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Class of quantum error-correcting codes saturating the quantum Hamming bound
Open accessAbstract
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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“Daniel Gottesman [5] developed methods for analyzing quantum error-correcting codes, and use these methods to construct an infinite class of codes saturating the quantum Hamming bound.”
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“Usually, LP bounds provide the strongest known bounds on code sizes for quantum codes [5], [12], [13], [14], [15], outperforming the quantum Hamming bounds and quantum Singleton bounds [16], [17], [18], [19].”
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BackgroundDesign of Short Codes for Quantum Channels with Asymmetric Pauli Errors
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A Note on Quantum Hamming BoundPreprint
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“quantum MDS code of arbitrary length in this range (see [9] for some numerical results). Another important bound for quantum codes is the quantum Hamming bound. The quantum Hamming bound states (see [6,7]) that: 1 Theorem 2(Quantum Hamming Bound). Any pure ((n,K,d)) q stabilizer code satisfies ⌊(d−X1)/2⌋ i=0 n i (q2 −1…”
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Flag fault-tolerant error correction for cyclic CSS codes
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“For example, the parity check matrix of the two-fold product of the [3,2,2] single parity check code with itself is 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 Now, think of Gallager’s construction of classical LDPC codes [14, Section…”
BackgroundInfluentialMulti-hop fault-tolerant teleportation of arbitrary two-qubit states with cluster channel
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“Finally, for the [8 , 3 , 3] code [23] we use the graph state from [24]).”
MethodA Brief Review and Challenges in Quantum Computing
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“The major quantum error correcting codes [18] include Shor code [19], Steane code [20], 5-qubit codes [21], CSS codes [20], [22], Stabilizer codes [23], Bacon-Shor codes [24], Surface code and Color code [25], [26], and Bosonic codes [27].”
Measurement-free reconstruction circuit of quantum secrets in quantum secret sharing
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“…sharing, in which the dealer encodes a 𝑘 -qudit quantum secret into a stabilizer codeword, each quantum symbol in the codeword is distributed to each participant as a quantum share, and finally a qualified set 𝐽 of participants collectively reconstructs the quantum secret from available shares.”
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Ahn, C., Wiseman, H. M., & Milburn, G. J. (2004) · IEEE International Workshop on Workload Characterization (WWC) · doi:10.1109/phycon.2003.1237011 · Cited by 1
New decoding algorithms for a class of subsystem codes and generalized shor codes
Sarvepalli, P. K., Klappenecker, A., & Rötteler, M. (2009) · IEEE International Symposium on Information Theory (ISIT) · doi:10.1109/isit.2009.5205650 · Cited by 1
“Then for every error e on n1n2 qudits, there exists an error e′ in the coset eS such that the support of bit flip errors in e′ is restricted to k2 columns.”
BackgroundQuantum Codes Derived from Self-Orthogonal Codes over Large Finite Rings
Lu, H., Dong, X., Liu, Z., & Zhang, M. (2016) · International Conference on Information Science and Control Engineering · doi:10.1109/icisce.2016.117 · Cited by 1
“Keywords-nonbinary quantum code; error basis; selforthogonal code I. INTRODUCTION In quantum communication and quantum computation, decoherence inevitably occurs because of environmental or operational noise.”
BackgroundBondal, A. I., & Zhdanovskiy, I. Y. (2021) · Успехи математических наук · doi:10.4213/rm9983 · Cited by 1
Discretization, the Road to Quantum Computing?
de Lacalle, J. G. L. (2021) · IntechOpen eBooks · doi:10.5772/intechopen.98827 · Cited by 2
Finding the disjointness of stabilizer codes is NP-complete
Bostanci, J., & Kubica, A. (2021) · Physical Review Research · doi:10.1103/physrevresearch.3.043192 · Cited by 1
“Although systematic approaches to finding transversal logical gates for generic quantum error-correcting codes are not known, for stabilizer codes [32] we can rule out the possibility of implementing certain logical operations.”
Ternary optimal quantum codes constructed from caps in $$PG(k,9)(k \ge 2)$$
Li, H., Li, R., Fu, Q., & Zhan, X. (2022) · Quantum Information Processing · doi:10.1007/s11128-022-03437-5 · Cited by 1
“2 (quantum Hamming bound [21, 22]) For any pure quantum code [[n, k, d]]q , t = ⌊ d−1 2 ⌋ , then”
Backgroundde Albuquerque, C. D., da Silva, E. B., & Soares, W. S. (2022) · SpringerBriefs in mathematics · doi:10.1007/978-3-031-06833-1_4 · Cited by 1
Fault-tolerant error correction for quantum Hamming codes with only two ancillary qudits
Zhao, L.-Y., Chen, X., Xu, G., Zhang, J., & Yang, Y. (2023) · Quantum Information Processing · doi:10.1007/s11128-022-03796-z · Cited by 1
“Gottesman [10] and Calderbank et al. [11] constructed several well-formed QECCs, such as stabilizer codes in two-dimensional systems.”
BackgroundPerfect and Quasi-Perfect Codes for the Bosonic Classical-Quantum Channel
Coll, A. B., & Fonollosa, J. R. (2023) · IEEE Transactions on Quantum Engineering · doi:10.1109/tqe.2023.3237986 · Cited by 1
“For more information, see http://creativecommons.org/licenses/by/4.0/ VOLUME 4, 2023 2100208 Engineeringuantum Transactions onIEEE Coll and Fonollosa: PERFECT AND QUASI-PERFECT CODES FOR THE BOSONIC CLASSICAL-QUANTUM CHANNEL stabilizer codes were introduced by Calderbank [13], [14] and Gottesman [15].”
MethodInfluentialHomology of Generic Stabilizer StatesPreprint
Wirthmüller, K. (2008) · arXiv (Cornell University) · doi:10.48550/arxiv.0809.3346 · Cited by 1
“Stabilizer states were first described in [4] in the context of error-correcting quantum codes, and the formalism has been repeatedly presented since [2,5,6,7,9,12].”
BackgroundDesign of Quantum Stabilizer Codes From Quadratic Residues SetsPreprint
Xie, Y., Yuan, J., Qifu, & Sun (2014) · arXiv (Cornell University) · doi:10.48550/arxiv.1407.8249 · Cited by 2
“n [[N,K, dmin]] d † ≤ 2n 1 [[5,1,3]] (4) 3 [[13,1,5]] 6 7 [[29,1,11]] 12 9 [[37,1, 12]] 12 13 [[53,1,15]] 16 15 [[61,1,17]] 18 25 [[101,1,21]] 22”
BackgroundInfluential
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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