Quantum error-correcting codes associated with graphs

By D. Schlingemann, Reinhard F. Werner. arXiv.org, 2000

359 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/6fba2231f7a53ed9e7d7c65f71dcdaae46527d91

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Citations in the corpus

Genuinely multipartite entangled states and orthogonal arrays (2014, background): “”Enrico Fermi” on ”Quantum Computers, Algorithms and Chaos”, Varenna, Italy (2005), arXiv:quant-ph/0602096 [45] D.”
Index Theory of One Dimensional Quantum Walks and Cellular Automata (2009, background): “We note that U1 is known in the quantum information literature as the interaction used to generate graph states [9,15].”
Measurement-based quantum computation on cluster states (2003, background): “The treatment here follows that of [53], adapted to our notation.”
Experimental entanglement of six photons in graph states (2006, neutral): “Furthermore, we use A0 = I −|HHH〉〈HHH|−|V V V 〉〈V V V |, A1 = |V V V 〉〈V V V |−|HHH〉〈HHH| and B1 = (M⊗3 (1) +M⊗3 (−1) )/2 √ 3, where M(i) is defined as for the GHZ state (see equation (3)).”
Stabilizer states and Clifford operations for systems of arbitrary dimensions and modular arithmetic (2004, background): “The n-qud it generalized Pauli group and Clifford group and the related concepts of stabilizer codes and states have been studied in various levels of detail in a number of papers [6, 7, 8, 9, 10, 11, 12, 13, 14, 15].”
Nonbinary Stabilizer Codes Over Finite Fields (2005, result): “More recently, some results were generalized to the case of nonbinary stabilizer codes [1,3,4,16,23,24,34,35,41,47,49,56,64,70,79,83,85, 86 ], but the theory is not nearly as complete as in the binary case.”