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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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Graph states of prime-power dimension from generalized CNOT quantum circuit
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How to Correct Small Quantum Errors
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Self-citationGeneration of cluster-type entangled coherent states
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Mutually Unbiased Bases in Low Dimensions
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“The error correcting properties of VΓ are summarized in the following result [16].”
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“Applying graph theory, some authors can also construct qMDS codes [40, 50, 86, 87].”
MethodGeneral framework for genuine multipartite entanglement detection
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Chen, X., & Jiang, L.-Z. (2011) · Physical Review A · doi:10.1103/physreva.83.052316 · Cited by 8
Some Ulam’s reconstruction problems for quantum states
Huber, F., & Simone Severini, B. (2018) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8121/aadd1e · Cited by 8
Univariate and Multivariate Merit Factors
Parker, M. G. (2005) · Lecture notes in computer science · doi:10.1007/11423461_4 · Cited by 8
“Moreover, stabilizer quantum error-correcting codes (QECCs) are exactly described using quadratic Boolean functions [27,28,29,30].”
MethodGeneralized concatenation for quantum codes
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“A one-dimensional stabilizer code [[n, 0, d]] can also be described by a graph with n vertices [23].”
BackgroundQuantum stabilizer codes and beyondPreprint
Sarvepalli, P. K. (2008) · arXiv (Cornell University) · doi:10.48550/arxiv.0810.2574 · Cited by 9
“lizer codes is that there exist links to classical coding theory that facilitate the construction of good codes. More recently, some results were generalized to the case of nonbinary stabilizer codes [1,10,11,28,39,40,53,54,62,71,73,86,102,109, 126,132,137,138], but the theory is not nearly as complete as in the binar…”
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Small quantum networks in the qudit stabilizer formalismPreprint
Miller, D. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1910.09551 · Cited by 8
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Multipartite entanglement and quantum error identification in D -dimensional cluster states
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Stabilizer codes and equientangled bases from phase states
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Dissipative engineering of a tripartite Greenberger–Horne–Zeilinger state for neutral atoms
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“More and more researchers begin to focus on unleashing the potential of multipartite entanglement in the context of measurementbased quantum computation [29–31], quantum error correction [32, 33], quantum networks [34–36], and condensed matter physics [37, 38].”
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“Therefore, some authors used graph theory to construct qMDS codes [39], [44], [69], [70].”
MethodGeometric properties of evolutionary graph states and their detection on a quantum computer
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On the error analysis of quantum repeaters with encoding
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“We remind the reader that they are quantum states associated with mathematical graphs [4, 26].”
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How to transform graph states using single-qubit operations: computational complexity and algorithms
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From graph states to two-graph states
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“Proof: It is known that a stabilizer state is locally-equivalent to a graph state [32, 14], and local-equivalence is reversible.”
BackgroundInfluentialQUANTUM KEY DISTRIBUTION AND REMOTE STATE PREPARATION VIA CLUSTER STATE
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Percolation in quantum computation and communicationPreprint
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“ence of later action on earlier measurement outcomes). It will be necessary – before we establish the percolationargument – to introduce a number of concepts that will be used later on. A graph state [40, 23, 42] is a quantumstate definedon lattice that is describedbyan undir ectedgraphG=(V,E), with Vbeing the set of v…”
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Evidence of a significant full tripartite entanglement in a mixed spin-(1/2,1) Heisenberg tetramer
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“For the application of error correction, Schlingemann [6] proposed the notion of Graph codes that makes use of measurement to design good quantum codes.”
MethodTeleportation of an Arbitrary Two-Particle State via a Single Cluster-Class State
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Interlace polynomials: Enumeration, unimodality and connections to codes
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“It is known that self-dual quantum codes, so called because they correspond to self-dual additive codes over F4 [7], can be represented as graphs [5, 16, 17, 23, 24, 27].”
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Schlingemann, D.-M. (2004) · Journal of Mathematical Physics · doi:10.1063/1.1797533 · Cited by 5
Self-citationTwo Infinite Families of Nonadditive Quantum Error-Correcting Codes
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“Despite their nonadditiveness the codes ((9, 12, 3)) and (10, 24, 3)) admit a stabilizer-like structure and can be most conveniently formulated by using the graph states [12], [17].”
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Linear optics quantum computing - construction of small networks and asymptotic scaling
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Quantum Dense Coding in Trapped Ions
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TRIPARTITE CONTROLLED TELEPORTATION OF AN ARBITRARY TWO-QUBIT STATE WITH MULTIPARTITE CLUSTER STATES
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“There exist quantum codes [[8, 2, 4]]p for all odd prime numbers p.”
