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

[1]
E. Culf, T. Vidick, and V. V. Albert, “Group coset monogamy games and an application to device-independent continuous-variable QKD”, (2022) arXiv:2212.03935
[2]
S. Yokoyama et al., “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain”, Nature Photonics 7, 982 (2013) arXiv:1306.3366 DOI
[3]
M. Chen, N. C. Menicucci, and O. Pfister, “Experimental Realization of Multipartite Entanglement of 60 Modes of a Quantum Optical Frequency Comb”, Physical Review Letters 112, (2014) arXiv:1311.2957 DOI
[4]
J. Yoshikawa et al., “Invited Article: Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing”, APL Photonics 1, (2016) arXiv:1606.06688 DOI
[5]
Y. Miwa et al., “Demonstration of a universal one-way quantum quadratic phase gate”, Physical Review A 80, (2009) arXiv:0906.3141 DOI
[6]
E. Eleftheriou and S. Olcer, “Low-density parity-check codes for digital subscriber lines”, 2002 IEEE International Conference on Communications. Conference Proceedings. ICC 2002 (Cat. No.02CH37333) DOI
[7]
K. Tsujino et al., “Quantum Receiver beyond the Standard Quantum Limit of Coherent Optical Communication”, Physical Review Letters 106, (2011) arXiv:1103.5592 DOI
[8]
C. Wittmann et al., “Demonstration of Near-Optimal Discrimination of Optical Coherent States”, Physical Review Letters 101, (2008) arXiv:0809.4953 DOI
[9]
M. L. Shcherbatenko et al., “Sub-shot-noise-limited fiber-optic quantum receiver”, Physical Review A 101, (2020) arXiv:1911.08932 DOI
[10]
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
[11]
M. T. DiMario et al., “Optimized communication strategies with binary coherent states over phase noise channels”, npj Quantum Information 5, (2019) arXiv:1907.12515 DOI
[12]
M. T. DiMario and F. E. Becerra, “Phase tracking for sub-shot-noise-limited receivers”, Physical Review Research 2, (2020) DOI
[13]
R. Yuan et al., “Optimally Displaced Threshold Detection for Discriminating Binary Coherent States Using Imperfect Devices”, (2020) arXiv:2007.11109
[14]
M. T. DiMario and F. E. Becerra, “Demonstration of optimal non-projective measurement of binary coherent states with photon counting”, npj Quantum Information 8, (2022) arXiv:2207.12234 DOI
[15]
C. Delaney et al., “Demonstration of a quantum advantage by a joint detection receiver for optical communication using quantum belief propagation on a trapped-ion device”, Physical Review A 106, (2022) arXiv:2102.13052 DOI
[16]
L. Egan et al., “Fault-Tolerant Operation of a Quantum Error-Correction Code”, (2021) arXiv:2009.11482
[17]
K. A. S. Immink. Codes for mass data storage systems. Shannon Foundation Publisher, 2004.
[18]
Cheung, K-M., and F. Pollara. "Phobos lander coding system: Software and analysis." The Telecommunications and Data Acquisition Report (1988).
[19]
International Telecommunication Union-T, Recommendation V.21: 300 bits per second duplex modem standardized for use in the general switched telephone network, 1984
[20]
International Telecommunication Union-T, Recommendation V.23: 600/1200-baud modem standardized for use in the general switched telephone network, 1988
[21]
C. J. Axline et al., “On-demand quantum state transfer and entanglement between remote microwave cavity memories”, Nature Physics 14, 705 (2018) arXiv:1712.05832 DOI
[22]
L. Hu et al., “Quantum error correction and universal gate set operation on a binomial bosonic logical qubit”, Nature Physics 15, 503 (2019) arXiv:1805.09072 DOI
[23]
Y. Xu et al., “Demonstration of Controlled-Phase Gates between Two Error-Correctable Photonic Qubits”, Physical Review Letters 124, (2020) arXiv:1810.04690 DOI
[24]
Z. Ni et al., “Beating the break-even point with a discrete-variable-encoded logical qubit”, Nature 616, 56 (2023) arXiv:2211.09319 DOI
[25]
V. V. Sivak et al., “Real-time quantum error correction beyond break-even”, Nature 616, 50 (2023) arXiv:2211.09116 DOI
[26]
M. Kudra et al., “Robust Preparation of Wigner-Negative States with Optimized SNAP-Displacement Sequences”, PRX Quantum 3, (2022) arXiv:2111.07965 DOI
[27]
V. G. Matsos et al., “Robust and Deterministic Preparation of Bosonic Logical States in a Trapped Ion”, Physical Review Letters 133, (2024) arXiv:2310.15546 DOI
[28]
S. Zhu, Z. Sun, and X. Kai, “A Class of Narrow-Sense BCH Codes”, IEEE Transactions on Information Theory 65, 4699 (2019) DOI
[29]
L. Sun et al., “Tracking photon jumps with repeated quantum non-demolition parity measurements”, Nature 511, 444 (2014) arXiv:1311.2534 DOI
[30]
N. Ofek et al., “Demonstrating Quantum Error Correction that Extends the Lifetime of Quantum Information”, (2016) arXiv:1602.04768
[31]
S. Rosenblum et al., “Fault-tolerant detection of a quantum error”, Science 361, 266 (2018) arXiv:1803.00102 DOI
[32]
P. Walther et al., “Experimental one-way quantum computing”, Nature 434, 169 (2005) arXiv:quant-ph/0503126 DOI
[33]
R. Ceccarelli et al., “Experimental Entanglement and Nonlocality of a Two-Photon Six-Qubit Cluster State”, Physical Review Letters 103, (2009) arXiv:0906.2233 DOI
[34]
M. V. Jabir et al., “Energy and bandwidth efficiency optimization of quantum-enabled optical communication channels”, npj Quantum Information 8, (2022) DOI
[35]
M. V. Jabir et al., “Experimental demonstration of the near-quantum optimal receiver”, OSA Continuum 3, 3324 (2020) DOI
[36]
I. A. Burenkov et al., “Time-Resolving Quantum Measurement Enables Energy-Efficient, Large-Alphabet Communication”, PRX Quantum 1, (2020) DOI
[37]
P. van Loock and S. L. Braunstein, “Unconditional teleportation of continuous-variable entanglement”, Physical Review A 61, (1999) arXiv:quant-ph/9907073 DOI
[38]
F. Grosshans and P. Grangier, “Continuous Variable Quantum Cryptography Using Coherent States”, Physical Review Letters 88, (2002) arXiv:quant-ph/0109084 DOI
[39]
F. Grosshans et al., “Quantum key distribution using gaussian-modulated coherent states”, Nature 421, 238 (2003) arXiv:quant-ph/0312016 DOI
[40]
D. H. Smith, L. A. Hughes, and S. Perkins, “A New Table of Constant Weight Codes of Length Greater than 28”, The Electronic Journal of Combinatorics 13, (2006) DOI
[41]
S. I. Mrutu, A. Sam, and N. H. Mvungi, “Forward Error Correction Convolutional Codes for RTAs’ Networks: An Overview”, International Journal of Computer Network and Information Security 6, 19 (2014) DOI
[42]
T. Halonen, J. Romero, and J. Melero, editors , “GSM, GPRS and EDGE Performance”, (2003) DOI
[43]
Butman, Deutsch, and Miller. Performance of concatenated codes for deep space missions. 1981.
