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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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Controlled Quantum Secure Direct Communication by Using Four Particle Cluster States
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Application of Hermitian self-orthogonal GRS codes to some quantum MDS codes
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Characterization of entanglement on superconducting quantum computers of up to 414 qubits
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“Graph states are additionally useful for quantum error correcting codes [54], quantum secure communication [55], quantum metrology [56], and probing Bell inequalities over multi-partite systems [57].”
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Synthesis of and compilation with time-optimal multi-qubit gates
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“They have been employed to realize several tasks in quantum information processing, including quantum metrology [44], quantum error correcting codes [45] and one-way quantum computing [46].”
MethodMa, P.-C., Chen, G.-B., Li, X., & Zhan, Y.-B. (2017) · Quantum Information Processing · doi:10.1007/s11128-017-1764-y · Cited by 24
Quantum State Sharing of an Arbitrary Two-Atom State by Using a Six-Atom Cluster State in Cavity QED
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Hierarchical quantum information splitting with eight-qubit cluster states
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“Briegel [33], Raussendorf [34], Schlingemann [35] and Walther [36] investigated that cluster state can be used in one way quantum computation and quantum error correction, and it has been experimentally proved [36].”
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“raph states illuminate the entanglement in measurement-based protocols [15], finding wide usage throughout quantum information: from measurement-based quantum computing [11], to error correcting codes [16] and quantum communications [17]. Different graph states are suited to different applications, so the ability to ge…”
BackgroundRobust Self-Testing of Multiparticle Entanglement
Wu, D., Zhao, Q., Gu, X., Zhong, H.-S., Zhou, Y., Peng, L.-C., Qin, J., Luo, Y.-H., Chen, K., Li, L., et al. (2021) · Physical Review Letters · doi:10.1103/physrevlett.127.230503 · Cited by 22
Novel Encoding method for Quantum Error Correction
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“They act on a quantum state as unitary linear operators on C p defined by Thus, Hence, the set of bit errors A quantum error e on an n -system state is a unitary linear operator on ( C p ) ⊗ n of the form Then the quantum error e in (2.1) can be denoted by e = ζ λ X ( a ) Z ( b ) and the set of all…”
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High-Efficient Arbitrated Quantum Signature Scheme Based on Cluster States
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Local complementation rule for continuous-variable four-mode unweighted graph states
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Non-additive quantum codes from Goethals and Preparata codes
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“, a stabilizer code with parameters [[n, 0, d]] (see [8], [15]).”
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Measurement-based quantum communication with resource states generated by entanglement purification
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Quantum communication in noisy environments
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Topological graph states and quantum error-correction codes
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Location of quantum information in additive graph codes
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“The five qubit code just mentioned is an example of a quantum code that is locally equivalent to an additive graph code [13], and the information location has an “all or nothing” character.”
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Self-citationScheme for tunable quantum phase gate and effective preparation of graph-state entanglement
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Hierarchical Controlled Remote State Preparation by Using a Four-Qubit Cluster State
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Measuring the Parity of an N -Qubit State
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Generation and Distribution of GHZ States in Quantum Networks
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“They are used in many multiparty applications of quantum information such as error correction [18], quantum secret sharing [19], anonymous transmission [20], quantum metrology [21], conference key agreement [22], clock synchronization [23], and extending the baseline of telescopes [24].”
BackgroundProgress in integrated and fiber optics for time-bin based quantum information processing
Montaut, N., George, A., Monika, M., Nosrati, F., Yu, H., Sciara, S., Crockett, B., Peschel, U., Wang, Z. M., Franco, R. L., et al. (2025) · Advanced Optical Technologies · doi:10.3389/aot.2025.1560084 · Cited by 22
Generalized graph states based on Hadamard matrices
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Applications of universal parity quantum computation
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The Quantum Entanglement of Binary and Bipolar Sequences
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“1 LDPC and Turbo-Decoding strategies also exploit ’low-density’ [19,12,3], and it is hoped that APF in conjunction with graph-based models will be useful for the construction and analysis of many-qubit entangled systems [27,9,34].”
BackgroundOptimized Distribution of Entanglement Graph States in Quantum Networks
Fan, X., Zhan, C., Gupta, H., & Ramakrishnan, C. R. (2025) · IEEE Transactions on Quantum Engineering · doi:10.1109/tqe.2025.3552006 · Cited by 13
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Qudit hypergraph states and their properties
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A direct approach to fault-tolerance in measurement-based quantum computation via teleportation
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Entropic uncertainty relations and the stabilizer formalism
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Bell inequality tests of four-photon six-qubit graph states
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“Methods of error correction for independent errors based on graph states have been examined for example in [1] and [2].”
