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

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 [1]. |

Binary Golay code | Used in the Voyager 1 and 2 spacecraft [2]. |

Binary repetition code | Repetition codesm in conjunction with other codes, were used in magnetic disks [3]. |

Binomial code | Realized in microwave cavities coupled to superconducting circuits [4]. |

Cat code | Two-legged (\(S=1\)) cat code has been realized in a superconducting circuit device by the Devoret group [5]. Exponential suppression of bit-flip errors achieved [6] up to a bit-flip time of 1 ms. A bit-flip time of up to 1 sec has been achieved while away from the logical-qubit regime [7].Four-legged (\(S=2\)) cat code has been realized in a superconducting circuit device [8]. This paper is the first to reach break-even error-correction, where the lifetime of a logical qubit is on par with the lifetime of the noisiest constituent of the system. |

Convolutional code | A type of convolutional code used in Real-time Application networks [9].Mobile and radio communications (3G networks) use convolutional codes concatenated with Reed-Solomon codes to obtain suitable performance [10].A convolutional code with rate 1/2 was used for deep-space and satellite communication [11] |

Covering code | Data compression both with or without compression [12].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 [13][14]. |

Fountain code | Designed for servers sending data to many recipients, such as during broadcasting or file distribution [15]. |

Generalized Reed-Solomon (GRS) code | Various cloud storage systems [16].Public-key cryptosystems [17]. Initial construction of McEliece Public Key Cryptosystem was based on Goppa codes which are subfield subcode of GRS codes. Public Key Cryptosystem designs based on GRS codes first proposed in Ref. [18], which replaced the generator matrix with the parity check matrix, were proven to be insecure [19] since the public key exposes algebraic structure of code. More recent works focus on methods to mask the algebraic structure using subcodes of GRS codes [17]. Lately a key-recovery attack was developed in [20] for a variant of masking method proposed by [21]. |

Goppa code | The binary version \( (q=2) \) is commonly used in post-quantum cryptosystems such as the McElise cryptosystem [22]. |

Gottesman-Kitaev-Preskill (GKP) code | GKP encoding realized in the motional degree of freedom of a trapped ion [23], followed by realization of dissipative stabilization scheme [24].A reduced form of GKP error correction, consisting of measuring only the direction of a displacement error with an ancillary transmon, realized in a microwave cavity coupled to superconducting circuits [25]. |

Hamming code | Commonly used when error rates are very low, for example, computer RAM or integrated circuits [26]. |

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 \(d=2\) [27][28] and \(d=3\) [29]. |

Homological bosonic code | No experimental realization. However, Ref. [30] describes a potential experimental optical procedure for the simplest non-trival code with 5 modes. |

Justesen code | Generating small-bias sample spaces, i.e., probability distributions that parity functions cannot typically distinguish from the uniform distribution [31]. |

Kitaev surface code | Distance-two surface codes have been implemented by Andersen et al. [32], Erhard et al. [33], Marques et al. [34], Google Quantum AI [35], and both planar and toric versions by Bluvstein et al. [36]. Distance-three surface code implemented at ETH Zurich [37] and on the Zuchongzhi 2.1 superconducting quantum processor [38]. Signatures of corresponding topological phase of matter detected in superconducting circuits [39] and two-dimensional arrays of Rydberg atoms [40]. |

Locally recoverable code (LRC) | An \((18,14,7)\) LRC code has beed used in the Windows Azure cloud storage system [41]; see also Sec. 31.3.1.2 in Ref. [42]. |

Low-density parity-check (LDPC) code | 5G NR cellular communication for the traffic channel [43].WiMAX (IEEE 802.16e) [44].Satelite transmission of digital television [45]. |

Maximum distance separable (MDS) code | The McEliece Public Key Cryptosystem [46] 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. |

Maximum-rank distance (MRD) code | Useful for error and erasure correction in network coding [47][48]. |

Monitored random-circuit code | Measurement induced quantum phases have been realized in a trapped-ion processor [49]. |

Niset-Andersen-Cerf code | Realized in Ref. [50] 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). |

Polar code | Code control channels for the 5G NR (New Radio) interfaces [51]. |

Quantum Tanner code | Used to obtain explicit lower bounds in the sum-of-squares game [52]. |

Quantum parity code (QPC) | The \([[m^2,1,m]]\) codes for \(m\leq 7\) have been realized in trapped-ion quantum devices [53].Non-determinisitic linear-optical encoding [54] whose success probability \(P_{E}\) is determined by the efficiency \(\eta\) of the photonic encoding circuit. A threshold \(\eta > 0.82 \) exists for the efficiency, above which \(P_{E}\to 1\) as \(m_1\to\infty\) given particular \(m_2\).Studied in the context of error-corrected quantum repeaters [55]. |

