Here is a list of all classical and quantum codes that have been realized in experimental or real-world devices.
Name | Realization(s) |
---|---|
120-cell code | Improved proofs of the Bell-Kochen-Specker (BKS) theorem [1]. |
600-cell code | Improved proofs of the Bell-Kochen-Specker (BKS) theorem [2]. |
Alamouti code | Wireless standards since: 3G, LTE, LTE-Advanced, and 5G.Wireless communication: IEEE 802.11n, IEEE 802.11ad, IEEE 802.11ay, etc. |
Analog stabilizer code | One-sided device-independent QKD [3]. |
Array-based LDPC (AB-LDPC) code | Certain AB-LDPC codes have been proposed to be used for DSL transmission [4]. |
BPSK c-q code | Linear-optical quantum receiver [5].Homodyne receiver [6].Kennedy receiver [6,7].Photon-number resolving detector [8].Communication over dephasing [9], time-varying phase-noise [10], and thermal-noise [11] channels.Adaptive decoder using displacements and photon detection [12].Superconducting circuits: a bit-flip time of 1s has been achieved for the two-legged cat code in the classical-bit regime [13]. |
Bacon-Shor code | Trapped-ion qubits: state preparation, logical measurement, and stabilizer measurement for nine-qubit Bacon-Shor code demonstrated on a 13-qubit device by M. Cetina and C. Monroe groups [14]. |
Balanced code | Balanced length-eight code, known as a 6b/8b encoding, used for balancing direct current in a communications system [15] |
Binary BCH code | Satellite communication [16] |
Binary PSK (BPSK) code | Telephone-line modems throughout 1950s and 1960s: Bell 103 and 202, as well as international standards V.21 [17] and V.23 [18]. |
Binomial code | Microwave cavities coupled to superconducting circuits: state transfer between a binomial codeword to another system [19], error-correction protocol nearly reaching break-even [20], and a teleported CNOT gate [21]. 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 \(16\%\) [22] (see also [23]). See Ref. [24] for another experiment.Motional degree of freedom of a trapped ion: binomial state preparation for \(S=2\) realized by Tan group [25]. |
Bose–Chaudhuri–Hocquenghem (BCH) code | DVDs, disk drives, and two-dimensional bar codes [26]. |
Cat code | Parity-syndrome measurement tested [27] and implemented for the four-component (\(S=1\)) cat code [28] in a microwave cavity coupled to a superconducting circuit. The latter work [28] is the first to reach break-even error-correction, where the lifetime of a logical qubit is on par with the cavity lifetime, despite protection against dephasing not being implemented. A fault-tolerant version of parity measurement has also been realized [29]. |
Classical Goppa code | Initial version of the McEliece public-key cryptosystem [30,31] and its variation by Niederreiter [32] where the generator matrix is replaced by the parity check matrix. Some of these were proven to be insecure since the public key exposes algebraic structure of code [33]. |
Coherent FSK (CFSK) c-q code | Time-resolving quantum receiver [34].Bondurant receiver [35].Bayesian inference [36]. |
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 Reed-Solomon 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]. |
Cycle LDPC code | Cycle LDPC codes have been proposed to be used for MIMO channels [47]. |
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 [48]. 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 [49], which is the same topological order as the \(G=D_4\) quantum double [50,51]. |
Dual-rail quantum code | The dual-rail code is ubiquitous in linear optical quantum devices [52].Superconducting circuit devices: error detection has been demonstrated in 3D cavities in Michel Devoret's group at Yale University [53] and Amazon Web Services [54] using transmon qubits, following earlier theoretical proposals [55,56]. Logical readout in 3D cavities has been demonstrated by Quantum Circuits Inc. [57]. |
Fibonacci string-net code | NMR: Implementation of braiding-based Hamamard gate on two qubits [58]. |
Five-qubit perfect code | NMR: Implementation of perfect error correcting code on 5 spin subsystem of labeled crotonic acid for quantum network benchmarking [59]. Single-qubit logical gates [60]. Magic-state distillation using 7-qubit device [61].Superconducting qubits [62].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 [63]. Real-time magic-state distillation [64].Nitrogen-vacancy centers in diamond: fault-tolerant single-qubit Clifford operations [65]. |
Floquet color code | Plaquette stabilizer measurement realized on the IBM Falcon superconducting-qubit device [66] |
Fountain code | Designed for servers sending data to many recipients, such as during broadcasting or file distribution [67,68]. |
Gabidulin code | Public-key cryptosystems [69,70].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 [71]. |
Generalized RS (GRS) code | Commonly used in mass storage systems such as CDs, DVDs, QR codes etc.Various cloud storage systems [72].Public-key cryptosystems generalizing those that used Goppa codes [30–32], some of which were proven to be insecure [33]. More recent works focus on methods to mask the algebraic structure using subcodes of GRS codes [69]. For example, a key-recovery attack was developed in Ref. [73] for a variant of masking method proposed in Ref. [74]. |
Generalized Shor code | The \([[m^2,1,m]]\) codes for \(m\leq 7\) have been realized in trapped-ion quantum devices [75]. |
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 [76].Extended Golay code used in American military standards for automatic link establishment in high frequency radio systems [77].Proofs of the quantum mechanical Kochen-Specker theorem [78]. |
Gold code | Used in for synchronization purposes in telecommunication [79]GPS C/A for satellite navigation [80]. |
Gray code | Three-dimensional imaging [81].Broadcasting and communication [82]. |
Group GKP code | Cryptographic applications stemming from the monogamy of entanglement of group GKP code and error words [3]. |
Hamming code | Commonly used when error rates are very low, for example, computer RAM or integrated circuits [83].Hamming-code based matrix embedding used in steganography [84,85]. |
Heavy-hexagon code | Logical state preparation and flag-qubit error correction realized in superconducting-circuit devices (specifically, fixed-frequency transmon qubit architectures) by IBM for \(d=2\) [86,87] and \(d=3\) [88]. |
Hessian polyhedron code | Quantum mechanical SIC-POVMs [89]. |
