Simple quantum error-correcting codes
Open accessAbstract
Methods of finding good quantum error-correcting codes are discussed, and many example codes are presented. The recipe ${\mathit{C}}_{2}^{\mathrm{\ensuremath{\perp}}}$\ensuremath{\subset}${\mathit{C}}_{1}$, where ${\mathit{C}}_{1}$ and ${\mathit{C}}_{2}$ are classical codes, is used to obtain codes for up to 16 information quantum bits (qubits) with correction of small numbers of errors. The results are tabulated. More efficient codes are obtained by allowing ${\mathit{C}}_{1}$ to have reduced distance, and introducing sign changes among the code words in a systematic manner. This systematic approach leads to single-error-correcting codes for 3, 4, and 5 information qubits with block lengths of 8, 10, and 11 qubits, respectively. \textcopyright{} 1996 The American Physical Society.
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In 1D yoked surface codes, parity checks are measured along each row of an array of surface codes.
“In 1D, these consist of parity checks along each row of an array of surface codes29.”
From Yoked surface codes (2025) · cites this paper as [29]
1D and square 2D parity check codes form families of [[n, n − 2, 2]] and [[n2, n2 − 4n + 2, 4]] codes, respectively.
“In particular, for 1D and square 2D codes, we obtain families of [[n, n − 2, 2]]29 and [[n2, n2 − 4n + 2, 4]] codes, respectively.”
From Yoked surface codes (2025) · cites this paper as [29]
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- Authors
- Steane, A. M.
- Affiliations
- University of Oxford (GB)
- Corresponding author
- Steane, A. M.
- ORCID
- Steane, A. M.
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 1996-12-01
- Volume
- 54
- Issue
- 6
- Pages
- 4741-4751
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W2063592138
- Cited by
- 522
- References
- 15
- Citations by year
- 2026: 28 · 2025: 50 · 2024: 34 · 2023: 37 · 2022: 37 · 2021: 21 · 2020: 37 · 2019: 20 · 2018: 17 · 2017: 12 · 2016: 15 · 2015: 17 · 2014: 14 · 2013: 14 · 2012: 11
- FWCI
- 16.25
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review A (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Royal Society
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Computer Science › Artificial Intelligence › Coding theory and cryptography
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Keywords
- quantum error-correcting codes, CSS construction, classical codes, qubits, single-error-correcting codes, code distance, systematic codes, block length, 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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