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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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“Theorem 1 [1, 5, 6] If there is an The quantum code Q constructed by this method is called a stabilizer code and is proposed independently by Gottesman [11] and Calderbank et al. [5, 6].”
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Quantum convolutional error-correcting codes
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A Subfield-Based Construction of Optimal Linear Codes Over Finite Fields
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“There are also sets of quantum gates that are known not to be universal: for example, the basispreserving gates, the 1-qubit gates, and most interestingly, the so-called stabilizer gates [11, 3] (that is, the CNOT, Hadamard, and π/4-Phase gates), as well as the stabilizer gates conjugated by 1qubit unitary transformat…”
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The Classification of Reversible Bit Operations
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BackgroundThree-qubit quantum error-correction scheme for collective decoherence
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MethodQuantum Error Correction Protects Quantum Search Algorithms Against Decoherence
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Generalized X states of N qubits and their symmetries
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Entanglement properties of topological color codes
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Channel-Optimized Quantum Error Correction
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Information Rates Achievable With Algebraic Codes on Quantum Discrete Memoryless Channels
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“A symplectic quantum code is a simultaneous eigenspace of a set of commuting operators, which is called a stabilizer.”
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“(30) du(j, t) dt = α(2) 12�x2 [+11u(j − 1, t)− 20u(j, t)+ 6u(j + 1, t)+ 4u(j + 2, t)− u(j + 3, t)] (j = 2)”
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“The encoding process maps a single logical qubit | 𝜓 ⟩ = 𝛼 | 0 ⟩ + 𝛽 | 1 ⟩ onto a three-qubit code space: CSS Codes [45] Quantum Stabilizer codes [46] Quantum BCH (QBCH) Codes [47] Quantum Reed Mullers Codes [48] Quantum Reed Solomon Codes [49] 2000 Quantum LPD Codes (QLPDC) [50] Quantum…”
q-Ary Chain-Containing Quantum Synchronizable Codes
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“For instance, the CalderbankShor-Steane (CSS) code [3]–[5], the stabilizer formalism and stabilizer codes [6].”
Efficient color code decoders in d ≥ 2 dimensions from toric code decoders
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“Stabilizer codes are quantum error correcting codes [34, 35] specified by the stabilizer group S, which is an Abelian subgroup of the Pauli group Pn generated by tensor products of Pauli operators on n qubits.”
MethodQuantum error correction with gauge symmetries
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QUANTUM ERROR-CORRECTING CODES
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“Die allgemeine Konstruktion von so genannten stabilizer codes [4] oder additiven Quantencodes [2] basiert auf klassischen Blockcodes über dem Körper GF(4) mit vier Elementen.”
BackgroundFault-Tolerant Code-Switching Protocols for Near-Term Quantum Processors
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Asymmetric Entanglement-Assisted Quantum Error-Correcting Codes and BCH Codes
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“The first steps in the construction of quantum errorcorrecting codes corresponded to the binary case [8], [9], [22] (see also [2], [3], [25]).”
BackgroundAly, S. A., & Klappenecker, A. (2008) · IEEE International Symposium on Information Theory (ISIT) · doi:10.1109/isit.2008.4595010 · Cited by 28
“[[8, 1, 5, 2]]3 [8, 6, 3]32 [[8, 4, 2, 2]]3 [8, 3, 6] 32 [[8, 5, 1, 2]]3 [8, 2, 7] 32 [[9, 1, 4, 3]]3 [9, 6, 4] †2 ,� = 3 [[9, 4 , 1, 3]]3 [9, 3, 7] †2 ,� = 6”
BackgroundMethodInfluentialUniversal quantum computation in decoherence-free subspaces with hot trapped ions
Aolita, L., Davidovich, L., Kim, K., & Häffner, H. (2007) · Physical Review A · doi:10.1103/physreva.75.052337 · Cited by 25
Spatially correlated qubit errors and burst-correcting quantum codes
Vatan, F., Roychowdhury, V., & Anantram, M. P. (1999) · IEEE Transactions on Information Theory · doi:10.1109/18.771250 · Cited by 28
“The theory of quantum error correction is now an active area of research [17, 5, 18, 10, 2, 8]. Various techniques regarding construction of quantum error correcting codes based on classical codes have been proposed [17, 5, 18, 8 ]. These papers typically assume the”
BackgroundMethodInfluentialPerturbative expansions for the fidelities and spatially correlated dissipation of quantum bits
Duan, L.-M., & Guo, G. (1997) · Physical Review A · doi:10.1103/physreva.56.4466 · Cited by 26
All the Stabilizer Codes of Distance 3
Yu, S., Bierbrauer, J., Dong, Y., Chen, Q., & Oh, C. H. (2013) · IEEE Transactions on Information Theory · doi:10.1109/tit.2013.2259138 · Cited by 27
“The redundancy counts the number of the independent generators of the stabilizer.”
BackgroundDistant entanglement protected through artificially increased local temperature
Carvalho, A. R. R., & Santos, M. F. (2011) · New Journal of Physics · doi:10.1088/1367-2630/13/1/013010 · Cited by 27
“In the last few decades, different approaches to overcome its effects have been explored including quantum error correction codes[1, 2, 3, 4], dynamical decoupling [5], or encoding the system in decoherence free subspaces [6, 7].”
BackgroundResearch problems on numerical ranges in quantum computing
Kribs, D. W., Pasieka, A., Laforest, M., Ryan, C. A., & Silva, M. P. D. (2009) · Linear and Multilinear Algebra · doi:10.1080/03081080802677441 · Cited by 27
“The stabilizer formalism [10] for quantum error correction is primarily concerned with constructing codes for such channels.”
BackgroundLópez, C. C., Bendersky, A., Paz, J. P., & Cory, D. G. (2010) · Physical Review A · doi:10.1103/physreva.81.062113 · Cited by 27
Particle-wave duality: a dichotomy between symmetry and asymmetry
Vaccaro, J. A. (2011) · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rspa.2011.0271 · Cited by 27
“n terms of the mutually unbiased bases (MUB) introduced by Schwinger (1960). The study of MUB is important for areas such as discrete Wigner functions (Gibbons et al. 2004), quantum error correction (Gottesman 1996), quantum cryptography (Miyadera & Imai 2006) as well as entangled systems (Kalev et al. 2009, Berta…”
BackgroundQuantum Error Correction and Fault Tolerance
Gottesman, D. (2006) · Elsevier eBooks · doi:10.1016/b0-12-512666-2/00273-x · Cited by 27
“An example of a stabilizer code is the 5-qubit code, a [[ 5 ,1,3]] code whose stabilizer can be generated by”
BackgroundSelf-citation
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
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