[44]
G. Cohen, I. Honkala, S. Litsyn, A. Lobstein, Covering Codes, Elsevier (1997).
[45]
H. Hamalainen et al., “Football Pools--A Game for Mathematicians”, The American Mathematical Monthly 102, 579 (1995) DOI
[46]
A. Barg, “At the Dawn of the Theory of Codes”, The Mathematical Intelligencer 15, 20 (1993) DOI
[47]
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
[48]
M. Blaum, P. G. Farrell, H. C. A. van Tilborg, 1998. Array codes. Handbook of coding theory, 2 (Part 2), pp. 1855-1909.
[49]
S. B. Wicker and V. K. Bhargava, Reed-Solomon Codes and Their Applications (IEEE, 1999) DOI
[50]
Ronghui Peng and Rong-Rong Chen, “Application of Nonbinary LDPC Cycle Codes to MIMO Channels”, IEEE Transactions on Wireless Communications 7, 2020 (2008) DOI
[51]
Borrelli, Chris. "IEEE 802.3 cyclic redundancy check." application note: Virtex Series and Virtex-II Family, XAPP209 (v1. 0) (2001).
[52]
M. Ergen, “Basics of Cellular Communication”, Mobile Broadband 19 (2008) DOI
[53]
A. R. Calderbank, S. N. Diggavi, and N. Al-Dhahir, “Space-time signaling based on Kerdock and Delsafte-Goethals codes”, 2004 IEEE International Conference on Communications (IEEE Cat. No.04CH37577) (2004) DOI
[54]
A. Thompson and R. Calderbank, “Compressed Neighbour Discovery using Sparse Kerdock Matrices”, (2018) arXiv:1801.04537
[55]
M. Iqbal et al., “Non-Abelian topological order and anyons on a trapped-ion processor”, Nature 626, 505 (2024) arXiv:2305.03766 DOI
[56]
B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 DOI
[57]
M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
[58]
L. Lootens et al., “Mapping between Morita-equivalent string-net states with a constant depth quantum circuit”, Physical Review B 105, (2022) arXiv:2112.12757 DOI
[59]
C. R. Myers and R. Laflamme, “Linear Optics Quantum Computation: an Overview”, (2005) arXiv:quant-ph/0512104
[60]
P. Kok et al., “Linear optical quantum computing with photonic qubits”, Reviews of Modern Physics 79, 135 (2007) arXiv:quant-ph/0512071 DOI
[61]
S. Slussarenko and G. J. Pryde, “Photonic quantum information processing: A concise review”, Applied Physics Reviews 6, (2019) arXiv:1907.06331 DOI
[62]
Y. Lu et al., “High-fidelity parametric beamsplitting with a parity-protected converter”, Nature Communications 14, (2023) DOI
[63]
A. Koottandavida et al., “Erasure detection of a dual-rail qubit encoded in a double-post superconducting cavity”, (2023) arXiv:2311.04423
[64]
H. Levine et al., “Demonstrating a Long-Coherence Dual-Rail Erasure Qubit Using Tunable Transmons”, Physical Review X 14, (2024) arXiv:2307.08737 DOI
[65]
J. D. Teoh et al., “Dual-rail encoding with superconducting cavities”, Proceedings of the National Academy of Sciences 120, (2023) arXiv:2212.12077 DOI
[66]
A. Kubica et al., “Erasure qubits: Overcoming the \(T_1\) limit in superconducting circuits”, (2022) arXiv:2208.05461
[67]
K. S. Chou et al., “Demonstrating a superconducting dual-rail cavity qubit with erasure-detected logical measurements”, (2023) arXiv:2307.03169
[68]
K. Alexander et al., “A manufacturable platform for photonic quantum computing”, (2024) arXiv:2404.17570
[69]
T. Unden et al., “Quantum Metrology Enhanced by Repetitive Quantum Error Correction”, Physical Review Letters 116, (2016) arXiv:1602.07144 DOI
[70]
T. Windeatt, “Boosted ECOC ensembles for face recognition”, International Conference on Visual Information Engineering (VIE 2003). Ideas, Applications, Experience (2003) DOI
[71]
J. Kittler et al., “Face verification using error correcting output codes”, Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001 DOI
[72]
R. Ghani, “Combining labeled and unlabeled data for text classification with a large number of categories”, Proceedings 2001 IEEE International Conference on Data Mining DOI
[73]
J. Zhou and C. Y. Suen, “Unconstrained numeral pair recognition using enhanced error correcting output coding: a holistic approach”, Eighth International Conference on Document Analysis and Recognition (ICDAR’05) (2005) DOI
[74]
Y. Fan et al., “Experimental realization of a topologically protected Hadamard gate via braiding Fibonacci anyons”, (2022) arXiv:2210.12145
[75]
S. Xu et al., “Non-Abelian braiding of Fibonacci anyons with a superconducting processor”, Nature Physics (2024) arXiv:2404.00091 DOI
[76]
Z. K. Minev et al., “Realizing string-net condensation: Fibonacci anyon braiding for universal gates and sampling chromatic polynomials”, (2024) arXiv:2406.12820
[77]
E. Knill et al., “Benchmarking Quantum Computers: The Five-Qubit Error Correcting Code”, Physical Review Letters 86, 5811 (2001) arXiv:quant-ph/0101034 DOI
[78]
J. Zhang, R. Laflamme, and D. Suter, “Experimental Implementation of Encoded Logical Qubit Operations in a Perfect Quantum Error Correcting Code”, Physical Review Letters 109, (2012) arXiv:1208.4797 DOI
[79]
A. M. Souza et al., “Experimental magic state distillation for fault-tolerant quantum computing”, Nature Communications 2, (2011) arXiv:1103.2178 DOI
[80]
M. Gong et al., “Experimental exploration of five-qubit quantum error-correcting code with superconducting qubits”, National Science Review 9, (2021) arXiv:1907.04507 DOI
[81]
C. Ryan-Anderson et al., “Implementing Fault-tolerant Entangling Gates on the Five-qubit Code and the Color Code”, (2022) arXiv:2208.01863
[82]
N. C. Brown et al., “Advances in compilation for quantum hardware -- A demonstration of magic state distillation and repeat-until-success protocols”, (2023) arXiv:2310.12106
[83]
M. H. Abobeih et al., “Fault-tolerant operation of a logical qubit in a diamond quantum processor”, Nature 606, 884 (2022) arXiv:2108.01646 DOI
[84]
Wootton, James R., “Quantum error correction: From the blackboard to the cloud”, University of Basel (2023) arXiv:2210.13154 DOI
[85]
A. Shokrollahi, “Raptor codes”, IEEE Transactions on Information Theory 52, 2551 (2006) DOI
[86]
E. Baik, A. Pande, and P. Mohapatra, “Cross-layer coordination for efficient contents delivery in LTE eMBMS traffic”, 2012 IEEE 9th International Conference on Mobile Ad-Hoc and Sensor Systems (MASS 2012) (2012) DOI
[87]
Y. Erlich and D. Zielinski, “DNA Fountain enables a robust and efficient storage architecture”, Science 355, 950 (2017) DOI
[88]
S. R. Blackburn, “Frameproof Codes”, SIAM Journal on Discrete Mathematics 16, 499 (2003) DOI
[89]
T. P. Berger and P. Loidreau, “How to Mask the Structure of Codes for a Cryptographic Use”, Designs, Codes and Cryptography 35, 63 (2005) DOI
[90]
T. Lau and C. Tan, “A New Technique in Rank Metric Code-Based Encryption”, Cryptography 2, 32 (2018) DOI
[91]
P. Lefevre, P. Carre, and P. Gaborit, “Watermarking and Rank Metric Codes”, 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (2018) DOI
[92]
H. Dau et al., “Repairing Reed-Solomon Codes With Multiple Erasures”, IEEE Transactions on Information Theory 64, 6567 (2018) arXiv:1612.01361 DOI
[93]
R. J. McEliece, A public-key cryptosystem based on algebraic coding theory, Technical report, Jet Propulsion Lab. DSN Progress Report (1978).
