Entanglement-assisted quantum codes from algebraic geometry codes

Francisco Fernandes Pereira, Ruud Pellikaan, Giuliano Gadioli La Guardia, Francisco Marcos de Assis

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Abstract

Quantum error correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error correcting codes to achieve higher rates than the one obtained via traditional stabilizer formalism. Such codes are called entanglement-assisted quantum (QUENTA) codes. In this paper, we use algebraic geometry codes to construct several families of QUENTA codes via Euclidean and Hermitian construction. Two of the families created has maximal entanglement and have quantum Singleton defect equal to zero or one. Comparing the other families with the codes with the respective quantum Gilbert-Varshamov bound, we show that our codes have a rate that surpasses such bound. In the end, asymptotically good tower of linear complementary dual codes is used to obtain an asymptotically good family of maximal entanglement QUENTA codes with nonzero rate, relative minimal distance, and relative entanglement. Furthermore, a simple comparison with the quantum Gilbert-Varshamov bound demonstrates that from our construction it is possible to create an asymptotically family of QUENTA codes that exceed such this bound.
Original languageEnglish
Article number1907.06357v1
Number of pages21
JournalarXiv
Publication statusPublished - 2019

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geometry
error correcting codes
redundancy
towers
formalism
defects

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Fernandes Pereira, F., Pellikaan, R., Gadioli La Guardia, G., & de Assis, F. M. (2019). Entanglement-assisted quantum codes from algebraic geometry codes. arXiv, [1907.06357v1].
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title = "Entanglement-assisted quantum codes from algebraic geometry codes",
abstract = "Quantum error correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error correcting codes to achieve higher rates than the one obtained via traditional stabilizer formalism. Such codes are called entanglement-assisted quantum (QUENTA) codes. In this paper, we use algebraic geometry codes to construct several families of QUENTA codes via Euclidean and Hermitian construction. Two of the families created has maximal entanglement and have quantum Singleton defect equal to zero or one. Comparing the other families with the codes with the respective quantum Gilbert-Varshamov bound, we show that our codes have a rate that surpasses such bound. In the end, asymptotically good tower of linear complementary dual codes is used to obtain an asymptotically good family of maximal entanglement QUENTA codes with nonzero rate, relative minimal distance, and relative entanglement. Furthermore, a simple comparison with the quantum Gilbert-Varshamov bound demonstrates that from our construction it is possible to create an asymptotically family of QUENTA codes that exceed such this bound.",
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Entanglement-assisted quantum codes from algebraic geometry codes. / Fernandes Pereira, Francisco; Pellikaan, Ruud; Gadioli La Guardia, Giuliano; de Assis, Francisco Marcos.

In: arXiv, 2019.

Research output: Contribution to journalArticleAcademic

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AU - Gadioli La Guardia, Giuliano

AU - de Assis, Francisco Marcos

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AB - Quantum error correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error correcting codes to achieve higher rates than the one obtained via traditional stabilizer formalism. Such codes are called entanglement-assisted quantum (QUENTA) codes. In this paper, we use algebraic geometry codes to construct several families of QUENTA codes via Euclidean and Hermitian construction. Two of the families created has maximal entanglement and have quantum Singleton defect equal to zero or one. Comparing the other families with the codes with the respective quantum Gilbert-Varshamov bound, we show that our codes have a rate that surpasses such bound. In the end, asymptotically good tower of linear complementary dual codes is used to obtain an asymptotically good family of maximal entanglement QUENTA codes with nonzero rate, relative minimal distance, and relative entanglement. Furthermore, a simple comparison with the quantum Gilbert-Varshamov bound demonstrates that from our construction it is possible to create an asymptotically family of QUENTA codes that exceed such this bound.

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