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Class of quantum error-correcting codes saturating the quantum Hamming bound
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I develop methods for analyzing quantum error-correcting codes, and use these methods to construct an infinite class of codes saturating the quantum Hamming bound. These codes encode k=n-j-2 quantum bits (qubits) in n=${2}^{\mathit{j}}$ qubits and correct t=1 error. \textcopyright{} 1996 The American Physical Society.
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“Without error corrections, the effect of noise will accumulate and ruin the entire computation[72, 17, 54, 52, 7], and hence the computation must be protected.”
From Fault-Tolerant Quantum Computation with Constant Error Rate (2008) · cites this paper as [52]
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Cites
Scheme for reducing decoherence in quantum computer memory
Shor, P. W. (1995) · Physical Review A · doi:10.1103/physreva.52.r2493 · Cited by 4,479
Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels
Bennett, C. H., Brassard, G., Popescu, S., Schumacher, B., Smolin, J. A., & Wootters, W. K. (1996) · Physical Review Letters · doi:10.1103/physrevlett.76.722 · Cited by 2,961
Good quantum error-correcting codes exist
Calderbank, A. R., & Shor, P. W. (1996) · Physical Review A · doi:10.1103/physreva.54.1098 · Cited by 2,516
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Verstraete, F., Cirac, J. I., & Latorre, J. I. (2009) · Physical Review A · doi:10.1103/physreva.79.032316 · Cited by 158
Vedral, V., & Plenio, M. B. (1998) · Progress in Quantum Electronics · doi:10.1016/s0079-6727(98)00004-4 · Cited by 159
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Gühne, O., Ekert, A., Hyllus, P., Bruss, D., Lewenstein, M., Macchiavello, C., & Sanpera, A. (2003) · Journal of Modern Optics · doi:10.1080/09500340308234554 · Cited by 156
Quantum information is physical
Divincenzo, D. P., & Loss, D. (1998) · Superlattices and Microstructures · doi:10.1006/spmi.1997.0520 · Cited by 148
Steane, A. M. (1999) · IEEE Transactions on Information Theory · doi:10.1109/18.771249 · Cited by 147
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Divincenzo, D. P. (1998) · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rspa.1998.0159 · Cited by 182
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Bombín, H. (2010) · Physical Review A · doi:10.1103/physreva.81.032301 · Cited by 118
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Efficient computations of encodings for quantum error correction
Cleve, R., & Gottesman, D. (1997) · Physical Review A · doi:10.1103/physreva.56.76 · Cited by 119
“The method is illustrated throughout the paper using an n = 8, d = 5, k = 3 code (that can handle a one-qubit error) [8–10]; however, it works for any stabilizer code.”
BackgroundMethodInfluentialSelf-citationZeno Effect for Quantum Computation and Control
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Bowen, G. (2002) · Physical Review A · doi:10.1103/physreva.66.052313 · Cited by 118
Fifteen Years of Quantum LDPC Coding and Improved Decoding Strategies
Babar, Z., Botsinis, P., Alanis, D., Ng, S. X., & Hanzo, L. (2015) · IEEE Access · doi:10.1109/access.2015.2503267 · Cited by 109
“QLDPC codes belong to the family of Quantum Stabilizer Codes (QSCs) [20], [21], which is a generalized formalism for designing quantum codes from any arbitrary classical binary and quaternary codes.”
BackgroundCatalytic Quantum Error Correction
Brun, T. A., Devetak, I., & Hsieh, M.-H. (2014) · IEEE Transactions on Information Theory · doi:10.1109/tit.2014.2313559 · Cited by 115
“The first quantum error-correcting (QEC) code was discovered by Shor [31], leading to an explosion of research in subsequent years [16, 2, 22, 21, 33, 9, 17, 7, 8].”
BackgroundMethodInformation-theoretic approach to quantum error correction and reversible measurement
Nielsen, M. A., Caves, C. M., Schumacher, B. W., & Barnum, H. (1998) · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rspa.1998.0160 · Cited by 122
Partial separability and entanglement criteria for multiqubit quantum states
Seevinck, M., & Uffink, J. (2008) · Physical Review A · doi:10.1103/physreva.78.032101 · Cited by 106
A Quantum Logic Array Microarchitecture: Scalable Quantum Data Movement and Computation
Metodi, T. S., Thaker, D., & Cross, A. W. (2006) · IEEE/ACM International Symposium on Microarchitecture (MICRO) · doi:10.1109/micro.2005.9 · Cited by 108
“Fault tolerance and quantum error correction constitute a significant field of research [13, 14, 15, 16, 17, 18] that has produced some very powerful quantum error correcting codes analogous, but fundamentally different from their classical counterparts.”