[94]
H. Janwa and O. Moreno, “McEliece public key cryptosystems using algebraic-geometric codes”, Designs, Codes and Cryptography 8, (1996) DOI
[95]
H. Niederreiter (1986). Knapsack-type cryptosystems and algebraic coding theory. Problems of Control and Information Theory. Problemy Upravlenija I Teorii Informacii. 15: 159–166.
[96]
V. M. SIDELNIKOV and S. O. SHESTAKOV, “On insecurity of cryptosystems based on generalized Reed-Solomon codes”, Discrete Mathematics and Applications 2, (1992) DOI
[97]
A. Couvreur et al., “Distinguisher-Based Attacks on Public-Key Cryptosystems Using Reed-Solomon Codes”, (2014) arXiv:1307.6458
[98]
M. Baldi et al., “Enhanced public key security for the McEliece cryptosystem”, (2014) arXiv:1108.2462
[99]
N. H. Nguyen et al., “Demonstration of Shor Encoding on a Trapped-Ion Quantum Computer”, Physical Review Applied 16, (2021) arXiv:2104.01205 DOI
[100]
E. C. Stone, “The Voyager 2 encounter with Uranus”, Journal of Geophysical Research: Space Physics 92, 14873 (1987) DOI
[101]
E. E. Johnson. An Efficient Golay Codec For MIL-STD-188-141A and FED-STD-1045. Department of Electrical and Computer Engineering, New Mexico State University, 1991.
[102]
M. Waegell and P. K. Aravind, “Golay codes and quantum contextuality”, Physical Review A 106, (2022) arXiv:2206.04209 DOI
[103]
Mujtaba Hamid and Andy Miller, Gold Code Generators in Virtex Devices, (2000)
[104]
J. J. SPILKER Jr., “GPS Signal Structure and Performance Characteristics”, Navigation 25, 121 (1978) DOI
[105]
H. Randriambololona, “The syzygy distinguisher”, (2024) arXiv:2407.15740
[106]
G. Sansoni et al., “Three-dimensional imaging based on Gray-code light projection: characterization of the measuring algorithm and development of a measuring system for industrial applications”, Applied Optics 36, 4463 (1997) DOI
[107]
S. L. Johnsson and C.-T. Ho, “Optimum broadcasting and personalized communication in hypercubes”, IEEE Transactions on Computers 38, 1249 (1989) DOI
[108]
M. Takita et al., “Experimental Demonstration of Fault-Tolerant State Preparation with Superconducting Qubits”, Physical Review Letters 119, (2017) arXiv:1705.09259 DOI
[109]
E. H. Chen et al., “Calibrated Decoders for Experimental Quantum Error Correction”, Physical Review Letters 128, (2022) arXiv:2110.04285 DOI
[110]
N. Sundaresan et al., “Demonstrating multi-round subsystem quantum error correction using matching and maximum likelihood decoders”, Nature Communications 14, (2023) arXiv:2203.07205 DOI
[111]
B. Hetényi and J. R. Wootton, “Creating entangled logical qubits in the heavy-hex lattice with topological codes”, (2024) arXiv:2404.15989
[112]
B. C. Stacey, “Sporadic SICs and Exceptional Lie Algebras”, (2019) arXiv:1911.05809
[113]
P. Campagne-Ibarcq et al., “Quantum error correction of a qubit encoded in grid states of an oscillator”, Nature 584, 368 (2020) arXiv:1907.12487 DOI
[114]
S. ten Brink, G. Kramer, and A. Ashikhmin, “Design of Low-Density Parity-Check Codes for Modulation and Detection”, IEEE Transactions on Communications 52, 670 (2004) DOI
[115]
R. Purnamasari, H. Wijanto, and I. Hidayat, “Design and implementation of LDPC(Low Density Parity Check) coding technique on FPGA (Field Programmable Gate Array) for DVB-S2 (Digital Video Broadcasting-Satellite)”, 2014 IEEE International Conference on Aerospace Electronics and Remote Sensing Technology (2014) DOI
[116]
ETSI, ETSI. "Digital video broadcasting (dvb); second generation framing structure, channel coding and modulation systems for broadcasting, interactive services, news gathering and other broadband satellite applications." Part II: S2-Extensions (DVB-S2X) (2005): 22-27.
[117]
Hui Jin, Aamod Khandekar, and Robert J. McEliece. "Serial concatenation of interleaved convolutional codes forming turbo-like codes." United States Patent Number 7116710B1 (2023).