BackgroundMethodInfluentialScheme for Implementing Controlled Dense Coding with Six-Atom Cluster State in Cavity QED
Li, Y., Li, C., Wang, X., Sang, M.-H., & Nie, Y.-Y. (2013) · Communications in Theoretical Physics · doi:10.1088/0253-6102/59/5/07 · Cited by 4
On the relation between a graph code and a graph state
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“[10], Schlingemann and Werner showed that a quantum error-correcting code can be constructed based on a mathematical graph and the graph-based quantum code called a graph code belongs to stabilizer codes.”
BackgroundInfluentialExclusivity structures and graph representatives of local complementation orbits
Cabello, A., Parker, M. G., Scarpa, G., & Simone Severini, B. (2013) · Centrum Wiskunde & Informatica (CWI), the national research institute for mathematics and computer science in the Netherlands · Cited by 4
Model for fractons, fluxons, and free vertex excitations
Kim, J., & Han, J. H. (2021) · Physical review. B./Physical review. B · doi:10.1103/physrevb.104.115128 · Cited by 4
An efficient algebraic representation for graph states for measurement-based quantum computing
Corli, S., & Prati, E. (2022) · IEEE International Conference on Rebooting Computing (ICRC) · doi:10.1109/icrc57508.2022.00015 · Cited by 4
Dinh, H. Q., Nguyen, B. T., Paravee, M., Thi, H. L., & Vo, T. M. (2023) · Quantum Information Processing · doi:10.1007/s11128-023-03958-7 · Cited by 4
“Therefore, some authors used graph theory to construct qMDS codes [42, 53, 89, 90].”
MethodQuantum random state generation with predefined entanglement constraint
Daskin, A., Grama, A., & Kais, S. (2014) · International Journal of Quantum Information · doi:10.1142/s0219749914500300 · Cited by 4
“…characterizations [9] of graph states have been studied, and their use has been demonstrated for different applications in quantum error correction [10], quantum communication, and one-way quantum computation [11], among others (Hein et al. [12] present an excellent review of the applications of…”
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Entanglement concentration for arbitrary four-particle linear cluster states
Song, T., Tan, X., & Wang, T. (2017) · Scientific Reports · doi:10.1038/s41598-017-02146-9 · Cited by 4
ESTIMATING PURITY AND ENTROPY IN STABILIZER STATE EXPERIMENTS
Wunderlich, H., & Plenio, M. B. (2010) · International Journal of Quantum Information · doi:10.1142/s0219749910006356 · Cited by 4
Scheme for constructing graphs associated with stabilizer quantum codesPreprint
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“…a more systematic procedure for constructing graphs associated with arbitrary binary stabilizer codes with special emphasis on the verification of their error correcting capabilities which, as pointed out earlier, constituted a major driving motivation for the introduction of graphs in QEC [5].”
BackgroundMethodInfluentialQuantum Random State Generation with Predefined Entanglement ConstraintPreprint
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Universal Graph Representation of Stabilizer Codes
Khesin, A. B., Lu, J. Z., & Shor, P. W. (2025) · PRX Quantum · doi:10.1103/1gjs-2rhx · Cited by 16
Some quantum MDS codes from GRS codes
Wang, G., & Tang, C. (2023) · Linear and Multilinear Algebra · doi:10.1080/03081087.2023.2181937 · Cited by 3
“…developments in the construction of quantum MDS codes with different tools, mainly based on cyclic codes [16,18], negacyclic codes [19], constacyclic codes [20–23], pseudo-cyclic codes [24], polynomials [25], combinatorial structure [26], graph theory [27–29] and (extended) GRS codes [9,10,30–36].”
BackgroundConstructing quantum error correcting code via logic function
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Undeniable quantum state sharing with a five-atom cluster state in cavity QED
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Identifying non-k-separability of a class of N-qubit complete graph states using correlation tensors
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Polylog-LDPC Capacity Achieving Codes for the Noisy Quantum Erasure Channel
Lloyd, S., Shor, P. W., & Thompson, K. (2019) · IEEE Transactions on Information Theory · doi:10.1109/tit.2019.2925100 · Cited by 3
“INTRODUCTION GRAPH States [17], [30] are a very useful set of stabilizer states that have found many applications in quantum information theory.”
BackgroundLearning quantum graph states with product measurements
Ouyang, Y., & Tomamichel, M. (2022) · IEEE International Symposium on Information Theory (ISIT) · doi:10.1109/isit50566.2022.9834440 · Cited by 3
“Graph states [13], [14], [15], [16] are quantum states that correspond directly to undirected simple graphs, and the set of all graph states is equivalent under local Clifford operations to stabilizer states [17], [18], [19].”