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 [90]. |
Honeycomb Floquet code | Plaquette stabilizer measurement realized on the IBM Falcon superconducting-qubit device [66] |
Interleaved RS (IRS) code | The cross-interleaved RS (CIRC), an IRS code using two shortened RS codes and two forms of interleaving, was used for compact discs (CDs) [91] (see Ref. [92], Sec. 5.6 and Ref. [93], Ch. 4). |
Irregular LDPC code | Satellite communication after concatenating with a modulation scheme [94]. |
Irregular repeat-accumulate (IRA) code | LDPC codes are used for digital satellite video broadcasting per the DVB-S2 standard [95,96] utilize IRA code features and are subject to ongoing litigation; see Ref. [97].Apple and Broadcom Wi-Fi devices utilize IRA encoding and decoding code features and are subject to ongoing litigation; see Ref. [97]. |
Justesen code | Generating small-bias sample spaces, i.e., probability distributions that parity functions cannot typically distinguish from the uniform distribution [98]. |
Kerdock code | Digital fingerprinting: \(t\)-collision secure schemes can be designed to detect a pirated copy once \(t\) users have colluded [99]. |
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 [100].Superconducting circuits: initialization [101], braiding [102] and detection [102,103] of defects has been simulated in devices that have a different notion of locality than a bona-fide fermionic system. |
Kitaev surface code | One cycle of syndrome readout on 19-qubit planar and 24-qubit toric codes realized in two-dimensional Rydberg atomic arrays [104]. Signatures of corresponding topological phase of matter detected in superconducting circuits [105] and two-dimensional Rydberg atomic arrays [106]. 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 [107]. |
Lloyd-Slotine nine-mode code | Optical network by the Furusawa group [108]. |
Locally recoverable code (LRC) | An \((18,14,7)\) LRC code has beed used in the Windows Azure cloud storage system [109]; see also [110; 31.3.1.2]. |
Low-rank parity-check (LRPC) code | Cryptosystem [111] that is a rank-metric analogue of NTRU [112] and MDPC [113] cryptosystems.Post-quantum cryptography [114]. |
Matching code | Braiding of defects has been demonstrated for a five-qubit version of code [115]. |
Maximum distance separable (MDS) code | The McEliece Public Key Cryptosystem [116] uses algebraic coding theory to secure communications against eavesdropping attack, in which private keys are generator matrices of linear codes, i.e., \(G\). Public Keys shared to outside world are scrambled and permutated versions of \(G\), i.e., \(G^\prime=SGP\). Data to be encrypted, \(u\), is multiplied by public key and added intentional errors \(e\), i.e., \(x=uG^\prime+e\). Upon receiving encrypted data, private key owner can apply inverse permutation \(P^{-1}\) to \(x\), decode the scrambled message given the presence of \(e\) errors, and finally unscramble to obtain \(u\). Security parameters of the system are \(n\) and \(e\), hence MDS codes, such as GRS codes may provide the same security level for shorter key sizes, and the private key owner can decode arguably fast enough using known decoding algorithms.Automatic repeat request (ARQ) data transmission protocols ([93], Ch. 7). |
Maximum-rank distance (MRD) code | Useful for error and erasure correction in network coding [117,118]. |
Monitored random-circuit code | Measurement induced quantum phases have been realized in a trapped-ion processor [119]. |
NTRU-GKP code | Public-key quantum communication protocol [120]. |
Niset-Andersen-Cerf code | Realized in Ref. [121] 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 [122] and TES receivers [123]. |
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 ([124], Ch. 27).Marking the state of a finite automaton [125].Used in machine-learning based classification tasks because one-hot encodings, as opposed to integer encodings, do not presume an order [126]. |
PPM c-q code | Conditional pulse nulling (CPN) receiver [127]. |
PSK c-q code | Unambiguous state discrimination using displacement-based receiver for 4-PSK [128].Multi-stage quantum receivers [129–132].Bayesian inference [133].Time resolving quantum receiver opertaing in the telecom C band [134].Displacements and photon detection [135].Adaptive decoder using linear-optical elements and photon detection [12]. |
Pair-cat code | Microwave cavities coupled to superconducting circuits by the Wang group [136]. |
Phase-shift keying (PSK) code | Telephone-line modems: 1967 Milgo 4400/48 and international standard V.27 used 8-PSK [137]. |
Polar code | Code control channels for the 5G NR (New Radio) interfaces [138]. |
Pulse-position modulation (PPM) code | Greek hydraulic semaphore system [139,140].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 [141].Telephone-line modems: 1962 Bell 201 and international standard V.24 [142]. |
Quadrature-amplitude modulation (QAM) code | Optical communication (e.g., Ref. [143]).Telephone-line modems: 1971 Codex 9600C and international standard V.29 used 16-QAM [144]. |
Quantum Tanner code | Used to obtain explicit lower bounds in the sum-of-squares game [145].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 [146] of the No low-energy trivial states (NLTS) conjecture [147]. |
Quantum parity code (QPC) | The \([[m^2,1,m]]\) codes for \(m\leq 7\) have been realized in trapped-ion quantum devices [75]. |
Quantum repetition code | NMR: 3-qubit phase-flip code [148,149], with up to two rounds of error correction in liquid-state NMR [150].Superconducting circuits: 3-qubit phase-flip and bit-flip code by Schoelkopf group [151]; 3-qubit bit-flip code [152]; 3-qubit phase-flip code up to 3 cycles of error correction [153]; IBM 15-qubit device [154]; IBM Rochester device using 43-qubit code [155]; Google system performing up to 8 error-correction cycles on 5 and 9 qubits [156]; Google Quantum AI Sycamore utilizing up to 11 physical qubits and running 50 correction rounds [157]; Google Quantum AI Sycamore utilizing up to 25 qubits for comparison of logical error scaling with a quantum code [158] (see also [159]).Continuous error correction protocols have been implemented on a 3-qubit superconducting qubit device [160].Semiconductor spin-qubit devices: 3-qubit devices at RIKEN [161] and Delft [162].Nitrogen-vacancy centers in diamond: 3-qubit phase-flip code [163,164] (see also Ref. [165]).Trapped-ion device: 3-qubit phase-flip algorithm implemented in 3 cycles on high fidelity gate operations [166]. Both phase- and bit-flip codes for 31 qubits and their stabilizer measurements have been realized by Quantinuum [167]. |