[118]
J. Naor and M. Naor, “Small-bias probability spaces: efficient constructions and applications”, Proceedings of the twenty-second annual ACM symposium on Theory of computing - STOC ’90 (1990) DOI
[119]
J.-S. Xu et al., “Simulating the exchange of Majorana zero modes with a photonic system”, Nature Communications 7, (2016) arXiv:1411.7751 DOI
[120]
K. J. Sung et al., “Simulating Majorana zero modes on a noisy quantum processor”, Quantum Science and Technology 8, 025010 (2023) arXiv:2206.00563 DOI
[121]
N. Harle, O. Shtanko, and R. Movassagh, “Observing and braiding topological Majorana modes on programmable quantum simulators”, Nature Communications 14, (2023) arXiv:2203.15083 DOI
[122]
X. Mi et al., “Noise-resilient edge modes on a chain of superconducting qubits”, Science 378, 785 (2022) arXiv:2204.11372 DOI
[123]
K. J. Satzinger et al., “Realizing topologically ordered states on a quantum processor”, Science 374, 1237 (2021) arXiv:2104.01180 DOI
[124]
G. Semeghini et al., “Probing topological spin liquids on a programmable quantum simulator”, Science 374, 1242 (2021) arXiv:2104.04119 DOI
[125]
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
[126]
Hörmann, F., Bartz, H., & Horlemann, A. L. (2022). Security considerations for Mceliece-like cryptosystems based on linearized Reed-Solomon codes in the sum-rank metric.
[127]
F. Hörmann, H. Bartz, and A.-L. Horlemann, “Distinguishing and Recovering Generalized Linearized Reed–Solomon Codes”, Lecture Notes in Computer Science 1 (2023) arXiv:2304.00627 DOI
[128]
C. Huang, H. Simitci, Y. Xu, A. Ogus, B. Calder, P. Gopalan, J. Li, and S. Yekhanin. Erasure coding in Windows Azure Storage. In Proc. USENIX Annual Technical Conference (ATC), pgs. 15-26, Boston, Massachusetts, June 2012.
[129]
V. Ramkumar, M. Vajha, S. B. Balaji, M. Nikhil Krishnan, B. Sasidharan, P. Vijay Kumar, "Codes for Distributed Storage." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[130]
Muralidhar, S., Lloyd, W., Roy, S., Hill, C., Lin, E., Liu, W., Pan, S., Shankar, S., Sivakumar, V., Tang, L. and Kumar, S., 2014. f4: Facebook's warm BLOB storage system. In 11th USENIX Symposium on Operating Systems Design and Implementation (OSDI 14) (pp. 383-398).
[131]
Y. Hong et al., “Entangling four logical qubits beyond break-even in a nonlocal code”, (2024) arXiv:2406.02666
[132]
N. Berthusen et al., “Experiments with the 4D Surface Code on a QCCD Quantum Computer”, (2024) arXiv:2408.08865
[133]
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).
[134]
J. Hoffstein, J. Pipher, and J. H. Silverman, “NTRU: A ring-based public key cryptosystem”, Lecture Notes in Computer Science 267 (1998) DOI
[135]
R. Misoczki et al., “MDPC-McEliece: New McEliece variants from Moderate Density Parity-Check codes”, 2013 IEEE International Symposium on Information Theory (2013) DOI
[136]
P. Gaborit et al., “RankSign: An Efficient Signature Algorithm Based on the Rank Metric”, Post-Quantum Cryptography 88 (2014) DOI
[137]
J. R. Wootton, “Demonstrating non-Abelian braiding of surface code defects in a five qubit experiment”, Quantum Science and Technology 2, 015006 (2017) arXiv:1609.07774 DOI
[138]
J. S. Plank and M. Blaum, “Sector-Disk (SD) Erasure Codes for Mixed Failure Modes in RAID Systems”, ACM Transactions on Storage 10, 1 (2014) DOI
[139]
R. Koetter and F. Kschischang, “Coding for Errors and Erasures in Random Network Coding”, (2008) arXiv:cs/0703061
[140]
D. Silva, F. R. Kschischang, and R. Koetter, “A Rank-Metric Approach to Error Control in Random Network Coding”, IEEE Transactions on Information Theory 54, 3951 (2008) arXiv:0711.0708 DOI
[141]
C. Reimer et al., “High-dimensional one-way quantum processing implemented on d-level cluster states”, Nature Physics 15, 148 (2018) DOI
[142]
C. Noel et al., “Measurement-induced quantum phases realized in a trapped-ion quantum computer”, Nature Physics 18, 760 (2022) arXiv:2106.05881 DOI
[143]
M. Milicevic et al., “Quasi-cyclic multi-edge LDPC codes for long-distance quantum cryptography”, npj Quantum Information 4, (2018) arXiv:1702.07740 DOI
[144]
H. Mani et al., “Multiedge-type low-density parity-check codes for continuous-variable quantum key distribution”, Physical Review A 103, (2021) DOI
[145]
A. A. E. Hajomer et al., “Continuous-variable quantum passive optical network”, (2024) arXiv:2402.16044
[146]
J. Conrad, J. Eisert, and J.-P. Seifert, “Good Gottesman-Kitaev-Preskill codes from the NTRU cryptosystem”, Quantum 8, 1398 (2024) arXiv:2303.02432 DOI
[147]
M. Lassen et al., “Quantum optical coherence can survive photon losses using a continuous-variable quantum erasure-correcting code”, Nature Photonics 4, 700 (2010) arXiv:1006.3941 DOI
[148]
R. L. Cook, P. J. Martin, and J. M. Geremia, “Optical coherent state discrimination using a closed-loop quantum measurement”, Nature 446, 774 (2007) DOI
[149]
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
[150]
Encyclopedia of Computer Science and Technology, Second Edition Volume I (CRC Press, 2017) DOI
[151]
S. Devadas and A. R. Newton, “Decomposition and factorization of sequential finite state machines”, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 8, 1206 (1989) DOI
[152]
J. Chen et al., “Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver”, Nature Photonics 6, 374 (2012) arXiv:1111.4017 DOI
[153]
F. E. Becerra, J. Fan, and A. Migdall, “Implementation of generalized quantum measurements for unambiguous discrimination of multiple non-orthogonal coherent states”, Nature Communications 4, (2013) DOI
[154]
F. E. Becerra et al., “Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination”, Nature Photonics 7, 147 (2013) DOI
[155]
S. Izumi et al., “Experimental Demonstration of a Quantum Receiver Beating the Standard Quantum Limit at Telecom Wavelength”, Physical Review Applied 13, (2020) arXiv:2001.05902 DOI
[156]
F. E. Becerra, J. Fan, and A. Migdall, “Photon number resolution enables quantum receiver for realistic coherent optical communications”, Nature Photonics 9, 48 (2014) DOI
[157]
A. R. Ferdinand, M. T. DiMario, and F. E. Becerra, “Multi-state discrimination below the quantum noise limit at the single-photon level”, npj Quantum Information 3, (2017) arXiv:1711.00074 DOI
[158]
I. A. Burenkov et al., “Experimental demonstration of time resolving quantum receiver for bandwidth and power efficient communications”, Conference on Lasers and Electro-Optics (2020) DOI
[159]
M. V. Jabir et al., “Versatile quantum-enabled telecom receiver”, AVS Quantum Science 5, (2023) DOI
[160]
S. Izumi, J. S. Neergaard-Nielsen, and U. L. Andersen, “Adaptive Generalized Measurement for Unambiguous State Discrimination of Quaternary Phase-Shift-Keying Coherent States”, PRX Quantum 2, (2021) arXiv:2009.02558 DOI
[161]
J. M. Gertler et al., “Experimental Realization and Characterization of Stabilized Pair Coherent States”, (2022) arXiv:2209.11643
[162]
International Telecommunication Union-T, Recommendation V.27ter: 4800/2400 Bits Per Second Modem Standardized For Use in the General Switched Telephone Network, 1984
[163]
3rd Generation Partnership Project (3GPP), Technical specification group radio access network, 3GPP TS 38.212 V.15.0.0, 2017.
