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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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“kgenerators, denoted Xˆ i, which commute with the stabilizer but not with all of the Zˆ i’s, to obey the relations [Zˆ i,Zˆ j] = 0 = [Xˆ i,Xˆ j] , [Zˆ i,Xˆ j] = 0 ,(i6= j) , Zˆ iXˆ i +Xˆ iZˆ i = 0 . (29) Since Xˆ i anticommutes with Zˆ i, it flips the eigenvalue of Zˆ i, and hence the value of qubit i. All of these ope…”
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Self-citationQuantum computations on a topologically encoded qubit
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“The distance d of the code is the minimum weight of any operator in N ( S ) \ S [ 12, 20 ].”
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“[70,38,4]).”
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“[27] Daniel Gottesman, “Class of quantum error-correcting”
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“[28] Daniel Gottesman, “Class of quantum error-correcting codes saturating the quantum hamming bound,” Phys.”
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Benchmarking Quantum Computers: The Five-Qubit Error Correcting Code
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Bosonic quantum codes for amplitude damping
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Performance of deterministic dynamical decoupling schemes: Concatenated and periodic pulse sequences
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Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes
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Causal and localizable quantum operations
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Statistical mechanics of quantum error correcting codes
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“[7] Daniel Gottesman, “Class of quantum error-correcting codes saturating the quantum Hamming bound,” Physical Review A 54, 1862–1868 (1996), arXiv:quantph/9604038 [quant-ph].”
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Self-organized error correction in random unitary circuits with measurement
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Optimal resources for topological two-dimensional stabilizer codes: Comparative study
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“A suitable formalism to study quantum error correction codes is the stabilizer formalism [4].”
MethodAnalog Quantum Error Correction
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Single-Shot Fault-Tolerant Quantum Error Correction
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“Stabilizer codes [26] are defined in systems composed of a number of physical qubits.”
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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