Quasi-cyclic LDPC (QC-LDPC) code | 5G NR cellular communication for the traffic channel [168,169].Wireless communication: WiMAX (IEEE 802.16e) [170–172], WiFi 4 (IEEE 802.11n) [173], and WPAN (IEEE 802.15.3c) [174]. |
Qubit CSS code | Fully homomorphic encryption [175].Entanglement purification protocols related to quantum key distribution (QKD) [176].Cryptographic applications stemming from the monogamy of entanglement of CSS code and error words [177]. |
Qubit c-q code | Quantum enhancement was demonstrated using a polarization-based non-error-correcting c-q encodings [178]. |
Random code | Distributed storage systems [179].Classical and quantum cryptography based on the learning-with-errors problem, which is related to decoding a random linear code [180]. |
Rank-metric code | Identity-Based Encryption [181].Digital watermarking [182].Network coding and streaming media broadcasting [183]. |
Rank-modulation Gray code (RMGC) | Electronic devices where charges can either increase in an individual cell or decrease in a block of adjacent cells, e.g., flash memories [184]. |
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 [185,186]. For example, the \((32, 6, 16)\) RM code was used for the Mariner 9 spacecraft. |
Reed-Solomon (RS) code | RS Product Code (RSPC) was used in DVDs (see Ref. [93], Ch. 4).DSL technologies and their variants against impluse noise [187].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 [188].RS codes as outer codes concatenated with convolutional codes are used indirectly in space exploration programs such as Voyager and Galileo. RS codes were part of a temetry channel coding standard issued by the Consultative Committee for Space Data Systems (see Ref. [93], Ch. 3).The ubiquity of RS codes has yielded off-the-shelf VLSI intergrated-circuit decoding hardware [189] (see also Ref. [93], Ch. 5 and 10).Automatic repeat request (ARQ) data transmission protocols (see Ref. [93], Ch. 7).Slow-frequency-hop spread-spectrum transmission (see Ref. [93], Chs. 8-9).Coded sharding designs in blockchains to increase efficiency [190].Private Information Retrieval [191].Used in QR-Codes to retrieve damaged barcodes [192].Wireless communication systems such as 3G, DVB, and WiMAX [193].Correcting pooled testing results for SARS-CoV-2 [194]. |
Regular binary Tanner code | First hardware implementation was done using a semi-systolic decoding architecture [195]. |
Repetition code | Repetition codes, in conjunction with other codes, were used in magnetic disks [196].Communication protocol such as FlexRay [197] is using repetition code' |
Residue AG code | Improvements over the McEliece public-key cryptosystem to linear AG codes on curves of arbitrary genus [198]. Only the subfield subcode proposal remains resilient to attacks [199; Sec. 15.7.5.3].Algebraic secret-sharing schemes [200]. |
Single parity-check (SPC) code | Can be realized on almost every communication device. SPCs are some of the earliest error-correcting codes ([124], 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) [201–203]. |
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 [204,205], 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 [206]. State preparation also realized by Tan group [25].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 [90]. Subsequent paper by Devoret group [23] (see also [207]) 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 [22]). See Ref. [24] for another experiment.GKP states and homodyne measurements have been realized in propagating telecom light by the Furusawa group [208].Single-qubit \(Z\)-gate has been demonstrated in the single-photon subspace of an infinite-mode space [209], in which time and frequency become bosonic conjugate variables of a single effective bosonic mode.In signal processing, GKP state position-state wavefunctions are related to Dirac combs [210]. |
Surface-17 code | Implemented at ETH Zurich by the Wallraff group [211] and on the Zuchongzhi 2.1 superconducting quantum processor [212]. Both experimental error rates are above the pseudothreshold for this code relative to a single qubit; see Physics viewpoint for a summary [213]. Magic state have been created on the latter processor [214]. |
Tensor-product code | Construction can be used in magnetic recording by taking the tensor product of a Reed-Solomon code and a parity-check code [215]. |
Ternary Golay code | Code used in football pools with at least one good bet [46,216]. In fact, the code was originally constructed by Juhani Virtakallio and published in the Finnish football pool magazine Veikkaaja [46,217].Proofs of the quantum mechanical Kochen-Specker theorem [78]. |
Two-component cat code | Lindbladian-based [218,219] and Hamiltonian-based 'Kerr-cat' [220] encodings have been achieved in superconducting circuit devices by the Devoret group; Ref. [219] 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 [221] and later 10 seconds [222], with the latter scheme repurposing a holonomic gate [223] as a measurement.T4C code realized in a superconducting circuit device by the Wang group [224]. |
Unary code | Neural networks [225].Birdsong production [226]. |
Very small logical qubit (VSLQ) code | Star-code autonomous correction scheme realized using superconducting circuits [227]. |
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 [217,228]. |
XZZX surface code | Superconducting circuits: Distance-five 25-qubit code implemented on a superconducting quantum processor by Google Quantum AI [158]. 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 [229]. Leakage errors have been handled in a separate work in a distance-three code [159]. |
Zero-pi qubit code | A related superconducting circuit has been realized by the Houck group [230]. |
Zetterberg code | Code used to provide better protection of data transmission with its double error correcting capacity [231]. |
\(A_2\) hexagonal lattice code | Wireless communication [232,233]. |
\([[2m,2m-2,2]]\) error-detecting code | Trapped-ion devices: 12-qubit device by Quantinuum [234]. Subsequent experiment performed Bayesian Quantum Phase Estimation on the \(m=3\) code [235]. |