[164]
Michael Lahanas. "Ancient Greek Communication Methods". https://web.archive.org/web/20141102224501/http://www.mlahanas.de/Greeks/Communication.htm. Archived from the original on 2014-11-02.
[165]
A. B. Raj and A. K. Majumder, “Historical perspective of free space optical communications: from the early dates to today’s developments”, IET Communications 13, 2405 (2019) DOI
[166]
K. Feher, “Modems for emerging digital cellular-mobile radio system”, IEEE Transactions on Vehicular Technology 40, 355 (1991) DOI
[167]
International Telecommunication Union-T, Recommendation V.24: List of definitions for interchange circuits between data terminal equipment (DTE) and data circuit-terminating equipment (DCE), 1988
[168]
F. Buchali et al., “Rate Adaptation and Reach Increase by Probabilistically Shaped 64-QAM: An Experimental Demonstration”, Journal of Lightwave Technology 34, 1599 (2016) DOI
[169]
International Telecommunication Union-T, Recommendation V.29: 9600 Bits Per Second Modem Standardized For Use on Point-to-Point 4-Wire Leased Telephone-Tpe Circuits, 1993
[170]
M. Hopkins and T.-C. Lin, “Explicit Lower Bounds Against \(Ω(n)\)-Rounds of Sum-of-Squares”, (2022) arXiv:2204.11469
[171]
A. Anshu, N. P. Breuckmann, and C. Nirkhe, “NLTS Hamiltonians from Good Quantum Codes”, Proceedings of the 55th Annual ACM Symposium on Theory of Computing (2023) arXiv:2206.13228 DOI
[172]
M. H. Freedman and M. B. Hastings, “Quantum Systems on Non-\(k\)-Hyperfinite Complexes: A Generalization of Classical Statistical Mechanics on Expander Graphs”, (2013) arXiv:1301.1363
[173]
P. A. Mishchenko and K. Xagawa, “Secure multi-party quantum computation based on triply-even quantum error-correcting codes”, (2022) arXiv:2206.04871
[174]
B. Douçot et al., “Protected qubits and Chern-Simons theories in Josephson junction arrays”, Physical Review B 71, (2005) arXiv:cond-mat/0403712 DOI
[175]
B. Douçot and L. B. Ioffe, “Physical implementation of protected qubits”, Reports on Progress in Physics 75, 072001 (2012) DOI
[176]
S. Gladchenko et al., “Superconducting nanocircuits for topologically protected qubits”, Nature Physics 5, 48 (2008) arXiv:0802.2295 DOI
[177]
D. G. Cory et al., “Experimental Quantum Error Correction”, Physical Review Letters 81, 2152 (1998) arXiv:quant-ph/9802018 DOI
[178]
O. Moussa et al., “Demonstration of Sufficient Control for Two Rounds of Quantum Error Correction in a Solid State Ensemble Quantum Information Processor”, Physical Review Letters 107, (2011) arXiv:1108.4842 DOI
[179]
J. Zhang et al., “Experimental quantum error correction with high fidelity”, Physical Review A 84, (2011) arXiv:1109.4821 DOI
[180]
J. Chiaverini et al., “Realization of quantum error correction”, Nature 432, 602 (2004) DOI
[181]
P. Schindler et al., “Experimental Repetitive Quantum Error Correction”, Science 332, 1059 (2011) DOI
[182]
S. A. Moses et al., “A Race-Track Trapped-Ion Quantum Processor”, Physical Review X 13, (2023) arXiv:2305.03828 DOI
[183]
L. Postler et al., “Demonstration of fault-tolerant Steane quantum error correction”, (2023) arXiv:2312.09745
[184]
L. DiCarlo et al., “Preparation and measurement of three-qubit entanglement in a superconducting circuit”, Nature 467, 574 (2010) arXiv:1004.4324 DOI
[185]
M. D. Reed et al., “Realization of three-qubit quantum error correction with superconducting circuits”, Nature 482, 382 (2012) arXiv:1109.4948 DOI
[186]
D. Ristè et al., “Detecting bit-flip errors in a logical qubit using stabilizer measurements”, Nature Communications 6, (2015) arXiv:1411.5542 DOI
[187]
J. Cramer et al., “Repeated quantum error correction on a continuously encoded qubit by real-time feedback”, Nature Communications 7, (2016) arXiv:1508.01388 DOI
[188]
J. R. Wootton and D. Loss, “Repetition code of 15 qubits”, Physical Review A 97, (2018) arXiv:1709.00990 DOI
[189]
J. R. Wootton, “Benchmarking near-term devices with quantum error correction”, Quantum Science and Technology 5, 044004 (2020) arXiv:2004.11037 DOI
[190]
J. Kelly et al., “State preservation by repetitive error detection in a superconducting quantum circuit”, Nature 519, 66 (2015) arXiv:1411.7403 DOI
[191]
“Exponential suppression of bit or phase errors with cyclic error correction”, Nature 595, 383 (2021) arXiv:2102.06132 DOI
[192]
R. Acharya et al., “Suppressing quantum errors by scaling a surface code logical qubit”, (2022) arXiv:2207.06431
[193]
K. C. Miao et al., “Overcoming leakage in quantum error correction”, Nature Physics 19, 1780 (2023) arXiv:2211.04728 DOI
[194]
R. Acharya et al., “Quantum error correction below the surface code threshold”, (2024) arXiv:2408.13687
[195]
