Construction of Polygonal Color Codes from Hyperbolic Tesselations
DOI:
https://doi.org/10.5540/tema.2020.021.01.43Palavras-chave:
Color codes, topological quantum codes, hyperbolic geometry.Resumo
This current work propose a technique to generate polygonal color codes in the hyperbolic geometry environment. The color codes were introduced by Bombin and Martin-Delgado in 2007, and the called triangular color codes have a higher degree of interest because they allow the implementation of the Clifford group, but they encode only one qubit. In 2018 Soares e Silva extended the triangular codes to the polygonal codes, which encode more qubits. Using an approach through hyperbolic tessellations we show that it is possible to generate Hyperbolic Polygonal codes, which encode more than one qubit with the capacity to implement the entire Clifford group and also having a better coding rate than the previously mentioned codes, for the color codes on surfaces with boundary with minimum distance d = 3.
Referências
W. G. Unruh, "Maintaining coherence in Quantum Computers", Physical Review A, vol. 51, pp. 992-997, June 1994.
C. E. Shannon, "A mathematical theory of communication," SIGMOBILE Mob. Comput. Commun. Rev., vol. 5, pp. 3-55, Jan. 2001.
P. W. Shor, "Scheme for reducing decoherence in quantum computer memory," Phys. Rev. A, vol. 52, pp. R2493-R2496, Oct 1995.
A. R. Calderbank and P. W. Shor, "Good Quantum Error-Correcting Codes Exist," Physical Review A, vol. 54, pp. 1098-1105, Apr. 1996.
A. Steane, "Simple Quantum Error Correcting Codes," Physical Review A, vol. 54, pp. 4741-4751, May 1996.
D. Gottesman, "A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound," Physical Review A, vol. 54, pp. 1862-1868, July 1996.
A. Kitaev, "Fault-tolerant quantum computation by anyons," Annals of
Physics, vol. 303, no. 1, pp. 2 - 30, 2003.
H. Bombin and M. A. Martin-Delgado, "Topological Quantum Error Correction with Optimal Encoding Rate," Physical Review A, vol. 73, May 2006.
C. D. de Albuquerque, R. P. Junior, and E. B. da Silva, "On toric quantum codes," Int. J. Pure Appl. Math, vol. 50, pp. 221-226, 2009.
C. D. Albuquerque, R. Palazzo, and E. B. Silva, "Topological quantum codes on compact surfaces with genus g>2," Journal of Mathematical Physics, vol. 50, pp. 023513-023513, Feb. 2009.
H. Bombin and M. A. Martin-Delgado, "Topological Quantum Distillation," Physical Review Letters, vol. 97, pp. 180501+, Mar. 2007.
W. S. Soares Jr and E. B. Silva, "Hyperbolic quantum color codes," Quantum Information and Computation, vol. 18, no. 3 and 4, pp. 308-320, 2018.
W. S. Soares Jr and E. B. Silva, "Construction of color codes from polygons," Journal of Physics Communications, vol. 2, no. 9, p. 095011, 2018.
N. P. Breuckmann and B. M. Terhal, "Constructions and noise threshold of hyperbolic surface codes," IEEE Transactions on Information Theory, vol. 62, pp. 3731-3744, June 2016.
N. Delfosse, "Tradeoffs for reliable quantum information storage in surface codes and color codes," pp. 917-921, Jan. 2013.
S. Katok, Fuchsian Groups. Chicago Lectures in Mathematics, University of Chicago Press, 1992.
D. Lidar and T. Brun, Quantum Error Correction. Cambridge University Press, 2013.
D. Gottesman and I. L. Chuang, "Quantum teleportation is a universal computational primitive," Nature, vol. 402, pp. 390-393, Aug. 1999.
H. Bombin and M. A. Martin-Delgado, "Computacion cuantica topologica y sistemas fuertemente correlacionados," Revista espanola de Fisica, vol. 21, no. 2, pp. 31 - 45, 2007.
A. Ramsay and R. D. Richtmyer, Constructions by Straightedge and Compass in the Hyperbolic Plane, pp. 254-282. New York, NY: Springer New York, 1995.
Downloads
Publicado
Como Citar
Edição
Seção
Licença
Direitos Autorais
Autores de artigos publicados no periódico Trends in Computational and Applied Mathematics mantêm os direitos autorais de seus trabalhos. O periódico utiliza a Atribuição Creative Commons (CC-BY) nos artigos publicados. Os autores concedem ao periódico o direito de primeira publicação.
Propriedade Intelectual e Termos de uso
O conteúdo dos artigos é de responsabilidade exclusiva dos autores. O periódico utiliza a Atribuição Creative Commons (CC-BY) nos artigos publicados. Esta licença permite que os artigos publicados sejam reutilizados sem permissão para qualquer finalidade, desde que o trabalho original seja corretamente citado.
O periódico encoraja os Autores a autoarquivar seus manuscritos aceitos, publicando-os em blogs pessoais, repositórios institucionais e mídias sociais acadêmicas, bem como postando-os em suas mídias sociais pessoais, desde que seja incluída a citação completa à versão do website da revista.