\([[4,2,2]]\) CSS code | \([[4,1,2]]\) subcode implemented using four-qubit graph state of photons [236].Trapped-ion device by IonQ [237].Logical state preparation and flag-qubit error correction realized in superconducting-circuit devices by IBM [86,87,238,239].The subcode \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [240] and \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) [157], 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 [241].Logical gates for the \(\{|\overline{00}\rangle,|\overline{11}\rangle\}\) subcode, treated as a planar code, realized in superconducting circuits [242].The CZ magic state has been realized on an IBM heavy-hex superconducting circuit device [243]. |
\([[7,1,3]]\) Steane code | Trapped-ion qubits: seven-qubit device in Blatt group [244], ten-qubit QCCD device by Quantinuum [245] realizing repeated rounds of error correction, real-time look-up-table decoding, and non-fault-tolerant magic-state distillations (see APS Physics Synopsis [246]). Fault-tolerant universal two-qubit gate set by Monz group [247]. Logical CNOT gate between two logical qubits, including rounds of correction and fault-tolerant primitives such as flag qubits and pieceable fault tolerance, on a 20-qubit device by Quantinuum [63]; logical fidelity interval of the combined preparation-CNOT-measurement procedure was higher than that of the unencoded physical qubits.Rydberg atom arrays: Lukin group [104]; transversal CNOT gate performed on distance \(3\), \(5\), and \(7\) codes, logical ten-qubit GHZ state initialized with break-even fidelity, fault-tolerant logical two-qubit GHZ state initialized [248]. |
\([[8,3,2]]\) CSS code | Trapped ions: one-qubit addition algorithm implemented fault-tolerantly on the Quantinuum H1-1 device [249].Superconducting circuits: fault-tolerant \(CZZ\) gate performed on IBM and IonQ devices [250].Rydberg atom arrays: Lukin group [251]. 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 [248]. |
\([[9,1,3]]\) 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 [75]. Variants of the code to handle coherent noise studied and realized by K. Brown and C. Monroe groups [252].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% [253]. All-photonic quantum repeater architecture tested on the same code [254]. |
References
- [1]
- M. Waegell and P. K. Aravind, “Parity Proofs of the Kochen–Specker Theorem Based on the 120-Cell”, Foundations of Physics 44, 1085 (2014) arXiv:1309.7530 DOI
- [2]
- M. Waegell and P. K. Aravind, “Critical noncolorings of the 600-cell proving the Bell–Kochen–Specker theorem”, Journal of Physics A: Mathematical and Theoretical 43, 105304 (2010) arXiv:0911.2289 DOI
- [3]
- 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
- [4]
- 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
- [5]
- K. Tsujino et al., “Quantum Receiver beyond the Standard Quantum Limit of Coherent Optical Communication”, Physical Review Letters 106, (2011) arXiv:1103.5592 DOI
- [6]
- C. Wittmann et al., “Demonstration of Near-Optimal Discrimination of Optical Coherent States”, Physical Review Letters 101, (2008) arXiv:0809.4953 DOI
- [7]
- M. L. Shcherbatenko et al., “Sub-shot-noise-limited fiber-optic quantum receiver”, Physical Review A 101, (2020) arXiv:1911.08932 DOI
- [8]
- 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
- [9]
- 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
- [10]
- M. T. DiMario and F. E. Becerra, “Phase tracking for sub-shot-noise-limited receivers”, Physical Review Research 2, (2020) DOI
- [11]
- R. Yuan et al., “Optimally Displaced Threshold Detection for Discriminating Binary Coherent States Using Imperfect Devices”, (2020) arXiv:2007.11109
- [12]
- 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
- [13]
- C. Berdou et al., “One Hundred Second Bit-Flip Time in a Two-Photon Dissipative Oscillator”, PRX Quantum 4, (2023) arXiv:2204.09128 DOI
- [14]
- L. Egan et al., “Fault-Tolerant Operation of a Quantum Error-Correction Code”, (2021) arXiv:2009.11482
- [15]
- K. A. S. Immink. Codes for mass data storage systems. Shannon Foundation Publisher, 2004.
- [16]
- Cheung, K-M., and F. Pollara. "Phobos lander coding system: Software and analysis." The Telecommunications and Data Acquisition Report (1988).
- [17]
- International Telecommunication Union-T, Recommendation V.21: 300 bits per second duplex modem standardized for use in the general switched telephone network, 1984
- [18]
- International Telecommunication Union-T, Recommendation V.23: 600/1200-baud modem standardized for use in the general switched telephone network, 1988
- [19]
- 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
- [20]
- 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
- [21]
- Y. Xu et al., “Demonstration of Controlled-Phase Gates between Two Error-Correctable Photonic Qubits”, Physical Review Letters 124, (2020) arXiv:1810.04690 DOI
- [22]
- Z. Ni et al., “Beating the break-even point with a discrete-variable-encoded logical qubit”, Nature 616, 56 (2023) arXiv:2211.09319 DOI
- [23]
- V. V. Sivak et al., “Real-time quantum error correction beyond break-even”, Nature 616, 50 (2023) arXiv:2211.09116 DOI
- [24]
- M. Kudra et al., “Robust Preparation of Wigner-Negative States with Optimized SNAP-Displacement Sequences”, PRX Quantum 3, (2022) arXiv:2111.07965 DOI
- [25]
- V. G. Matsos et al., “Robust and Deterministic Preparation of Bosonic Logical States in a Trapped Ion”, (2023) arXiv:2310.15546
- [26]
- S. Zhu, Z. Sun, and X. Kai, “A Class of Narrow-Sense BCH Codes”, IEEE Transactions on Information Theory 65, 4699 (2019) DOI
- [27]
- L. Sun et al., “Tracking photon jumps with repeated quantum non-demolition parity measurements”, Nature 511, 444 (2014) arXiv:1311.2534 DOI
- [28]
- N. Ofek et al., “Demonstrating Quantum Error Correction that Extends the Lifetime of Quantum Information”, (2016) arXiv:1602.04768
- [29]
- S. Rosenblum et al., “Fault-tolerant detection of a quantum error”, Science 361, 266 (2018) arXiv:1803.00102 DOI
- [30]
- R. J. McEliece, A public-key cryptosystem based on algebraic coding theory, Technical report, Jet Propulsion Lab. DSN Progress Report (1978).
- [31]
- H. Janwa and O. Moreno, “McEliece public key cryptosystems using algebraic-geometric codes”, Designs, Codes and Cryptography 8, (1996) DOI
- [32]
- H. Niederreiter (1986). Knapsack-type cryptosystems and algebraic coding theory. Problems of Control and Information Theory. Problemy Upravlenija I Teorii Informacii. 15: 159–166.