Quantum repetition code | NMR: 3-qubit device [56].Superconducting circuits: IBM 15-qubit device [57], Google Quantum AI Sycamore utilizing from 5 to 21 qubits [35].Semiconductor spin-qubit devices: 3-qubit devices at RIKEN [58] and Delft [59].Nitrogen-vacancy centers in diamond: 3-qubit device [60].Continuous error correction protocols have been implemented on a 3-qubit superconducting circuit device [61].Liquid-state NMR: 3-qubit phase-flip code [56].See Table S6 in Ref. [35] for a history of earlier implementations.Repetition codes can be used to benchmark device performance [62]. |

Random code | Distributed storage systems [63]. |

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 [64]. |

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-Solomon (RS) code | Cross-interleaved RS code (CIRC) is adopted in Compact Discs (CDs) and RS Product Code (RSPC) in DVDs; see Ch. 4 of Ref. [65].In DSL technologies and its variants against impluse noise [66].RS codes as outer codes concatenated with convolutional codes are used indirectly in solar exploration programs; see Ch. 3 of Ref. [65].Coded sharding designs in blockchains to increase efficiency [67].Private Information Retrieval [68].Used in QR-Codes to retrieve damaged barcodes [69].Wireless communication systems such as 3G, DVB, and WiMAX [70].Correcting pooled testing results for SARS-CoV-2 [71]. |

Shor \([[9,1,3]]\) 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 [53]. Variants of the code to handle coherent noise studied and realized by K. Brown and C. Monroe groups [72].All-photonic quantum repeater architecture [73]. |

Single parity-check code | Can be realized on almost every communication device. |

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) [74][75]. |

Steane \([[7,1,3]]\) code | Trapped-ion qubits: seven-qubit device in Blatt group [76], ten-qubit QCCD device by Quantinuum [77], fault-tolerant universal two-qubit gate realized by Monz group [78].Rydberg atom arrays: Lukin group [36]. |

Zetterberg code | Code used to provide better protection of data transmission with its double error correcting capacity [79]. |

\([[4,2,2]]\) CSS code | Trapped-ion device by IonQ [80].Logical state preparation and flag-qubit error correction realized in superconducting-circuit devices by IBM [27][28]. |

\([[5,1,3]]\) perfect code | First realized in NMR [81].Demonstration with superconducting qubits [82]. |