BackgroundEntanglement-assisted quantum error-correcting codes over arbitrary finite fields
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Optimum quantum error recovery using semidefinite programming
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Rains, E. M., Hardin, R. H., Shor, P. W., & Sloane, N. J. A. (1997) · Physical Review Letters · doi:10.1103/physrevlett.79.953 · Cited by 106
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Haah, J. (2013) · Communications in Mathematical Physics · doi:10.1007/s00220-013-1810-2 · Cited by 112
“The first formula is a rephrasing of the fact that the number of encoded qubits is the total number of qubits minus the number of independent stabilizer generators [10,22].”
BackgroundUncertainty relations for MUBs and SIC-POVMs in terms of generalized entropies
Rastegin, A. E. (2013) · The European Physical Journal D · doi:10.1140/epjd/e2013-40453-2 · Cited by 106
“Mutually unbiased bases have found use in quantum state reconstruction [12], quantum error correction codes [13, 14], detection of quantum entanglement [15], the mean king’s problem [16, 17].”
BackgroundPhysical optimization of quantum error correction circuits
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Holographic tensor network models and quantum error correction: a topical review
Jahn, A., & Eisert, J. (2021) · Quantum Science and Technology · doi:10.1088/2058-9565/ac0293 · Cited by 123
“This code is optimal in a variety of ways: it saturates the quantum Hamming bound [64] as well as the quantum Singleton bound [58], which follows from conditions on reconstructability after erasures [65] and is given by”
Fault-tolerant logical gates in quantum error-correcting codes
Pastawski, F., & Yoshida, B. (2015) · Physical Review A · doi:10.1103/physreva.91.012305 · Cited by 99
EFFICIENT QUANTUM CIRCUITS FOR NON-QUBIT QUANTUM ERROR-CORRECTING CODES
Grassl, M., Rötteler, M., & Beth, T. (2003) · International Journal of Foundations of Computer Science · doi:10.1142/s0129054103002011 · Cited by 103
“A particular class of quantum error-correcting codes are so-called stabilizer codes (see [2, 4, 9]).”
BackgroundMethodAharonov, D. (1999) · WORLD SCIENTIFIC eBooks · doi:10.1142/9789812815569_0007 · Cited by 131
“Codes that obey this structure are called stabilizer codes[105, 106], and their group theoretical structure gives a recipe for constructing more quantum codes.”
Constructions of new families of nonbinary quantum codes
Guardia, G. G. L. (2009) · Physical Review A · doi:10.1103/physreva.80.042331 · Cited by 98
Group representations, error bases and quantum codes
Lab., N. (. S. L. A. N., Knill, E. H., & USDOE, W. (1996) · doi:10.2172/378680 · Cited by 106
Commuting Pauli Hamiltonians as maps between free modulesPreprint
Haah, J. (2012) · Cited by 109
Divincenzo, D. P. (1997) · Springer eBooks · doi:10.1007/978-94-015-8839-3_18 · Cited by 130
“The good approach is a group-theoretic one[28, 29]: suppose there is a set of operators Mi ∈ M each of which leave the code vectors invariant:”
BackgroundByrd, M., & Lidar, D. A. (2002) · Physical Review Letters · doi:10.1103/physrevlett.89.047901 · Cited by 97
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Qubit devices and the issue of quantum decoherence
Brandt, H. E. (1999) · Progress in Quantum Electronics · doi:10.1016/s0079-6727(99)00003-8 · Cited by 102
Quantum annealing correction with minor embedding
Vinci, W., Albash, T., Paz-Silva, G. A., Hen, I., & Lidar, D. A. (2015) · Physical Review A · doi:10.1103/physreva.92.042310 · Cited by 96
Conditional efficient multiuser quantum cryptography network
Xue, P., Li, C., & Guo, G. (2002) · Physical Review A · doi:10.1103/physreva.65.022317 · Cited by 96
Generalization of Quantum Error Correction via the Heisenberg Picture
Bény, C., Kempf, A., & Kribs, D. W. (2007) · Physical Review Letters · doi:10.1103/physrevlett.98.100502 · Cited by 91
Structure of the sets of mutually unbiased bases for N qubits
Romero, J. L., Bjork, G. G., Klimov, A. B., & Sánchez-Soto, L. L. (2005) · Physical Review A · doi:10.1103/physreva.72.062310 · Cited by 93
Experimental Investigation of a Two-Qubit Decoherence-Free Subspace
Altepeter, J. B., Hadley, P. G., Wendelken, S. M., Berglund, A. J., & Kwiat, P. G. (2004) · Physical Review Letters · doi:10.1103/physrevlett.92.147901 · Cited by 95
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He, Y., Luo, M.-X., Zhang, E., Wang, H.-K., & Wang, X. (2017) · International Journal of Theoretical Physics · doi:10.1007/s10773-017-3389-4 · Cited by 104
Homological error correction: Classical and quantum codes
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“[10] D. Gottesman, “Class of quantum error-correcting codes saturating the quantum Hamming bound”.”