- [33]
- V. M. SIDELNIKOV and S. O. SHESTAKOV, “On insecurity of cryptosystems based on generalized Reed-Solomon codes”, Discrete Mathematics and Applications 2, (1992) 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]
- Ronghui Peng and Rong-Rong Chen, “Application of Nonbinary LDPC Cycle Codes to MIMO Channels”, IEEE Transactions on Wireless Communications 7, 2020 (2008) DOI
- [48]
- M. Iqbal et al., “Creation of Non-Abelian Topological Order and Anyons on a Trapped-Ion Processor”, (2023) arXiv:2305.03766
- [49]
- B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 DOI
- [50]
- M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
- [51]
- 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
- [52]
- P. Kok et al., “Linear optical quantum computing with photonic qubits”, Reviews of Modern Physics 79, 135 (2007) arXiv:quant-ph/0512071 DOI
- [53]
- A. Koottandavida et al., “Erasure detection of a dual-rail qubit encoded in a double-post superconducting cavity”, (2023) arXiv:2311.04423
- [54]
- H. Levine et al., “Demonstrating a long-coherence dual-rail erasure qubit using tunable transmons”, (2023) arXiv:2307.08737
- [55]
- J. D. Teoh et al., “Dual-rail encoding with superconducting cavities”, Proceedings of the National Academy of Sciences 120, (2023) arXiv:2212.12077 DOI
- [56]
- A. Kubica et al., “Erasure qubits: Overcoming the \(T_1\) limit in superconducting circuits”, (2022) arXiv:2208.05461
- [57]
- K. S. Chou et al., “Demonstrating a superconducting dual-rail cavity qubit with erasure-detected logical measurements”, (2023) arXiv:2307.03169
- [58]
- Y. Fan et al., “Experimental realization of a topologically protected Hadamard gate via braiding Fibonacci anyons”, (2022) arXiv:2210.12145
- [59]
- E. Knill et al., “Benchmarking Quantum Computers: The Five-Qubit Error Correcting Code”, Physical Review Letters 86, 5811 (2001) arXiv:quant-ph/0101034 DOI
- [60]
- 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
- [61]
- A. M. Souza et al., “Experimental magic state distillation for fault-tolerant quantum computing”, Nature Communications 2, (2011) arXiv:1103.2178 DOI
- [62]
- 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
- [63]
- C. Ryan-Anderson et al., “Implementing Fault-tolerant Entangling Gates on the Five-qubit Code and the Color Code”, (2022) arXiv:2208.01863
- [64]
- 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
- [65]
- 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
- [66]
- J. R. Wootton, “Measurements of Floquet code plaquette stabilizers”, (2022) arXiv:2210.13154
- [67]
- A. Shokrollahi, “Raptor codes”, IEEE Transactions on Information Theory 52, 2551 (2006) DOI
- [68]
- 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
- [69]
- 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
- [70]
- T. Lau and C. Tan, “A New Technique in Rank Metric Code-Based Encryption”, Cryptography 2, 32 (2018) DOI
- [71]
- 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
- [72]
- H. Dau et al., “Repairing Reed-Solomon Codes With Multiple Erasures”, IEEE Transactions on Information Theory 64, 6567 (2018) arXiv:1612.01361 DOI
- [73]
- A. Couvreur et al., “Distinguisher-Based Attacks on Public-Key Cryptosystems Using Reed-Solomon Codes”, (2014) arXiv:1307.6458
- [74]
- M. Baldi et al., “Enhanced public key security for the McEliece cryptosystem”, (2014) arXiv:1108.2462
- [75]
- N. H. Nguyen et al., “Demonstration of Shor Encoding on a Trapped-Ion Quantum Computer”, Physical Review Applied 16, (2021) arXiv:2104.01205 DOI
- [76]
- E. C. Stone, “The Voyager 2 encounter with Uranus”, Journal of Geophysical Research: Space Physics 92, 14873 (1987) DOI
- [77]
- 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.
- [78]
- M. Waegell and P. K. Aravind, “Golay codes and quantum contextuality”, Physical Review A 106, (2022) arXiv:2206.04209 DOI
- [79]
- Mujtaba Hamid and Andy Miller, Gold Code Generators in Virtex Devices, (2000)
- [80]
- J. J. SPILKER Jr., “GPS Signal Structure and Performance Characteristics”, Navigation 25, 121 (1978) DOI
- [81]
- 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
- [82]
- S. L. Johnsson and C.-T. Ho, “Optimum broadcasting and personalized communication in hypercubes”, IEEE Transactions on Computers 38, 1249 (1989) DOI
- [83]
- 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
- [84]
- Crandall, Ron. "Some notes on steganography." Posted on steganography mailing list 1998 (1998): 1-6.
- [85]
- A. Westfeld, “F5—A Steganographic Algorithm”, Information Hiding 289 (2001) DOI
- [86]
- M. Takita et al., “Experimental Demonstration of Fault-Tolerant State Preparation with Superconducting Qubits”, Physical Review Letters 119, (2017) arXiv:1705.09259 DOI
- [87]
- E. H. Chen et al., “Calibrated Decoders for Experimental Quantum Error Correction”, Physical Review Letters 128, (2022) arXiv:2110.04285 DOI
- [88]
- 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
- [89]
- B. C. Stacey, “Sporadic SICs and Exceptional Lie Algebras”, (2019) arXiv:1911.05809
- [90]
- 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
- [91]
- Odaka K., Sako Y., Iwamoto I., Doi T.; Vries L.B.; SONY: Error correctable data transmission method (Patent US4413340) filing date May 21, 1980.
- [92]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [93]
- S. B. Wicker and V. K. Bhargava, Reed-Solomon Codes and Their Applications (IEEE, 1999) DOI
- [94]
- 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
- [95]
- 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
- [96]
- 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.
- [97]
- Hui Jin, Aamod Khandekar, and Robert J. McEliece. "Serial concatenation of interleaved convolutional codes forming turbo-like codes." United States Patent Number 7116710B1 (2023).
