs or frequency combs. |

Squeezed cat code | Squeezed cat code has been realized in a superconducting circuit device by the Gao group [246]. |

Subspace code | Packet-based transmission over networks [247]. |

Subsystem QECC | A two-qubit unitarily correctable subsystem code recovery has been realized in an optical system [248]. |

Surface-17 code | Implemented at ETH Zurich by the Wallraff group [249] and on the Zuchongzhi 2.1 superconducting quantum processor [250]. Both experimental error rates are above the pseudo-threshold for this code relative to a single qubit; see Physics viewpoint for a summary [251]. Magic state have been created on the latter processor [252]. |

Tensor-product code | Construction can be used in magnetic recording by taking the tensor product of an RS code and a parity-check code [253]. |

Ternary Golay code | Code used in football pools with at least one good bet [46,254]. In fact, the code was originally constructed by Juhani Virtakallio and published in the Finnish football pool magazine Veikkaaja [46,255].Proofs of the quantum mechanical Kochen-Specker theorem [102]. |

Toric code | One cycle of syndrome readout on 19-qubit planar and 24-qubit toric codes realized in two-dimensional Rydberg atomic arrays [256]. |

Traceability code | Broadcast messages, pay-per-view movies, and protecting copyrighted online material [257]. |

Turbo code | Recommended by Consultative Committee on Space Data Systems (CCDJS) for telemetry and telecommand [258,259].Several standards related to wireless communication, including W-CDMA, DVB-RCS, TD-SCDMA, 802.16, and CDMA2000 [260]. |

Twist-defect surface code | 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 [261]. Logical Clifford gates for \([[8,2,2]]\) and \([[10,2,3]]\) twist-defect surface codes realized in a trapped ion device by Quantinuum [262]. |

Two-component cat code | Lindbladian-based [263,264] and Hamiltonian-based 'Kerr-cat' [265] encodings have been achieved in superconducting circuit devices by the Devoret group; Ref. [264] 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 up to a bit-flip time of 1ms [266]. A bit-flip time of up to 10s has been achieved for the two-legged cat code in the classical-bit regime [267–269]. A holonomic gate has been repurposed as a logical measurement [270]. The 'Kerr-cat' encoding and a \(\pi/2\) gate have been realized with the help of a band-block filter, yielding a bit-flip lifetime of 1 ms in the 10-photon regime [271] (see also Ref. [272]).T4C code realized in a superconducting circuit device by the Wang group [273].Squeezed cat code has been realized in superconducting circuit device by the Gao group [246]. |

Unary code | Neural networks [274].Birdsong production [275]. |

Very small logical qubit (VSLQ) code | Star-code autonomous correction scheme realized using superconducting circuits [276]. |

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 [255,277]. |

XZZX surface code | Superconducting circuits: Distance-five 25-qubit code implemented on a superconducting quantum processor by Google Quantum AI [192]. 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 [278]. Leakage errors have been handled in a separate work in a distance-three code [193]. Google Quantum AI follow-up experiment realizing distance-5 and distance-7 codes with 100 rounds of correction using the Libra and transformer-based decoders. The logical error rate is suppressed by a factor of \(\approx 2\), demonstrating beyond-break-even error correction with a block quantum code [194]. Rydberg atom arrays: Lukin group [279]. Transversal CNOT gates performed on distance \(3\), \(5\), and \(7\) codes. |

Zero-pi qubit code | A related superconducting circuit has been realized by the Houck group [280]. |

Zetterberg code | Code used to provide better protection of data transmission with its double error correcting capacity [281]. |

\(A_2\) hexagonal lattice code | Wireless communication [282,283]. |

\([2^m,m+1,2^{m-1}]\) First-order RM code | The \([32, 6, 16]\) RM\((1,5)\) code was used for the 1971 Mariner 9 spacecraft [284]. |

\([2^r-1,2^r-r-1,3]\) Hamming code | Commonly used when error rates are very low, for example, computer RAM or integrated circuits [285].Hamming-code based matrix embedding used in steganography [286,287]. |

\([[10,1,2]]\) CSS code | Trapped-ion devices: fault-tolerant universal gate set via code switching between the Steane code and the \([[10,1,2]]\) code on a device from the Monz group [288]. |

