Construction of Polygonal Color Codes from Hyperbolic Tesselations
DOI:
https://doi.org/10.5540/tema.2020.021.01.43Keywords:
Color codes, topological quantum codes, hyperbolic geometry.Abstract
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.
References
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
Published
How to Cite
Issue
Section
License
Copyright
Authors of articles published in the journal Trends in Computational and Applied Mathematics retain the copyright of their work. The journal uses Creative Commons Attribution (CC-BY) in published articles. The authors grant the TCAM journal the right to first publish the article.
Intellectual Property and Terms of Use
The content of the articles is the exclusive responsibility of the authors. The journal uses Creative Commons Attribution (CC-BY) in published articles. This license allows published articles to be reused without permission for any purpose as long as the original work is correctly cited.
The journal encourages Authors to self-archive their accepted manuscripts, publishing them on personal blogs, institutional repositories, and social media, as long as the full citation is included in the journal's website version.
