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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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“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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Dangniam, N., Han, Y.-G., & Zhu, H. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.043323 · Cited by 43
MEASUREMENT-BASED QUANTUM COMPUTATION WITH CLUSTER STATES
Raussendorf, R. (2009) · International Journal of Quantum Information · doi:10.1142/s0219749909005699 · Cited by 47
“The treatment here follows that of [53], adapted to our notation.”
BackgroundInfluentialA Simple Protocol for Certifying Graph States and Applications in Quantum Networks
Markham, D., & Krause, A. (2020) · Cryptography · doi:10.3390/cryptography4010003 · Cited by 43
“They are incredibly useful resources across quantum information, acting as the key entanglement resource for error correction [2], measurement based quantum computation [3], quantum secret sharing [4] and more [1].”
BackgroundOptimal quantum-state reconstruction for cold trapped ions
Klimov, A. B., Muñoz, C., Fernández, A., & Saavedra, C. (2008) · Physical Review A · doi:10.1103/physreva.77.060303 · Cited by 41
Optimal preparation of graph states
Cabello, A., Danielsen, L. E., López-Tarrida, A. J., & Portillo, J. R. (2011) · Physical Review A · doi:10.1103/physreva.83.042314 · Cited by 41
Noisy evolution of graph-state entanglement
Aolita, L., Cavalcanti, D., Chaves, R., Dhara, C., Davidovich, L., & Acín, A. (2010) · Physical Review A · doi:10.1103/physreva.82.032317 · Cited by 41
Photonic resource state generation from a minimal number of quantum emitters
Li, B., Economou, S. E., & Barnes, E. (2022) · npj Quantum Information · doi:10.1038/s41534-022-00522-6 · Cited by 40
Greenberger-Horne-Zeilinger Paradoxes from Qudit Graph States
Tang, W., Yu, S., & Oh, C. H. (2013) · Physical Review Letters · doi:10.1103/physrevlett.110.100403 · Cited by 40
Photonic graph state generation from quantum dots and color centers for quantum communications
Russo, A., Barnes, E., & Economou, S. E. (2018) · Physical review. B./Physical review. B · doi:10.1103/physrevb.98.085303 · Cited by 50
Entanglement in eight-qubit graph states
Cabello, A., López-Tarrida, A. J., Moreno, P., & Portillo, J. R. (2009) · Physics Letters A · doi:10.1016/j.physleta.2009.04.055 · Cited by 39
Local unitary versus local Clifford equivalence of stabilizer and graph states
Zeng, B., Chung, H., Cross, A. W., & Chuang, I. L. (2007) · Physical Review A · doi:10.1103/physreva.75.032325 · Cited by 39
Optical generation of matter qubit graph states
Benjamin, S. C., Eisert, J., & Stace, T. M. (2005) · CERN Document Server (European Organization for Nuclear Research) · Cited by 39
Solutions of Modular Bootstrap Constraints from Quantum Codes
Dymarsky, A. Y., & Shapere, A. D. (2021) · Physical Review Letters · doi:10.1103/physrevlett.126.161602 · Cited by 38
“We start by reviewing graph codes, a particular family of real self-dual stabilizer codes [18, 19].”
BackgroundMethod
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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