BackgroundTransformations of Stabilizer States in Quantum Networks
Englbrecht, M., Kraft, T., & Kraus, B. (2022) · Quantum · doi:10.22331/q-2022-10-25-846 · Cited by 11
Quantum Secret Sharing with Graph States
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Some new Results for Additive Self-Dual Codes over GF(4)
Varbanov, Z. (2007) · Serdica Journal of Computing · doi:10.55630/sjc.2007.1.213-227 · Cited by 11
“There are at least two nonequivalent [[17, 0, 7]] QECC, at least two nonequivalent [[18, 0, 7]] QECC, at least one [[18, 0, 8]] QECC, at least 17 nonequivalent [[19, 0, 7]] QECC, and at least two nonequivalent [[21, 0, 8]] QECC.”
BackgroundComplexity of Graph State PreparationPreprint
Mhalla, M., & Perdrix, S. (2004) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/0412071 · Cited by 11
Commuting Local Hamiltonians on Expanders, Locally Testable Quantum codes, and the qPCP conjecturePreprint
Aharonov, D., & Eldar, L. (2013) · arXiv (Cornell University) · doi:10.4230/lipics.tqc.2025.12 · Cited by 11
Bordbar, M., Naderi, N., & Chamgordani, M. A. (2020) · Indian Journal of Physics · doi:10.1007/s12648-020-01755-x · Cited by 10
Li, Y., Liu, J.-C., & Nie, Y.-Y. (2011) · Communications in Theoretical Physics · doi:10.1088/0253-6102/55/3/10 · Cited by 11
Calculating concentratable entanglement in graph states
Cullen, A. R., & Kok, P. (2022) · Physical Review A · doi:10.1103/physreva.106.042411 · Cited by 11
Variations on Encoding Circuits for Stabilizer Quantum Codes
Grassl, M. (2011) · Lecture notes in computer science · doi:10.1007/978-3-642-20901-7_9 · Cited by 10
“The second method uses a connection to graph codes introduced by Schlingemann and Werner [ 15 ].”
BackgroundMethodKlimov, A. B., Muñoz, C., & Sánchez-Soto, L. L. (2012) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/45/21/215303 · Cited by 10
A direct approach to fault-tolerance in measurement-based quantum computation via teleportation
Silva, M. P. D., Vincent Danos, A., Kashefi, E., & Ollivier, H. (2007) · HAL · Cited by 10
Efficient and Economic Schemes for Remotely Preparing a Four-Qubit Cluster-Type Entangled State
Zhao, S.-Y., Fu, H., Li, X., Chen, G.-B., Ma, P.-C., & Zhan, Y.-B. (2014) · International Journal of Theoretical Physics · doi:10.1007/s10773-014-2047-3 · Cited by 10
Experiments on Multiphoton Entanglement
Kiesel, N. (2007) · doi:10.5282/edoc.7729 · Cited by 10
Generalized graph states and mutually unbiased bases from multi-qudits phase states
Achkir, O., Mohammed Daoud, A., & Mansour, M. (2017) · Modern Physics Letters B · doi:10.1142/s0217984917501834 · Cited by 9
Coherent driving of direct and indirect excitons in a quantum dot molecule
Bopp, F. W., Schall, J., Bart, N., Vögl, F., Cullip, C., Sbresny, F., Boos, K., Thalacker, C., Lienhart, M., Rodt, S., et al. (2023) · Physical review. B./Physical review. B · doi:10.1103/physrevb.107.165426 · Cited by 9
“Furthermore, other specific forms of graph states such as photonic cluster states promise realization of measurement-based quantum computing [4] as well as quantum error correction [5, 6].”
BackgroundEntanglement-enhanced quantum error-correcting codes
Dong, Y., Deng, X., Jiang, M., Chen, Q., & Yu, S. (2009) · Physical Review A · doi:10.1103/physreva.79.042342 · Cited by 9
Dinh, H. Q., Le, H. T., Nguyen, B. T., & Tansuchat, R. (2021) · Quantum Information Processing · doi:10.1007/s11128-021-03306-7 · Cited by 10
“Applying graph theory, some authors can also construct qMDS codes [44,54,90,91].”
MethodFeasible schemes for preparing all five-photon graph states
Zhang, J., Zhou, X., & Tao, Y. (2007) · Chinese Physics · doi:10.1088/1009-1963/16/8/012 · Cited by 9
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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