W. P. Livingston et al., “Experimental demonstration of continuous quantum error correction”, Nature Communications 13, (2022) arXiv:2107.11398 DOI
[196]
K. Takeda et al., “Quantum error correction with silicon spin qubits”, Nature 608, 682 (2022) arXiv:2201.08581 DOI
[197]
F. van Riggelen et al., “Phase flip code with semiconductor spin qubits”, (2022) arXiv:2202.11530
[198]
G. Waldherr et al., “Quantum error correction in a solid-state hybrid spin register”, Nature 506, 204 (2014) arXiv:1309.6424 DOI
[199]
T. Nakazato et al., “Quantum error correction of spin quantum memories in diamond under a zero magnetic field”, Communications Physics 5, (2022) DOI
[200]
T. H. Taminiau et al., “Universal control and error correction in multi-qubit spin registers in diamond”, Nature Nanotechnology 9, 171 (2014) arXiv:1309.5452 DOI
[201]
T. Richardson and S. Kudekar, “Design of Low-Density Parity Check Codes for 5G New Radio”, IEEE Communications Magazine 56, 28 (2018) DOI
[202]
M. V. Patil, S. Pawar, and Z. Saquib, “Coding Techniques for 5G Networks: A Review”, 2020 3rd International Conference on Communication System, Computing and IT Applications (CSCITA) (2020) DOI
[203]
LDPC coding for OFDMA PHY. 802.16REVe Sponsor Ballot Recirculation comment, July 2004. IEEE C802.16e04/141r2
[204]
T. Brack et al., “A Synthesizable IP Core for WIMAX 802.16E LDPC Code Decoding”, 2006 IEEE 17th International Symposium on Personal, Indoor and Mobile Radio Communications (2006) DOI
[205]
G. Falcão et al., “High coded data rate and multicodeword WiMAX LDPC decoding on Cell/BE”, Electronics Letters 44, 1415 (2008) DOI
[206]
Xiao Han, Kai Niu, and Zhiqiang He, “Implementation of IEEE 802.11n LDPC codes based on general purpose processors”, 2013 15th IEEE International Conference on Communication Technology (2013) DOI
[207]
W. Zhang et al., “A full layer parallel QC-LDPC decoder for WiMAX and Wi-Fi”, 2015 IEEE 11th International Conference on ASIC (ASICON) (2015) DOI
[208]
G. Alagic et al., “Quantum Fully Homomorphic Encryption with Verification”, Advances in Cryptology – ASIACRYPT 2017 438 (2017) arXiv:1708.09156 DOI
[209]
A. Coladangelo et al., “Hidden Cosets and Applications to Unclonable Cryptography”, (2022) arXiv:2107.05692
[210]
R. J. Chapman et al., “Beating the classical limits of information transmission using a quantum decoder”, Physical Review A 97, (2018) arXiv:1704.07036 DOI
[211]
Yunfeng Lin, Ben Liang, and Baochun Li, “Priority Random Linear Codes in Distributed Storage Systems”, IEEE Transactions on Parallel and Distributed Systems 20, 1653 (2009) DOI
[212]
O. Regev, “On lattices, learning with errors, random linear codes, and cryptography”, Journal of the ACM 56, 1 (2009) DOI
[213]
H. Chen et al., “Secure Computation from Random Error Correcting Codes”, Advances in Cryptology - EUROCRYPT 2007 291 (2007) DOI
[214]
P. Gaborit et al., “Identity-Based Encryption from Codes with Rank Metric”, Advances in Cryptology – CRYPTO 2017 194 (2017) DOI
[215]
P. Lefèvre, P. Carré, and P. Gaborit, “Application of rank metric codes in digital image watermarking”, Signal Processing: Image Communication 74, 119 (2019) DOI
[216]
D. Silva and F. R. Kschischang, “Rank-Metric Codes for Priority Encoding Transmission with Network Coding”, 2007 10th Canadian Workshop on Information Theory (CWIT) (2007) DOI
[217]
Anxiao Jiang et al., “Rank Modulation for Flash Memories”, IEEE Transactions on Information Theory 55, 2659 (2009) DOI
[218]
J. L. Massey, “Deep-space communications and coding: A marriage made in heaven”, Advanced Methods for Satellite and Deep Space Communications 1 DOI
[219]
E.C. Posner, Combinatorial Structures in Planetary Reconnaissance in Error Correcting Codes, ed. H.B. Mann, Wiley, NY 1968.
[220]
D. Zhang, K. Ho-Van, and T. Le-Ngoc, “Impulse noise detection techniques for retransmission to reduce delay in DSL systems”, 2012 IEEE International Conference on Communications (ICC) (2012) DOI
[221]
A. Kiayias and M. Yung, “Cryptographic Hardness Based on the Decoding of Reed-Solomon Codes”, Automata, Languages and Programming 232 (2002) DOI
[222]
Anvin, H. Peter. "The mathematics of RAID-6." (2007).
[223]
S. T. Position. (2009) Common raid disk data format specification. [Online]. Available: http://www.snia.org/tech activities/standards/curr standards/ddf
[224]
S. Li et al., “PolyShard: Coded Sharding Achieves Linearly Scaling Efficiency and Security Simultaneously”, (2020) arXiv:1809.10361
[225]
International Organization for Standardization, Information Technology: Automatic Identification and Data Capture Techniques-QR Code 2005 Bar Code Symbology Specification, 2nd ed., IEC18004 (ISO, 2006).