BackgroundThe Prospects of Quantum Secure Communication for Secret Sharing
Chou, Y., Tseng, K.-C., Kuo, S., Kuo, S., & Sheu, B. J. (2023) · IEEE Nanotechnology Magazine · doi:10.1109/mnano.2023.3249520 · Cited by 4
“another form of threshold QSS uses the QEcc class [27] with graphs; later, markham and Sand-”
MethodInflated graph states refuting communication-assisted local-hidden-variable models
Meyer, U. I., Grosshans, F., & Markham, D. (2023) · Physical Review A · doi:10.1103/physreva.108.012402 · Cited by 3
“Their application ranges from universal resources for quantum computation [3], to error correction and fault tolerance [4, 5], quantum metrology [6] and to quantum network protocols such as secret sharing [7] and anonymous communication [8].”
BackgroundLecture notes on quantum entanglement: From stabilizer states to stabilizer channels
Arab, A. R. (2024) · Frontiers of Physics · doi:10.1007/s11467-024-1397-4 · Cited by 5
Demonstration of teleportation across a quantum network code
Rall, H., & Tame, M. (2024) · Physical Review A · doi:10.1103/physreva.110.022602 · Cited by 4
Harnessing graph state resources for robust quantum magnetometry under noise
Nguyen, P. T., Le, T. K., Nguyen, H. Q., & Ho, L. B. (2024) · Scientific Reports · doi:10.1038/s41598-024-71365-8 · Cited by 3
Recovery from an eavesdropping attack on a qubit of a graph state
Raina, A., & Garani, S. S. (2019) · Quantum Information Processing · doi:10.1007/s11128-019-2387-2 · Cited by 3
Stabilizer quantum codes: A unified view based on Forney-style factor graphs
Vontobel, P. O. (2008) · International Symposium on Turbo Codes and Iterative Information Processing · doi:10.1109/turbocoding.2008.4658700 · Cited by 3
Zhao, C., & Ye, L. (2011) · Science China Physics Mechanics and Astronomy · doi:10.1007/s11433-011-4259-3 · Cited by 3
Topos logic in measurement-based quantum computation
Loveridge, L., Dridi, R., & Raussendorf, R. (2015) · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rspa.2014.0716 · Cited by 3
“The state is altered by the local measurements (in particular, any entanglement is destroyed); the initial state is thus viewed as a resource, and for suitable choices such as cluster states [28,29] MBQC is universal for quantum computation.”
BackgroundGA-based approach to find the stabilizers of a given sub-space
Houshmand, M., Zamani, M. S., Sedighi, M., & Houshmand, M. (2014) · Genetic Programming and Evolvable Machines · doi:10.1007/s10710-014-9219-z · Cited by 3
Determining X -chains in graph states
Wu, J.-Y., Kampermann, H., & Bruß, D. (2015) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/49/5/055302 · Cited by 3
Quantum Error-Correcting Codes
Zeng, B., Chen, X., Zhou, D. L., & Wen, X.-G. (2019) · Quantum science and technology · doi:10.1007/978-1-4939-9084-9_3 · Cited by 3
Shaded Tangles for the Design and Verification of Quantum Programs (Extended Abstract)
Reutter, D., & Vicary, J. (2018) · Electronic Proceedings in Theoretical Computer Science · doi:10.4204/eptcs.266.21 · Cited by 4
Shaded tangles for the design and verification of quantum circuits
Reutter, D., & Vicary, J. (2019) · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rspa.2018.0338 · Cited by 3
“2 Cluster chains Another important class of entangled states are the cluster states or graph states [11, 27, 55] and their qudit generalizations associated to Hadamard matrices [18].”
Generalized Concatenation for Quantum Codes
Grassl, M., Shor, P. W., & Zeng, B. (2009) · DSpace@MIT (Massachusetts Institute of Technology) · Cited by 3
A proposal for preparation of cluster states with linear optics*
Ju, L., Yang, M., & Xue, P. (2020) · Chinese Physics B · doi:10.1088/1674-1056/abd74b · Cited by 3
Dür, W., & Briegel, H. J. (2006) · doi:10.1002/9783527618637.ch11 · Cited by 3
Quantum entanglement in non-local games, graph parameters and zero-error information theory
Scarpa, G. (2013) · Centrum Wiskunde & Informatica (CWI), the national research institute for mathematics and computer science in the Netherlands · Cited by 3
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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