The DOI matches its Crossref record; 21 of the 21 references it cites were found in a registry or an index.
Quantum error-correcting codes associated with graphs
Open accessAbstract
We present a construction for quantum error correcting codes. The basic ingredients are a graph and a finite Abelian group, from which the code can explicitly be obtained. We prove necessary and sufficient conditions for the graph such that the resulting code corrects a certain number of errors. This allows a simple verification of the one-error correcting property of codes of length 5 in any dimension. As examples, we construct a large class of maximum distance separable codes, i.e. codes saturating the Singleton bound, as well as a code of length 10 detecting three errors.
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“However, the computational model that emerges for the QC C [7] makes no reference to the concept of unitary evolution”
From Measurement-based quantum computation on cluster states (2003) · cites this paper as [7]
In the citing paper
“In [7] it has been shown, as stated in Section III A, that the required classical processing increases the computation time only marginally (logarithmically in the number n of logical qubits) and thus there is no exponential overhead in either classical or quantum resources.”
From Measurement-based quantum computation on cluster states (2003) · cites this paper as [7]
In the citing paper
“In conclusion, instead of describing a quantum algorithm as a network of gates applied to some fiducial input state, a quantum algorithm may (arguably more effectively) be characterized by a graph specifying the quantum resource and the structure [7] for the processing of the measurement outcomes.”
From Measurement-based quantum computation on cluster states (2003) · cites this paper as [7]
In the citing paper
“Concerning the resources for the classical processing of the measurement outcomes in a QC C -computation, we point out that this processing increases the total time of computation only marginally [7].”
From Measurement-based quantum computation on cluster states (2003) · cites this paper as [7]
In the citing paper
“As shown in [7], it turns out that there is none.”
From Measurement-based quantum computation on cluster states (2003) · cites this paper as [7]
In the citing paper
“If the network algorithm runs on n qubits then the classical data that the QC C has to keep track of is entirely contained in a 2ncomponent binary valued vector, which we have called the information flow vector I(t) [7].”
From Measurement-based quantum computation on cluster states (2003) · cites this paper as [7]
In the citing paper
“However, as we have explained in [7] and also briefly sketched in Section III A, the QC C has no quantum register.”
From Measurement-based quantum computation on cluster states (2003) · cites this paper as [7]
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Citations
Cites
A single quantum cannot be cloned
Wootters, W. K., & Zurek, W. H. (1982) · Nature · doi:10.1038/299802a0 · Cited by 5,406 · cited 1×
“On the other hand, the no-cloning theorem [1] seemed to forbid at least the most naive approach to classical error correction for noisy channels, e.”
BackgroundMixed-state entanglement and quantum error correction
Bennett, C. H., DiVincenzo, D. P., Smolin, J. A., & Wootters, W. K. (1996) · Physical Review A · doi:10.1103/physreva.54.3824 · Cited by 5,272 · cited 1×
“In the simplest case, the fivefold code [9–11] (for qubits as well as higher dimensional systems), it can be verified in a few lines that any two errors are corrected.”
BackgroundGood 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 · cited 1×
“It was therefore an important step to realize [2] that quantum mechanics had other, more subtle ways of “distributing” quantum information over several channels to stabilize against errors.”
BackgroundQuantum error correction via codes over GF(4)
Calderbank, A. R., Rains, E. M., Shor, P. W., & Sloane, N. J. A. (1998) · IEEE Transactions on Information Theory · doi:10.1109/18.681315 · Cited by 1,831 · cited 2×
“One problem with the known schemes of quantum error correction (e.g. [3,4]), however, is that they tend to be subtle indeed, and the verification of their error correcting capabilities often requires a lengthy computation.”
BackgroundMethodTheory of quantum error-correcting codes
Knill, E., & Laflamme, R. (1997) · Physical Review A · doi:10.1103/physreva.55.900 · Cited by 1,593
Class of quantum error-correcting codes saturating the quantum Hamming bound
Gottesman, D. (1996) · Physical Review A · doi:10.1103/physreva.54.1862 · Cited by 1,227 · cited 1×
“There are various efficient methods for constructing stabilizer codes [3,4,12–16].”