[226]
I. Shakeel et al., “Reed-Solomon coding for cooperative wireless communication”, 21st Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (2010) DOI
[227]
N. Shental et al., “Efficient high-throughput SARS-CoV-2 testing to detect asymptomatic carriers”, Science Advances 6, (2020) DOI
[228]
R. N. Grass et al., “Robust Chemical Preservation of Digital Information on DNA in Silica with Error‐Correcting Codes”, Angewandte Chemie International Edition 54, 2552 (2015) DOI
[229]
K. Karplus and H. Krit, “A semi-systolic decoder for the PDSC-73 error-correcting code”, Discrete Applied Mathematics 33, 109 (1991) DOI
[230]
T. Klove and M. Miller, “The Detection of Errors After Error-Correction Decoding”, IEEE Transactions on Communications 32, 511 (1984) DOI
[231]
“High-Performance Embedded Computing”, (2014) DOI
[232]
H. Janwa and O. Moreno, Designs, Codes and Cryptography 8, 293 (1996) DOI
[233]
A. Couvreur, H. Randriambololona, "Algebraic Geometry Codes and Some Applications." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[234]
H. Chen and R. Cramer, “Algebraic Geometric Secret Sharing Schemes and Secure Multi-Party Computations over Small Fields”, Lecture Notes in Computer Science 521 (2006) DOI
[235]
R. Vidhya Lavanya and M. Madheswaran, “Wimax (IEEE 802.16 m) system based on space time block code and discrete multiwavelet transform and implementation in FPGA”, Telecommunication Systems 56, 327 (2013) DOI
[236]
S. P. Alex and L. M. A. Jalloul, “Performance Evaluation of MIMO in IEEE802.16e/WiMAX”, IEEE Journal of Selected Topics in Signal Processing 2, 181 (2008) DOI
[237]
“Advances in smart antennas - MIMO-OFDM wireless systems: basics, perspectives, and challenges”, IEEE Wireless Communications 13, 31 (2006) DOI
[238]
C. Flühmann et al., “Encoding a qubit in a trapped-ion mechanical oscillator”, Nature 566, 513 (2019) arXiv:1807.01033 DOI
[239]
C. Flühmann and J. P. Home, “Direct Characteristic-Function Tomography of Quantum States of the Trapped-Ion Motional Oscillator”, Physical Review Letters 125, (2020) arXiv:1907.06478 DOI
[240]
B. de Neeve et al., “Error correction of a logical grid state qubit by dissipative pumping”, (2020) arXiv:2010.09681
[241]
V. G. Matsos et al., “Universal Quantum Gate Set for Gottesman-Kitaev-Preskill Logical Qubits”, (2024) arXiv:2409.05455
[242]
D. Lachance-Quirion et al., “Autonomous quantum error correction of Gottesman-Kitaev-Preskill states”, (2023) arXiv:2310.11400
[243]
S. Konno et al., “Logical states for fault-tolerant quantum computation with propagating light”, Science 383, 289 (2024) arXiv:2309.02306 DOI
[244]
N. Fabre et al., “Generation of a time-frequency grid state with integrated biphoton frequency combs”, Physical Review A 102, (2020) arXiv:1904.01351 DOI
[245]
É. Descamps, A. Keller, and P. Milman, “Gottesman-Kitaev-Preskill encoding in continuous modal variables of single photons”, (2024) arXiv:2310.12618
[246]
X. Pan et al., “Protecting the quantum interference of cat states by phase-space compression”, (2022) arXiv:2212.01271
[247]
F. R. Kschischang, "Network Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[248]
K. M. Schreiter et al., “Optical implementation of a unitarily correctable code”, Physical Review A 80, (2009) arXiv:0909.1584 DOI
[249]
S. Krinner et al., “Realizing repeated quantum error correction in a distance-three surface code”, Nature 605, 669 (2022) arXiv:2112.03708 DOI
[250]
Y. Zhao et al., “Realization of an Error-Correcting Surface Code with Superconducting Qubits”, Physical Review Letters 129, (2022) arXiv:2112.13505 DOI
[251]
L. Frunzio and S. Singh, “Error-Correcting Surface Codes Get Experimental Vetting”, Physics 15, (2022) DOI
[252]
Y. Ye et al., “Logical Magic State Preparation with Fidelity Beyond the Distillation Threshold on a Superconducting Quantum Processor”, (2023) arXiv:2305.15972
[253]
P. Chaichanavong and P. H. Siegel, “Tensor-product parity code for magnetic recording”, IEEE Transactions on Magnetics 42, 350 (2006) DOI
[254]
H. Hämäläinen and S. Rankinen, “Upper bounds for football pool problems and mixed covering codes”, Journal of Combinatorial Theory, Series A 56, 84 (1991) DOI
[255]
T. Thompson, From Error-Correcting Codes Through Sphere Packings to Simple Groups (American Mathematical Society, 1983) DOI
[256]
D. Bluvstein et al., “A quantum processor based on coherent transport of entangled atom arrays”, Nature 604, 451 (2022) arXiv:2112.03923 DOI
[257]
J. N. Staddon, D. R. Stinson, and Ruizhong Wei, “Combinatorial properties of frameproof and traceability codes”, IEEE Transactions on Information Theory 47, 1042 (2001) DOI
[258]
Divsalar, Dariush, and Fabrizio Pollara. "Turbo codes for deep-space communications." The Telecommunications and Data Acquisition Report 29-39 (SEE N95-21532 06-32) (1995).
[259]
T. Yamada, “CCSDS Telemetry/Tele-command Standards Restructured as Communications Protocols”, SpaceOps 2002 Conference (2002) DOI
[260]
K. Gracie and M.-H. Hamon, “Turbo and Turbo-Like Codes: Principles and Applications in Telecommunications”, Proceedings of the IEEE 95, 1228 (2007) DOI
[261]
S. Xu et al., “Digital Simulation of Projective Non-Abelian Anyons with 68 Superconducting Qubits”, Chinese Physics Letters 40, 060301 (2023) arXiv:2211.09802 DOI
[262]
S. Burton, E. Durso-Sabina, and N. C. Brown, “Genons, Double Covers and Fault-tolerant Clifford Gates”, (2024) arXiv:2406.09951
[263]
Z. Leghtas et al., “Confining the state of light to a quantum manifold by engineered two-photon loss”, Science 347, 853 (2015) arXiv:1412.4633 DOI
[264]
S. Touzard et al., “Coherent Oscillations inside a Quantum Manifold Stabilized by Dissipation”, Physical Review X 8, (2018) arXiv:1705.02401 DOI
[265]
A. Grimm et al., “Stabilization and operation of a Kerr-cat qubit”, Nature 584, 205 (2020) arXiv:1907.12131 DOI
[266]
R. Lescanne et al., “Exponential suppression of bit-flips in a qubit encoded in an oscillator”, Nature Physics 16, 509 (2020) arXiv:1907.11729 DOI
[267]
C. Berdou et al., “One Hundred Second Bit-Flip Time in a Two-Photon Dissipative Oscillator”, PRX Quantum 4, (2023) arXiv:2204.09128 DOI
[268]
U. Réglade et al., “Quantum control of a cat qubit with bit-flip times exceeding ten seconds”, Nature 629, 778 (2024) arXiv:2307.06617 DOI
[269]
A. Marquet et al., “Autoparametric Resonance Extending the Bit-Flip Time of a Cat Qubit up to 0.3 s”, Physical Review X 14, (2024) arXiv:2307.06761 DOI
[270]
V. V. Albert et al., “Holonomic Quantum Control with Continuous Variable Systems”, Physical Review Letters 116, (2016) arXiv:1503.00194 DOI
[271]
A. Hajr et al., “High-Coherence Kerr-cat qubit in 2D architecture”, (2024) arXiv:2404.16697
[272]
N. E. Frattini et al., “The squeezed Kerr oscillator: spectral kissing and phase-flip robustness”, (2022) arXiv:2209.03934
[273]
J. M. Gertler et al., “Protecting a bosonic qubit with autonomous quantum error correction”, Nature 590, 243 (2021) arXiv:2004.09322 DOI
[274]
S. Kak, “Unary Coding for Neural Network Learning”, (2010) arXiv:1009.4495
[275]
Fiete, I. R., & Seung, H. S. (2007). Neural network models of birdsong production, learning, and coding. New Encyclopedia of Neuroscience. Eds. L. Squire, T. Albright, F. Bloom, F. Gage, and N. Spitzer. Elsevier.