BackgroundInfluentialQuantum error correction for continuously detected errors
Ahn, C., Wiseman, H. M., & Milburn, G. J. (2003) · Physical Review A · doi:10.1103/physreva.67.052310 · Cited by 92
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Fang, K., & Liu, Z.-W. (2020) · Physical Review Letters · doi:10.1103/physrevlett.125.060405 · Cited by 104
“[50] Daniel Gottesman, “Class of quantum error-correcting codes saturating the quantum hamming bound,” Phys.”
BackgroundOptical Pumping into Many-Body Entanglement
Cho, J., Bose, S., & Kim, M. S. (2011) · Physical Review Letters · doi:10.1103/physrevlett.106.020504 · Cited by 90
Divincenzo, D. P. (1997) · CERN Document Server (European Organization for Nuclear Research) · Cited by 91
Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes
Yoder, T. J., Takagi, R., & Chuang, I. L. (2016) · Physical Review X · doi:10.1103/physrevx.6.031039 · Cited by 95
Asymptotically good quantum codes
Ashikhmin, A., Litsyn, S. N., & Tsfasman, M. A. (2001) · Physical Review A · doi:10.1103/physreva.63.032311 · Cited by 91
“ar X iv :q ua nt -p h/ 00 06 06 1v 1 1 quantum codes with asymptotically non-zero rate and relative distance.”
BackgroundEntanglement increases the error-correcting ability of quantum error-correcting codes
Lai, C., & Brun, T. A. (2013) · Physical Review A · doi:10.1103/physreva.88.012320 · Cited by 90
“…procedure were found to have promising results: we found [[7 , 1 , 5; 2]] and [[7 , 1 , 5; 3]] EAQEC codes from quantum BCH codes, and [[9 , 1 , 7; 4]] and [[9 , 1 , 7; 5]] EAQEC codes from Shor’s 9 qubit code, together with a family of [[ n, 1 , n ; n − 1]] EAQEC codes for n odd, all of which…”
BackgroundMethodInfluentialLi, Z., Xing, L., & Wang, X. (2008) · Physical Review A · doi:10.1103/physreva.77.012308 · Cited by 91
Decoherence-free subspaces for multiple-qubit errors. I. Characterization
Lidar, D. A., Bacon, D., Kempe, J., & Whaley, K. B. (2001) · Physical Review A · doi:10.1103/physreva.63.022306 · Cited by 87
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Yoshida, B. (2011) · Annals of Physics · doi:10.1016/j.aop.2011.06.001 · Cited by 87
“Stabilizer code Here, we give a brief review of stabilizer codes which are quantum codes possessing Hamiltonians to support logical qubits in the ground space with a finite energy gap [27].”
BackgroundMethodThe black-hole/qubit correspondence: an up-to-date review
Borsten, L., Duff, M. J., & Lévay, P. (2012) · Classical and Quantum Gravity · doi:10.1088/0264-9381/29/22/224008 · Cited by 88
Quantum accuracy threshold for concatenated distance-3 codesPreprint
Aliferis, P., Gottesman, D., & Preskill, J. (2005) · Quantum information & computation · doi:10.48550/arxiv.quant-ph/0504218 · Cited by 564
“ore, it will suffice to verify that the 1-gadgets satisfy properties 0–4. 8.1 Stabilizer codes Before proceeding to explicit gadget constructions, we briefly review the theory of binary stabilizer codes [23, 24], which are well suited for applications to fault-tolerant quantum computing. The code space of a stabilizer co…”
MethodInfluentialSelf-citationQuantum error correction in the presence of spontaneous emission
Plenio, M. B., Vedral, V., & Knight, P. L. (1997) · Physical Review A · doi:10.1103/physreva.55.67 · Cited by 82
Achievable rates for the Gaussian quantum channel
Harrington, J., & Preskill, J. P. (2001) · Physical Review A · doi:10.1103/physreva.64.062301 · Cited by 86
The ZX calculus is a language for surface code lattice surgery
de Beaudrap, N., & Horsman, D. (2020) · Quantum · doi:10.22331/q-2020-01-09-218 · Cited by 83
“stabilizer notation [27]).”