## References

- [1]
- Laird Egan et al., “Fault-Tolerant Operation of a Quantum Error-Correction Code”. 2009.11482
- [2]
- E. C. Stone, “The Voyager 2 encounter with Uranus”, Journal of Geophysical Research: Space Physics 92, 14873 (1987). DOI
- [3]
- T. Klove and M. Miller, “The Detection of Errors After Error-Correction Decoding”, IEEE Transactions on Communications 32, 511 (1984). DOI
- [4]
- L. Hu et al., “Quantum error correction and universal gate set operation on a binomial bosonic logical qubit”, Nature Physics 15, 503 (2019). DOI
- [5]
- Z. Leghtas et al., “Confining the state of light to a quantum manifold by engineered two-photon loss”, Science 347, 853 (2015). DOI; 1412.4633
- [6]
- R. Lescanne et al., “Exponential suppression of bit-flips in a qubit encoded in an oscillator”, Nature Physics 16, 509 (2020). DOI; 1907.11729
- [7]
- C. Berdou et al., “One hundred second bit-flip time in a two-photon dissipative oscillator”. 2204.09128
- [8]
- N. Ofek et al., “Extending the lifetime of a quantum bit with error correction in superconducting circuits”, Nature 536, 441 (2016). DOI
- [9]
- 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
- [10]
- T. Halonen, J. Romero, and J. Melero, editors , GSM, GPRS and EDGE Performance (Wiley, 2003). DOI
- [11]
- Butman, Deutsch, and Miller. Performance of concatenated codes for deep space missions. 1981.
- [12]
- G. Cohen, I. Honkala, S. Litsyn, A. Lobstein, Covering Codes, Elsevier (1997).
- [13]
- H. Hamalainen et al., “Football Pools--A Game for Mathematicians”, The American Mathematical Monthly 102, 579 (1995). DOI
- [14]
- A. Barg, “At the Dawn of the Theory of Codes”, The Mathematical Intelligencer 15, 20 (1993). DOI
- [15]
- A. Shokrollahi, “Raptor codes”, IEEE Transactions on Information Theory 52, 2551 (2006). DOI
- [16]
- H. Dau et al., “Repairing Reed-Solomon Codes With Multiple Erasures”, IEEE Transactions on Information Theory 64, 6567 (2018). DOI; 1612.01361
- [17]
- 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
- [18]
- L. Socha, “Some remarks about open-loop control in stochastic quasilinear systems”, Optimal Control Applications and Methods 7, 105 (1986). DOI
- [19]
- V. M. SIDELNIKOV and S. O. SHESTAKOV, “On insecurity of cryptosystems based on generalized Reed-Solomon codes”, Discrete Mathematics and Applications 2, (1992). DOI
- [20]
- Alain Couvreur et al., “Distinguisher-Based Attacks on Public-Key Cryptosystems Using Reed-Solomon Codes”. 1307.6458
- [21]
- Marco Baldi et al., “Enhanced public key security for the McEliece cryptosystem”. 1108.2462
- [22]
- H. Janwa and O. Moreno, “McEliece public key cryptosystems using algebraic-geometric codes”, Designs, Codes and Cryptography 8, (1996). DOI
- [23]
- C. Flühmann et al., “Encoding a qubit in a trapped-ion mechanical oscillator”, Nature 566, 513 (2019). DOI; 1807.01033
- [24]
- Brennan de Neeve et al., “Error correction of a logical grid state qubit by dissipative pumping”. 2010.09681
- [25]
- P. Campagne-Ibarcq et al., “Quantum error correction of a qubit encoded in grid states of an oscillator”. 1907.12487
- [26]
- 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
- [27]
- M. Takita et al., “Experimental Demonstration of Fault-Tolerant State Preparation with Superconducting Qubits”, Physical Review Letters 119, (2017). DOI; 1705.09259
- [28]
- Edward H. Chen et al., “Calibrated decoders for experimental quantum error correction”. 2110.04285
- [29]
- Neereja Sundaresan et al., “Matching and maximum likelihood decoding of a multi-round subsystem quantum error correction experiment”. 2203.07205
- [30]
- P. Hayden et al., “Spacetime replication of continuous variable quantum information”, New Journal of Physics 18, 083043 (2016). DOI; 1601.02544
- [31]
- 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
- [32]
- C. K. Andersen et al., “Repeated quantum error detection in a surface code”, Nature Physics 16, 875 (2020). DOI; 1912.09410
- [33]
- A. Erhard et al., “Entangling logical qubits with lattice surgery”, Nature 589, 220 (2021). DOI; 2006.03071
- [34]
- J. F. Marques et al., “Logical-qubit operations in an error-detecting surface code”, Nature Physics 18, 80 (2021). DOI; 2102.13071
- [35]
- Zijun Chen et al., “Exponential suppression of bit or phase flip errors with repetitive error correction”. 2102.06132
- [36]
- Dolev Bluvstein et al., “A quantum processor based on coherent transport of entangled atom arrays”. 2112.03923
- [37]
- Sebastian Krinner et al., “Realizing Repeated Quantum Error Correction in a Distance-Three Surface Code”. 2112.03708
- [38]
- Youwei Zhao et al., “Realization of an Error-Correcting Surface Code with Superconducting Qubits”. 2112.13505
- [39]
- K. J. Satzinger et al., “Realizing topologically ordered states on a quantum processor”, Science 374, 1237 (2021). DOI; 2104.01180
- [40]
- G. Semeghini et al., “Probing topological spin liquids on a programmable quantum simulator”, Science 374, 1242 (2021). DOI; 2104.04119
- [41]