[276]
Z. Li et al., “Autonomous error correction of a single logical qubit using two transmons”, (2023) arXiv:2302.06707
[277]
J. Svigals, “IBM 7070 data processing system”, Papers presented at the the March 3-5, 1959, western joint computer conference on XX - IRE-AIEE-ACM ’59 (Western) (1959) DOI
[278]
T. I. Andersen et al., “Non-Abelian braiding of graph vertices in a superconducting processor”, (2023) arXiv:2210.10255
[279]
D. Bluvstein et al., “Logical quantum processor based on reconfigurable atom arrays”, Nature (2023) arXiv:2312.03982 DOI
[280]
A. Gyenis et al., “Experimental Realization of a Protected Superconducting Circuit Derived from the 0 – π Qubit”, PRX Quantum 2, (2021) arXiv:1910.07542 DOI
[281]
J. Meggitt, “Error correcting codes and their implementation for data transmission systems”, IEEE Transactions on Information Theory 7, 234 (1961) DOI
[282]
C. Campopiano and B. Glazer, “A Coherent Digital Amplitude and Phase Modulation Scheme”, IEEE Transactions on Communications 10, 90 (1962) DOI
[283]
G. Foschini, R. Gitlin, and S. Weinstein, “Optimization of Two-Dimensional Signal Constellations in the Presence of Gaussian Noise”, IEEE Transactions on Communications 22, 28 (1974) DOI
[284]
E. Arikan, “A survey of reed-muller codes from polar coding perspective”, IEEE Information Theory Workshop 2010 (ITW 2010) (2010) DOI
[285]
R. Hentschke et al., “Analyzing area and performance penalty of protecting different digital modules with Hamming code and triple modular redundancy”, Proceedings. 15th Symposium on Integrated Circuits and Systems Design DOI
[286]
Crandall, Ron. "Some notes on steganography." Posted on steganography mailing list 1998 (1998): 1-6.
[287]
A. Westfeld, “F5—A Steganographic Algorithm”, Information Hiding 289 (2001) DOI
[288]
I. Pogorelov et al., “Experimental fault-tolerant code switching”, (2024) arXiv:2403.13732
[289]
M. P. da Silva et al., “Demonstration of logical qubits and repeated error correction with better-than-physical error rates”, (2024) arXiv:2404.02280
[290]
B. W. Reichardt et al., “Demonstration of quantum computation and error correction with a tesseract code”, (2024) arXiv:2409.04628
[291]
C. N. Self, M. Benedetti, and D. Amaro, “Protecting expressive circuits with a quantum error detection code”, Nature Physics (2024) arXiv:2211.06703 DOI
[292]
C.-Y. Lu et al., “Experimental quantum coding against qubit loss error”, Proceedings of the National Academy of Sciences 105, 11050 (2008) arXiv:0804.2268 DOI
[293]
B. A. Bell et al., “Experimental demonstration of a graph state quantum error-correction code”, Nature Communications 5, (2014) arXiv:1404.5498 DOI
[294]
N. M. Linke et al., “Fault-tolerant quantum error detection”, Science Advances 3, (2017) arXiv:1611.06946 DOI
[295]
Quantum Information and Computation 18, (2018) arXiv:1705.08957 DOI
[296]
R. Harper and S. T. Flammia, “Fault-Tolerant Logical Gates in the IBM Quantum Experience”, Physical Review Letters 122, (2019) arXiv:1806.02359 DOI
[297]
C. K. Andersen et al., “Repeated quantum error detection in a surface code”, Nature Physics 16, 875 (2020) arXiv:1912.09410 DOI
[298]
A. Erhard et al., “Entangling logical qubits with lattice surgery”, Nature 589, 220 (2021) arXiv:2006.03071 DOI
[299]
J. F. Marques et al., “Logical-qubit operations in an error-detecting surface code”, Nature Physics 18, 80 (2021) arXiv:2102.13071 DOI
[300]
R. S. Gupta et al., “Encoding a magic state with beyond break-even fidelity”, Nature 625, 259 (2024) arXiv:2305.13581 DOI
[301]
J. Roffe et al., “Protecting quantum memories using coherent parity check codes”, Quantum Science and Technology 3, 035010 (2018) arXiv:1709.01866 DOI
[302]
K. Yamamoto et al., “Demonstrating Bayesian quantum phase estimation with quantum error detection”, Physical Review Research 6, (2024) arXiv:2306.16608 DOI
[303]
D. Nigg et al., “Quantum computations on a topologically encoded qubit”, Science 345, 302 (2014) arXiv:1403.5426 DOI
[304]
C. Ryan-Anderson et al., “Realization of real-time fault-tolerant quantum error correction”, (2021) arXiv:2107.07505
[305]
P. Ball, “Real-Time Error Correction for Quantum Computing”, Physics 14, (2021) DOI
[306]
L. Postler et al., “Demonstration of fault-tolerant universal quantum gate operations”, Nature 605, 675 (2022) arXiv:2111.12654 DOI
[307]
K. Mayer et al., “Benchmarking logical three-qubit quantum Fourier transform encoded in the Steane code on a trapped-ion quantum computer”, (2024) arXiv:2404.08616
[308]
C. Ryan-Anderson et al., “High-fidelity and Fault-tolerant Teleportation of a Logical Qubit using Transversal Gates and Lattice Surgery on a Trapped-ion Quantum Computer”, (2024) arXiv:2404.16728
[309]
D. Miller et al., “Experimental measurement and a physical interpretation of quantum shadow enumerators”, (2024) arXiv:2408.16914
[310]
Y. Wang et al., “Fault-tolerant one-bit addition with the smallest interesting color code”, Science Advances 10, (2024) arXiv:2309.09893 DOI
[311]
D. Honciuc Menendez, A. Ray, and M. Vasmer, “Implementing fault-tolerant non-Clifford gates using the [[8,3,2]] color code”, Physical Review A 109, (2024) arXiv:2309.08663 DOI
[312]
D. M. Debroy et al., “Optimizing Stabilizer Parities for Improved Logical Qubit Memories”, Physical Review Letters 127, (2021) arXiv:2105.05068 DOI
[313]
Y.-H. Luo et al., “Quantum teleportation of physical qubits into logical code spaces”, Proceedings of the National Academy of Sciences 118, (2021) arXiv:2009.06242 DOI
[314]
R. Zhang et al., “Loss-tolerant all-photonic quantum repeater with generalized Shor code”, Optica 9, 152 (2022) arXiv:2203.07979 DOI
[315]
T. Aoki et al., “Quantum error correction beyond qubits”, (2008) arXiv:0811.3734