Robust quantum metrological schemes based on protection of quantum Fisher information
Lu, X.-M., Yu, S., & Oh, C. H. (2015) · Nature Communications · doi:10.1038/ncomms8282 · Cited by 87
“We show below that concrete robust metrological schemes can be easily constructed based on the stabilizer formalism [34].”
BackgroundMaslov, D. A. (2007) · Physical Review A · doi:10.1103/physreva.76.052310 · Cited by 74
Experimental Violation of Multipartite Bell Inequalities with Trapped Ions
Lanyon, B. P., Zwerger, M. J., Jurcevic, P., Hempel, C., Dür, W., Briegel, H. J., Blatt, R., & Roos, C. F. (2014) · Physical Review Letters · doi:10.1103/physrevlett.112.100403 · Cited by 82
Beyond stabilizer codes .I. Nice error bases
Klappenecker, A., & Rötteler, M. (2002) · IEEE Transactions on Information Theory · doi:10.1109/tit.2002.800471 · Cited by 91
“We classify all nice error bases of small degree, and all nice error bases with abelian index groups.”
MethodTopological order in 1D Cluster state protected by symmetry
Son, W., Luigi Amico, A., & Vedral, V. (2011) · Quantum Information Processing · doi:10.1007/s11128-011-0346-7 · Cited by 79
“= σ z i−1σ x i σ z i+1, Si |C〉 = |C〉 ,∀i with [ Si , S j ] = 0 ,∀i, j [23].”
Codes and Protocols for Distilling T , controlled- S , and Toffoli Gates
Haah, J., & Hastings, M. B. (2018) · Quantum · doi:10.22331/q-2018-06-07-71 · Cited by 79
“To give some numbers when these codes are used in a distillation protocol, consider a in = 10−3 input error rate using the [[912, 112, 6]] code.”
MethodInfluentialEntanglement entropy from tensor network states for stabilizer codes
He, H., Zheng, Y., Bernevig, B. A., & Regnault, N. (2018) · Physical review. B./Physical review. B · doi:10.1103/physrevb.97.125102 · Cited by 81
Advantages of versatile neural-network decoding for topological codes
Maskara, N., Kubica, A., & Jochym-O’Connor, T. (2019) · Physical Review A · doi:10.1103/physreva.99.052351 · Cited by 83
Mutually unbiased bases and symmetric informationally complete measurements in Bell experiments
Tavakoli, A., Farkas, M., Rosset, D., Bancal, J.-D., & Kaniewski, J. (2021) · Science Advances · doi:10.1126/sciadv.abc3847 · Cited by 119
“[32] D.”
BackgroundQuantum error correcting codes from the compression formalism
Choi, M.-D., Kribs, D. W., & Życzkowski, K. (2006) · Reports on Mathematical Physics · doi:10.1016/s0034-4877(06)80041-8 · Cited by 78
Spin squeezing and entanglement for an arbitrary spin
Vitagliano, G., Apellaniz, I., Egusquiza, I. L., & Tóth, G. (2014) · Physical Review A · doi:10.1103/physreva.89.032307 · Cited by 77
Nonstabilizerness via Matrix Product States in the Pauli Basis
Tarabunga, P. S., Tirrito, E., Bañuls, M. C., & Dalmonte, M. (2024) · Physical Review Letters · doi:10.1103/physrevlett.133.010601 · Cited by 86
Introduction to Quantum AlgorithmsPreprint
Shor, P. W. (2000) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/0005003 · Cited by 81
Self-correcting quantum computers
Bombín, H., Chhajlany, R. W., Horodecki, M., & Martin-Delgado, M. A. (2013) · New Journal of Physics · doi:10.1088/1367-2630/15/5/055023 · Cited by 94
“An important class of error correcting codes is that of stabilizer codes[25, 26], which are defined in terms of an abelian subgroup S ⊂ P of the Pauli group P of n qubits.”