- 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.
- [42]
- W. C. Huffman, J.-L. Kim, and P. Solé, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021). DOI
- [43]
- 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
- [44]
- LDPC coding for OFDMA PHY. 802.16REVe Sponsor Ballot Recirculation comment, July 2004. IEEE C802.16e04/141r2
- [45]
- 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
- [46]
- McEliece, R.J.: A public-key cryptosystem based on algebraic coding theory. DSN Progress Report pp. 114–116 (1978).
- [47]
- Ralf Koetter and Frank Kschischang, “Coding for Errors and Erasures in Random Network Coding”. cs/0703061
- [48]
- 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). DOI; 0711.0708
- [49]
- Crystal Noel et al., “Observation of measurement-induced quantum phases in a trapped-ion quantum computer”. 2106.05881
- [50]
- M. Lassen et al., “Quantum optical coherence can survive photon losses using a continuous-variable quantum erasure-correcting code”, Nature Photonics 4, 700 (2010). DOI
- [51]
- 3rd Generation Partnership Project (3GPP), Technical specification group radio access network, 3GPP TS 38.212 V.15.0.0, 2017.
- [52]
- Max Hopkins and Ting-Chun Lin, “Explicit Lower Bounds Against $Ω(n)$-Rounds of Sum-of-Squares”. 2204.11469
- [53]
- N. H. Nguyen et al., “Demonstration of Shor Encoding on a Trapped-Ion Quantum Computer”, Physical Review Applied 16, (2021). DOI; 2104.01205
- [54]
- T. C. Ralph, A. J. F. Hayes, and A. Gilchrist, “Loss-Tolerant Optical Qubits”, Physical Review Letters 95, (2005). DOI; quant-ph/0501184
- [55]
- S. Muralidharan et al., “Ultrafast and Fault-Tolerant Quantum Communication across Long Distances”, Physical Review Letters 112, (2014). DOI; 1310.5291
- [56]
- D. G. Cory et al., “Experimental Quantum Error Correction”, Physical Review Letters 81, 2152 (1998). DOI; quant-ph/9802018
- [57]
- J. R. Wootton and D. Loss, “Repetition code of 15 qubits”, Physical Review A 97, (2018). DOI; 1709.00990
- [58]
- Kenta Takeda et al., “Quantum error correction with silicon spin qubits”. 2201.08581
- [59]
- F. van Riggelen et al., “Phase flip code with semiconductor spin qubits”. 2202.11530
- [60]
- T. Nakazato et al., “Quantum error correction of spin quantum memories in diamond under a zero magnetic field”, Communications Physics 5, (2022). DOI
- [61]
- William P. Livingston et al., “Experimental demonstration of continuous quantum error correction”. 2107.11398
- [62]
- Teague Tomesh et al., “SupermarQ: A Scalable Quantum Benchmark Suite”. 2202.11045
- [63]
- 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
- [64]
- Anxiao Jiang et al., “Rank Modulation for Flash Memories”, IEEE Transactions on Information Theory 55, 2659 (2009). DOI
- [65]
- S. B. Wicker and V. K. Bhargava, Reed-solomon Codes and Their Applications (IEEE, 1999). DOI
- [66]
- 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
- [67]
- Songze Li et al., “PolyShard: Coded Sharding Achieves Linearly Scaling Efficiency and Security Simultaneously”. 1809.10361
- [68]
- B. Sasidharan and E. Viterbo, “Private Data Access in Blockchain Systems Employing Coded Sharding”, 2021 IEEE International Symposium on Information Theory (ISIT) (2021). DOI
- [69]
- International Organization for Standardization, Information Technology: Automatic Identification and Data Capture Techniques-QR Code 2005 Bar Code Symbology Specification, 2nd ed., IEC18004 (ISO, 2006).
- [70]
- 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
- [71]
- N. Shental et al., “Efficient high-throughput SARS-CoV-2 testing to detect asymptomatic carriers”, Science Advances 6, (2020). DOI
- [72]
- D. M. Debroy et al., “Optimizing Stabilizer Parities for Improved Logical Qubit Memories”, Physical Review Letters 127, (2021). DOI; 2105.05068
- [73]
- R. Zhang et al., “Loss-tolerant all-photonic quantum repeater with generalized Shor code”, Optica 9, 152 (2022). DOI; 2203.07979
- [74]
- 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
- [75]
- 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
- [76]
- D. Nigg et al., “Quantum computations on a topologically encoded qubit”, Science 345, 302 (2014). DOI; 1403.5426
- [77]
- C. Ryan-Anderson et al., “Realization of real-time fault-tolerant quantum error correction”. 2107.07505
- [78]
- Lukas Postler et al., “Demonstration of fault-tolerant universal quantum gate operations”. 2111.12654
- [79]
- J. Meggitt, “Error correcting codes and their implementation for data transmission systems”, IEEE Transactions on Information Theory 7, 234 (1961). DOI
- [80]
- N. M. Linke et al., “Fault-tolerant quantum error detection”, Science Advances 3, (2017). DOI; 1611.06946
- [81]
- E. Knill et al., “Benchmarking Quantum Computers: The Five-Qubit Error Correcting Code”, Physical Review Letters 86, 5811 (2001). DOI
- [82]
- M. Gong et al., “Experimental exploration of five-qubit quantum error-correcting code with superconducting qubits”, National Science Review 9, (2021). DOI; 1907.04507