- [98]
- 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
- [99]
- T. H. Helleseth et al., “On the<tex>$”, IEEE Transactions on Information Theory 50, 3312 (2004) DOI
- [100]
- J.-S. Xu et al., “Simulating the exchange of Majorana zero modes with a photonic system”, Nature Communications 7, (2016) arXiv:1411.7751 DOI
- [101]
- 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
- [102]
- 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
- [103]
- X. Mi et al., “Noise-resilient edge modes on a chain of superconducting qubits”, Science 378, 785 (2022) arXiv:2204.11372 DOI
- [104]
- D. Bluvstein et al., “A quantum processor based on coherent transport of entangled atom arrays”, Nature 604, 451 (2022) arXiv:2112.03923 DOI
- [105]
- K. J. Satzinger et al., “Realizing topologically ordered states on a quantum processor”, Science 374, 1237 (2021) arXiv:2104.01180 DOI
- [106]
- G. Semeghini et al., “Probing topological spin liquids on a programmable quantum simulator”, Science 374, 1242 (2021) arXiv:2104.04119 DOI
- [107]
- S. Xu et al., “Digital simulation of projective non-Abelian anyons with 68 superconducting qubits”, Chinese Physics Letters (2023) arXiv:2211.09802 DOI
- [108]
- T. Aoki et al., “Quantum error correction beyond qubits”, (2008) arXiv:0811.3734
- [109]
- 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.
- [110]
- 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
- [111]
- 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).
- [112]
- J. Hoffstein, J. Pipher, and J. H. Silverman, “NTRU: A ring-based public key cryptosystem”, Lecture Notes in Computer Science 267 (1998) DOI
- [113]
- R. Misoczki et al., “MDPC-McEliece: New McEliece variants from Moderate Density Parity-Check codes”, 2013 IEEE International Symposium on Information Theory (2013) DOI
- [114]
- P. Gaborit et al., “RankSign: An Efficient Signature Algorithm Based on the Rank Metric”, Post-Quantum Cryptography 88 (2014) DOI
- [115]
- 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
- [116]
- McEliece, R.J.: A public-key cryptosystem based on algebraic coding theory. DSN Progress Report pp. 114–116 (1978).
- [117]
- R. Koetter and F. Kschischang, “Coding for Errors and Erasures in Random Network Coding”, (2008) arXiv:cs/0703061
- [118]
- 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
- [119]
- C. Noel et al., “Measurement-induced quantum phases realized in a trapped-ion quantum computer”, Nature Physics 18, 760 (2022) arXiv:2106.05881 DOI
- [120]
- J. Conrad, J. Eisert, and J.-P. Seifert, “Good Gottesman-Kitaev-Preskill codes from the NTRU cryptosystem”, (2023) arXiv:2303.02432
- [121]
- 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
- [122]
- 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
- [123]
- 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
- [124]
- Encyclopedia of Computer Science and Technology, Second Edition Volume I (CRC Press, 2017) DOI
- [125]
- 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
- [126]
- K. Potdar, T. S., and C. D., “A Comparative Study of Categorical Variable Encoding Techniques for Neural Network Classifiers”, International Journal of Computer Applications 175, 7 (2017) DOI
- [127]
- 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
- [128]
- 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
- [129]
- 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
- [130]
- 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
- [131]
- 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
- [132]
- 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
- [133]
- 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
- [134]
- M. V. Jabir et al., “Versatile quantum-enabled telecom receiver”, AVS Quantum Science 5, 015001 (2023) DOI
- [135]
- 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
- [136]
- J. M. Gertler et al., “Experimental Realization and Characterization of Stabilized Pair Coherent States”, (2022) arXiv:2209.11643
- [137]
- International Telecommunication Union-T, Recommendation V.27ter: 4800/2400 Bits Per Second Modem Standardized For Use in the General Switched Telephone Network, 1984
- [138]
- 3rd Generation Partnership Project (3GPP), Technical specification group radio access network, 3GPP TS 38.212 V.15.0.0, 2017.
- [139]
- 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.
- [140]
- 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
- [141]
- K. Feher, “Modems for emerging digital cellular-mobile radio system”, IEEE Transactions on Vehicular Technology 40, 355 (1991) DOI
- [142]
- 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
- [143]
- 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
- [144]
- 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
- [145]
- M. Hopkins and T.-C. Lin, “Explicit Lower Bounds Against \(Ω(n)\)-Rounds of Sum-of-Squares”, (2022) arXiv:2204.11469
- [146]
- 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
- [147]
- 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
- [148]
- D. G. Cory et al., “Experimental Quantum Error Correction”, Physical Review Letters 81, 2152 (1998) arXiv:quant-ph/9802018 DOI
- [149]
- 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
- [150]
- J. Zhang et al., “Experimental quantum error correction with high fidelity”, Physical Review A 84, (2011) arXiv:1109.4821 DOI
- [151]
- M. D. Reed et al., “Realization of three-qubit quantum error correction with superconducting circuits”, Nature 482, 382 (2012) arXiv:1109.4948 DOI
- [152]
- D. Ristè et al., “Detecting bit-flip errors in a logical qubit using stabilizer measurements”, Nature Communications 6, (2015) arXiv:1411.5542 DOI
- [153]
- 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
- [154]
- J. R. Wootton and D. Loss, “Repetition code of 15 qubits”, Physical Review A 97, (2018) arXiv:1709.00990 DOI
- [155]
- J. R. Wootton, “Benchmarking near-term devices with quantum error correction”, Quantum Science and Technology 5, 044004 (2020) arXiv:2004.11037 DOI
- [156]
- J. Kelly et al., “State preservation by repetitive error detection in a superconducting quantum circuit”, Nature 519, 66 (2015) arXiv:1411.7403 DOI
- [157]
- “Exponential suppression of bit or phase errors with cyclic error correction”, Nature 595, 383 (2021) arXiv:2102.06132 DOI
- [158]
- R. Acharya et al., “Suppressing quantum errors by scaling a surface code logical qubit”, (2022) arXiv:2207.06431
- [159]
- K. C. Miao et al., “Overcoming leakage in scalable quantum error correction”, (2022) arXiv:2211.04728
- [160]
- W. P. Livingston et al., “Experimental demonstration of continuous quantum error correction”, Nature Communications 13, (2022) arXiv:2107.11398 DOI
- [161]
