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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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In the citing paper
“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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Extracting cluster states from indistinguishability resource
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Atomic hyperentangled complex graph states engineering for secure quantum communication
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Quantum AMDS Negacyclic Codes of Length 2 p s Over F p m
Dinh, H. Q., Nguyen, B. T., Nguyen, N. V., Thi, H. L., & Yamaka, W. (2025) · IEEE Access · doi:10.1109/access.2025.3595428
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Hybrid reward-driven reinforcement learning for efficient quantum circuit synthesis
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“Before going into the technical details of the algorithm used to build efficient circuits, let us first discuss a few basic concepts which will be used in the remaining part of the paper.”
Robustness of continuous variable quantum key distribution under strong polarization drift
Almeida, M., Pinto, A. N., & Silva, N. A. (2025) · EPJ Quantum Technology · doi:10.1140/epjqt/s40507-025-00417-3 · Cited by 1
Studies of properties of bipartite graphs with quantum programming
Gnatenko, K. P. (2025) · Physics Letters A · doi:10.1016/j.physleta.2025.131191
Efficient construction of stabilizer codes via ZX-calculus
Wu, Z., Cheng, S., & Zeng, B. (2026) · Academia quantum. · doi:10.20935/acadquant8237
Quantum Information Techniques for Quantum Metrology
Shettell, N. (2021) · theses.fr (ABES) · doi:10.70675/8853f6b9zb5e4z450cz9d3ezadfa13e09f3e
Continuous Variable Multimode Quantum States at Telecommunication Wavelengths for Quantum Networks
Fainsin, D. (2023) · theses.fr (ABES) · doi:10.70675/59e0ff53z15b6z40e5z8de0zd3e382a51b50
Phase Transitions and Noise Robustness of Quantum Graph States
Numajiri, T., Yamashika, S., Tanizawa, T., Yoshii, R., Takeuchi, Y., & Tsuchiya, S. (2026) · Quantum · doi:10.22331/q-2026-05-05-2094
“Graph states are a class of entangled states that serve as essential resources for various quantum information processing tasks, including quantum metrology [1], quantum communication [2], quantum error correction [3], and measurement-based quantum computation (MBQC) [4, 5].”
Bipartite entanglement under frequency comb pumping in parametric Josephson circuits
Vartiainen, M., Lilja, I., Mukhanova, E., Petrovnin, K. V., Paraoanu, G. S., & Hakonen, P. J. (2026) · Physical review. B./Physical review. B · doi:10.1103/51mq-qrxd
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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