\([[12,2,4]]\) carbon code | Trapped-ion devices: Three rounds of error correction and post-selected fault-tolerant logical Bell-state preparation with logical error rates at least 5 times lower than physical rate on a quantum charge-coupled device (QCCD) [182] by Microsoft and Quantinuum [289]. |

\([[16,6,4]]\) Tesseract color code | Trapped-ion devices: logical graph and GHZ states of up to 12 logical qubits constructed using three copies of the tesseract color code, along with five rounds of post-selected fault-tolerant error correction in a device by Quantinuum [290]. |

\([[2m,2m-2,2]]\) error-detecting code | Trapped-ion devices: the \(m=5\) code has been realized on a 12-qubit device by Quantinuum [291]. |

\([[4,2,2]]\) Four-qubit code | \([[4,1,2]]\) subcodes implemented in linear optical networks [292,293].Trapped-ion device by IonQ [294].Logical state preparation and flag-qubit error correction realized in superconducting-circuit devices by IBM [108,109,295,296].The subcode \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [297] and \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) [191], 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 [298].Logical gates for the \(\{|\overline{00}\rangle,|\overline{11}\rangle\}\) subcode, treated as a planar code, realized in superconducting circuits [299].The CZ magic state has been realized on an IBM heavy-hex superconducting circuit device [300].Logical Clifford gates for a twist-defect surface code that is single-qubit Clifford equivalent to a \([[4,1,2]]\) realized in a trapped ion device by Quantinuum [262].CPC gadgets for the \([[4,2,2]]\) code have been implemented on the IBM 5Q superconducting device [301]. |

\([[6,4,2]]\) error-detecting code | Trapped-ion devices: Bayesian quantum phase estimation on a device by Quantinuum [302]. |

\([[7,1,3]]\) Steane code | Trapped-ion devices: seven-qubit device in Blatt group [303]. Ten-qubit QCCD device by Quantinuum [304] realizing repeated syndrome extraction, real-time look-up-table decoding (yielding lower logical SPAM error rate than physical SPAM), and non-fault-tolerant magic-state distillation (see APS Physics Synopsis [305]). Fault-tolerant universal two-qubit gate set using T injection by Monz group [306]. Logical CNOT gate and Bell-pair creation between two logical qubits (yielding a logical fidelity higher than physical), including rounds of correction and fault-tolerant primitives such as flag qubits and pieceable fault tolerance, on a 20-qubit device by Quantinuum [81]; logical fidelity interval of the combined preparation-CNOT-measurement procedure was higher than that of the unencoded physical qubits. Multiple rounds of Steane error correction [183]. Fault-tolerant universal gate set via code switching between the Steane code and the \([[10,1,2]]\) code [288]. Post-selected fault-tolerant logical Bell-state preparation with logical error rates at least 10 times lower than physical rate on a device by Quantinuum [289]. The quantum Fourier transform on three code blocks [307]. Fault-tolerant transversal and lattice-surgery state teleportation protocols as well as Knill error correction [308]. Rains shadow enumerators have been measured [309].Rydberg atom arrays: Lukin group [256]; ten logical qubits, transversal CNOT gate performed, logical ten-qubit GHZ state initialized with break-even fidelity, and fault-tolerant logical two-qubit GHZ state initialized [279]. |

\([[8,3,2]]\) CSS code | Trapped ions: one-qubit addition algorithm implemented fault-tolerantly on the Quantinuum H1-1 device [310].Superconducting circuits: fault-tolerant \(CZZ\) gate performed on IBM and IonQ devices [311].Rydberg atom arrays: Lukin group [279]. 48 logical qubits, 228 logical two-qubit gates, 48 logical CCZ gates, and error detection peformed in 16 blocks. Circuit outcomes were sampled and cross-entropy (XEB) was calculated to verify quantumness. Logical entanglement entropy was measured [279]. |

\([[9,1,3]]\) 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 [99]. Variants of the code to handle coherent noise studied and realized by K. Brown and C. Monroe groups [312].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% [313]. All-photonic quantum repeater architecture tested on the same code [314]. |

\([[9,1,3]]_{\mathbb{R}}\) Lloyd-Slotine code | Optical network by the Furusawa group [315]. |

## References

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