- K. Takeda et al., “Quantum error correction with silicon spin qubits”, Nature 608, 682 (2022) arXiv:2201.08581 DOI
- [162]
- F. van Riggelen et al., “Phase flip code with semiconductor spin qubits”, (2022) arXiv:2202.11530
- [163]
- G. Waldherr et al., “Quantum error correction in a solid-state hybrid spin register”, Nature 506, 204 (2014) arXiv:1309.6424 DOI
- [164]
- T. Nakazato et al., “Quantum error correction of spin quantum memories in diamond under a zero magnetic field”, Communications Physics 5, (2022) DOI
- [165]
- 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
- [166]
- P. Schindler et al., “Experimental Repetitive Quantum Error Correction”, Science 332, 1059 (2011) DOI
- [167]
- S. A. Moses et al., “A Race Track Trapped-Ion Quantum Processor”, (2023) arXiv:2305.03828
- [168]
- T. Richardson and S. Kudekar, “Design of Low-Density Parity Check Codes for 5G New Radio”, IEEE Communications Magazine 56, 28 (2018) DOI
- [169]
- 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
- [170]
- LDPC coding for OFDMA PHY. 802.16REVe Sponsor Ballot Recirculation comment, July 2004. IEEE C802.16e04/141r2
- [171]
- 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
- [172]
- G. Falcão et al., “High coded data rate and multicodeword WiMAX LDPC decoding on Cell/BE”, Electronics Letters 44, 1415 (2008) DOI
- [173]
- 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
- [174]
- 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
- [175]
- G. Alagic et al., “Quantum Fully Homomorphic Encryption with Verification”, Advances in Cryptology – ASIACRYPT 2017 438 (2017) arXiv:1708.09156 DOI
- [176]
- P. W. Shor and J. Preskill, “Simple Proof of Security of the BB84 Quantum Key Distribution Protocol”, Physical Review Letters 85, 441 (2000) arXiv:quant-ph/0003004 DOI
- [177]
- A. Coladangelo et al., “Hidden Cosets and Applications to Unclonable Cryptography”, (2022) arXiv:2107.05692
- [178]
- 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
- [179]
- 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
- [180]
- O. Regev, “On lattices, learning with errors, random linear codes, and cryptography”, Journal of the ACM 56, 1 (2009) DOI
- [181]
- P. Gaborit et al., “Identity-Based Encryption from Codes with Rank Metric”, Advances in Cryptology – CRYPTO 2017 194 (2017) DOI
- [182]
- P. Lefèvre, P. Carré, and P. Gaborit, “Application of rank metric codes in digital image watermarking”, Signal Processing: Image Communication 74, 119 (2019) DOI
- [183]
- 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
- [184]
- Anxiao Jiang et al., “Rank Modulation for Flash Memories”, IEEE Transactions on Information Theory 55, 2659 (2009) DOI
- [185]
- J. L. Massey, “Deep-space communications and coding: A marriage made in heaven”, Advanced Methods for Satellite and Deep Space Communications 1 DOI
- [186]
- E.C. Posner, Combinatorial Structures in Planetary Reconnaissance in Error Correcting Codes, ed. H.B. Mann, Wiley, NY 1968.
- [187]
- 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
- [188]
- A. Kiayias and M. Yung, “Cryptographic Hardness Based on the Decoding of Reed-Solomon Codes”, Automata, Languages and Programming 232 (2002) DOI
- [189]
- D. V. Sarwate and N. R. Shanbhag, “High-speed architectures for Reed-Solomon decoders”, IEEE Transactions on Very Large Scale Integration (VLSI) Systems 9, 641 (2001) DOI
- [190]
- S. Li et al., “PolyShard: Coded Sharding Achieves Linearly Scaling Efficiency and Security Simultaneously”, (2020) arXiv:1809.10361
- [191]
- B. Sasidharan and E. Viterbo, “Private Data Access in Blockchain Systems Employing Coded Sharding”, 2021 IEEE International Symposium on Information Theory (ISIT) (2021) DOI
- [192]
- International Organization for Standardization, Information Technology: Automatic Identification and Data Capture Techniques-QR Code 2005 Bar Code Symbology Specification, 2nd ed., IEC18004 (ISO, 2006).
- [193]
- 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
- [194]
- N. Shental et al., “Efficient high-throughput SARS-CoV-2 testing to detect asymptomatic carriers”, Science Advances 6, (2020) DOI
- [195]
- K. Karplus and H. Krit, “A semi-systolic decoder for the PDSC-73 error-correcting code”, Discrete Applied Mathematics 33, 109 (1991) DOI
- [196]
- T. Klove and M. Miller, “The Detection of Errors After Error-Correction Decoding”, IEEE Transactions on Communications 32, 511 (1984) DOI
- [197]
- High-Performance Embedded Computing (Elsevier, 2014) DOI
- [198]
- H. Janwa and O. Moreno, Designs, Codes and Cryptography 8, 293 (1996) DOI
- [199]
- A. Couvreur, H. Randriambololona, "Algebraic Geometry Codes and Some Applications." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [200]
- 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
- [201]
- 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
- [202]
- 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
- [203]
- “Advances in smart antennas - MIMO-OFDM wireless systems: basics, perspectives, and challenges”, IEEE Wireless Communications 13, 31 (2006) DOI
- [204]
- C. Flühmann et al., “Encoding a qubit in a trapped-ion mechanical oscillator”, Nature 566, 513 (2019) arXiv:1807.01033 DOI
- [205]
- 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
- [206]
- B. de Neeve et al., “Error correction of a logical grid state qubit by dissipative pumping”, (2020) arXiv:2010.09681
- [207]
- D. Lachance-Quirion et al., “Autonomous quantum error correction of Gottesman-Kitaev-Preskill states”, (2023) arXiv:2310.11400
- [208]
- S. Konno et al., “Propagating Gottesman-Kitaev-Preskill states encoded in an optical oscillator”, (2023) arXiv:2309.02306
- [209]
- 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
- [210]
- H. G. Feichtinger and T. Strohmer, editors , Gabor Analysis and Algorithms (Birkhäuser Boston, 1998) DOI
- [211]
- S. Krinner et al., “Realizing repeated quantum error correction in a distance-three surface code”, Nature 605, 669 (2022) arXiv:2112.03708 DOI
- [212]
- Y. Zhao et al., “Realization of an Error-Correcting Surface Code with Superconducting Qubits”, Physical Review Letters 129, (2022) arXiv:2112.13505 DOI
- [213]
- L. Frunzio and S. Singh, “Error-Correcting Surface Codes Get Experimental Vetting”, Physics 15, (2022) DOI
- [214]
- Y. Ye et al., “Logical Magic State Preparation with Fidelity Beyond the Distillation Threshold on a Superconducting Quantum Processor”, (2023) arXiv:2305.15972
- [215]
- P. Chaichanavong and P. H. Siegel, “Tensor-product parity code for magnetic recording”, IEEE Transactions on Magnetics 42, 350 (2006) DOI
- [216]
- 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
- [217]
- T. M. Thompson, From Error-Correcting Codes Through Sphere Packings To Simple Groups, Mathematical Association of America, 1983.