MethodJacobson, N. S. (1964) · Graduate texts in mathematics · doi:10.1007/978-1-4612-9872-4 · Cited by 1,051
Perfect Quantum Error Correcting Code
Laflamme, R., Miquel, C., Paz, J. P., & Zurek, W. H. (1996) · Physical Review Letters · doi:10.1103/physrevlett.77.198 · Cited by 966
Quantum Error Correction and Orthogonal Geometry
Calderbank, A. R., Rains, E. M., Shor, P. W., & Sloane, N. J. A. (1997) · Physical Review Letters · doi:10.1103/physrevlett.78.405 · Cited by 811 · cited 1×
“[3,4]), however, is that they tend to be subtle indeed, and the verification of their error correcting capabilities often requires a lengthy computation.”
BackgroundStabilizer Codes and Quantum Error Correction
Gottesman, D. (1997) · arXiv · doi:10.48550/arxiv.quant-ph/9705052 · Cited by 753
Research works printed but not linked: 1
- R. Matsumoto, 2000, IEICE Trans. Fundamentals, E83-A, 10
Cited by
Sorted by citations
Items 401–405 of 405
Quantum hypergraph states: a review
Poderini, D., Bruß, D., & Macchiavello, C. (2026) · Reports on Progress in Physics · doi:10.1088/1361-6633/ae7735
“Graph states are fundamental in quantum information processing, specifically in protocols such as measurement-based quantum computation (MBQC) [1, 3] and quantum error correction [4], but also in communication protocols, for instance for quantum secret sharing [4].”
InfluentialAhmad, M., Ali, L., Rameez‐ul‐Islam, Imran, M., Alomar, M., & Ahmad, I. (2026) · Optical and Quantum Electronics · doi:10.1007/s11082-026-08944-8
“…used to study the intricate physical or biological phenomena such as quantum networks Zou ( 2021); Gaertner et al. (2007), quantum error correction Schlingemann and Werner ( 2001), quantum communication Hein et al. (2004 ); West ( 2001), quantum computation Raussendorf et al. ( 2003 ) and many…”
Game-theoretic discovery of quantum error-correcting codes through Nash equilibria
Anonymous (2026) · Physical Review A · doi:10.1103/x1ph-v78g
Faster than MIP Decoding of Graph Codes with a Single Logical Qubit
Basak, N., & Paul, G. (2026) · IEEE International Conference on Quantum Control, Computing and Learning (qCCL) · doi:10.1109/qccl70331.2026.11668573
“Graph codes [21]–[26] are a class of quantum error-correcting codes, where quantum information is encoded using the stabilizer formalism [27], [28] defined by an underlying graph structure, yielding an (cid:74) n, k, d (cid:75) code that encodes k logical qubits into n physical qubits with distance…”
Analysis of establishment time of GHZ entanglement over remote nodes
Mišić, J., & Mišić, V. B. (2026) · International Conference on Future and Intelligent Networking (FINE) · doi:10.1109/fine71157.2026.00037
“Quantum computing requires interconnection between quantum registers in order to perform applications such as synchronization of clocks [20], quantum anonymous transmission and voting [6], [27], multi-user key exchange [15], [16], and error detection and correction [23], [25].”
Metadata
- Authors
- Schlingemann, D., & Werner, R. F.
- Affiliations
- Technische Universität Braunschweig (DE)
- ORCID
- Werner, R. F.
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2001-12-11
- Volume
- 65
- Issue
- 1
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W1503834475
- Cited by
- 406
- References
- 19
- Citations by year
- 2026: 13 · 2025: 18 · 2024: 16 · 2023: 20 · 2022: 15 · 2021: 16 · 2020: 21 · 2019: 13 · 2018: 10 · 2017: 10 · 2016: 15 · 2015: 11 · 2014: 16 · 2013: 22 · 2012: 24
- FWCI
- 8.19
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review A (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- European Commission
- 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, graph codes, finite abelian groups, code construction, error correction capability, MDS codes, Singleton bound
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