BackgroundEfficient Quantum TransformsPreprint
Høyer, P. (1997) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/9702028 · Cited by 104
“Fundamental concepts of quantum error-correction rely on it; see for example [9, 10, 18, 31, 32].”
BackgroundMethodInfluentialAn Introduction to Quantum Error CorrectionPreprint
Gottesman, D. (2000) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/0004072 · Cited by 96
“The stabilizer has 6 generators on 7 qubits, so it encodes 1 qubit — it is a [[7 , 1 , 3]] code.”
BackgroundSelf-citationUpper bounds on the size of quantum codes
Ashikhmin, A., & Litsyu, S. (1999) · IEEE Transactions on Information Theory · doi:10.1109/18.761270 · Cited by 73
Djordjević, I. B. (2012) · Elsevier eBooks · doi:10.1016/b978-0-12-385491-9.00007-1 · Cited by 84
Graphical nonbinary quantum error-correcting codes
Hu, D. M., Tang, W., Zhao, M., Chen, Q., Yu, S., & Oh, C. H. (2008) · Physical Review A · doi:10.1103/physreva.78.012306 · Cited by 74
Multilinear formulas and skepticism of quantum computing
Aaronson, S. (2004) · ACM Symposium on Theory of Computing (STOC) · doi:10.1145/1007352.1007378 · Cited by 75
“Coset states arise as codewords in the class of quantum error-correcting codes known as stabilizer codes [11, 20, 37].”
BackgroundClifford gates by code deformation
Bombín, H. (2011) · New Journal of Physics · doi:10.1088/1367-2630/13/4/043005 · Cited by 74
“Geometry and gauge generators In stabilizer codes [26], the system is a collection of n qubits and the code subspace is defined in terms of a subgroup S of the Pauli group Pn := 〈i1, X1, Z1, .”
BackgroundDigital quantum simulation with Rydberg atoms
Weimer, H., Müller, M., Büchler, H. P., & Lesanovsky, I. (2011) · Quantum Information Processing · doi:10.1007/s11128-011-0303-5 · Cited by 70
“This model is paradigmatic for a whole class of so-called stabilizer codes [10,18].”
MethodHomological Error Correction: Classical and Quantum Codes
Bombín, H., & Martín-Delgado, M. A. (2006) · CERN Document Server (European Organization for Nuclear Research) · Cited by 74
Fault-Tolerant Holonomic Quantum Computation
Oreshkov, O., Brun, T. A., & Lidar, D. A. (2009) · Physical Review Letters · doi:10.1103/physrevlett.102.070502 · Cited by 70
Structure of 2D Topological Stabilizer Codes
Bombín, H. (2014) · Communications in Mathematical Physics · doi:10.1007/s00220-014-1893-4 · Cited by 69
“A stabilizer code [15,19] on n qubits is defined by a subgroup of Pauli operators S ⊂ P , called stabilizer group, such that−1 ∈ S.”
BackgroundBinary LCD Codes and Self-Orthogonal Codes From a Generic Construction
Zhou, Z., Li, X., Tang, C., & Ding, C. (2018) · IEEE Transactions on Information Theory · doi:10.1109/tit.2018.2823704 · Cited by 71
“Self-orthogonal codes can be used to construct quantum error-correcting codes, which can protect quantum information in quantum computations and quantum communications [3], [15].”
Background
Metadata
- Authors
- Gottesman, D.
- Affiliations
- California Institute of Technology (US)
- Corresponding author
- Gottesman, Daniel
- ORCID
- Gottesman, Daniel
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 1996-09-01
- Volume
- 54
- Issue
- 3
- Pages
- 1862-1868
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W2069711632
- Cited by
- 1,227
- References
- 3
- Citations by year
- 2026: 26 · 2025: 75 · 2024: 48 · 2023: 56 · 2022: 45 · 2021: 65 · 2020: 44 · 2019: 36 · 2018: 34 · 2017: 36 · 2016: 33 · 2015: 32 · 2014: 41 · 2013: 38 · 2012: 34
- FWCI
- 38.07
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review A (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- U.S. Department of Energy (FG0392ER40701)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Mathematics › Discrete Mathematics and Combinatorics › Coding theory and cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Keywords
- quantum error-correcting codes, code construction, single error correction, qubit encoding, code rate, quantum information
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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