- [218]
- 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
- [219]
- S. Touzard et al., “Coherent Oscillations inside a Quantum Manifold Stabilized by Dissipation”, Physical Review X 8, (2018) arXiv:1705.02401 DOI
- [220]
- A. Grimm et al., “Stabilization and operation of a Kerr-cat qubit”, Nature 584, 205 (2020) arXiv:1907.12131 DOI
- [221]
- 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
- [222]
- U. Réglade et al., “Quantum control of a cat-qubit with bit-flip times exceeding ten seconds”, (2023) arXiv:2307.06617
- [223]
- V. V. Albert et al., “Holonomic Quantum Control with Continuous Variable Systems”, Physical Review Letters 116, (2016) arXiv:1503.00194 DOI
- [224]
- J. M. Gertler et al., “Protecting a bosonic qubit with autonomous quantum error correction”, Nature 590, 243 (2021) arXiv:2004.09322 DOI
- [225]
- S. Kak, “Unary Coding for Neural Network Learning”, (2010) arXiv:1009.4495
- [226]
- 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.
- [227]
- Z. Li et al., “Autonomous error correction of a single logical qubit using two transmons”, (2023) arXiv:2302.06707
- [228]
- 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
- [229]
- T. I. Andersen et al., “Non-Abelian braiding of graph vertices in a superconducting processor”, (2023) arXiv:2210.10255
- [230]
- 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
- [231]
- J. Meggitt, “Error correcting codes and their implementation for data transmission systems”, IEEE Transactions on Information Theory 7, 234 (1961) DOI
- [232]
- C. Campopiano and B. Glazer, “A Coherent Digital Amplitude and Phase Modulation Scheme”, IEEE Transactions on Communications 10, 90 (1962) DOI
- [233]
- 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
- [234]
- C. N. Self, M. Benedetti, and D. Amaro, “Protecting Expressive Circuits with a Quantum Error Detection Code”, (2022) arXiv:2211.06703
- [235]
- K. Yamamoto et al., “Demonstrating Bayesian Quantum Phase Estimation with Quantum Error Detection”, (2023) arXiv:2306.16608
- [236]
- B. A. Bell et al., “Experimental demonstration of a graph state quantum error-correction code”, Nature Communications 5, (2014) arXiv:1404.5498 DOI
- [237]
- N. M. Linke et al., “Fault-tolerant quantum error detection”, Science Advances 3, (2017) arXiv:1611.06946 DOI
- [238]
- Quantum Information and Computation 18, (2018) arXiv:1705.08957 DOI
- [239]
- R. Harper and S. T. Flammia, “Fault-Tolerant Logical Gates in the IBM Quantum Experience”, Physical Review Letters 122, (2019) arXiv:1806.02359 DOI
- [240]
- C. K. Andersen et al., “Repeated quantum error detection in a surface code”, Nature Physics 16, 875 (2020) arXiv:1912.09410 DOI
- [241]
- A. Erhard et al., “Entangling logical qubits with lattice surgery”, Nature 589, 220 (2021) arXiv:2006.03071 DOI
- [242]
- J. F. Marques et al., “Logical-qubit operations in an error-detecting surface code”, Nature Physics 18, 80 (2021) arXiv:2102.13071 DOI
- [243]
- R. S. Gupta et al., “Encoding a magic state with beyond break-even fidelity”, (2023) arXiv:2305.13581
- [244]
- D. Nigg et al., “Quantum computations on a topologically encoded qubit”, Science 345, 302 (2014) arXiv:1403.5426 DOI
- [245]
- C. Ryan-Anderson et al., “Realization of real-time fault-tolerant quantum error correction”, (2021) arXiv:2107.07505
- [246]
- P. Ball, “Real-Time Error Correction for Quantum Computing”, Physics 14, (2021) DOI
- [247]
- L. Postler et al., “Demonstration of fault-tolerant universal quantum gate operations”, Nature 605, 675 (2022) arXiv:2111.12654 DOI
- [248]
- D. Bluvstein et al., “Logical quantum processor based on reconfigurable atom arrays”, Nature (2023) DOI
- [249]
- Y. Wang et al., “Fault-Tolerant One-Bit Addition with the Smallest Interesting Colour Code”, (2023) arXiv:2309.09893
- [250]
- D. H. Menendez, A. Ray, and M. Vasmer, “Implementing fault-tolerant non-Clifford gates using the [[8,3,2]] color code”, (2023) arXiv:2309.08663
- [251]
- Q. Xu et al., “Constant-Overhead Fault-Tolerant Quantum Computation with Reconfigurable Atom Arrays”, (2023) arXiv:2308.08648
- [252]
- D. M. Debroy et al., “Optimizing Stabilizer Parities for Improved Logical Qubit Memories”, Physical Review Letters 127, (2021) arXiv:2105.05068 DOI
- [253]
- 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
- [254]
- R. Zhang et al., “Loss-tolerant all-photonic quantum repeater with generalized Shor code”, Optica 9, 152 (2022) arXiv:2203.07979 DOI