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The Interplay Between Color Codes and Toric Codes

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Quantum Codes for Topological Quantum Computation

Part of the book series: SpringerBriefs in Mathematics ((BRIEFSMATH))

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Abstract

In each of the two previous chapters, we dealt with mathematical structures and properties related to topological toric codes and topological color codes individually.

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References

  1. H. Bombin, and M. A. Martin-Delgado. Topological quantum distillation. Phys. Rev. Letters, 97:180501, 2006.

    Article  Google Scholar 

  2. H. Bombin, and M.A. Martin-Delgado. Exact topological quantum order in D = 3 and beyond: Branyons and brane-net condensates. Phys. Rev. B, 75(7):75103, 2007.

    Google Scholar 

  3. H. Bombin, and M.A. Martin-Delgado. Topological computation without braiding. Phys. Rev. Lett., 98(16):160502, 2007.

    Google Scholar 

  4. H. Bombin, and M.A. Martin-Delgado. Quantum measurements and gates by code deformation. J. Phys. A: Math. and Theor., 42:095302, 2009.

    Article  MathSciNet  Google Scholar 

  5. H. Bombin, M. Kargarian and M.A. Martin-Delgado. Quantum 2-body Hamiltonian for topological color codes. Fortschr. Phys., 57:1103, 2009.

    Article  MathSciNet  Google Scholar 

  6. H. Bombin. Topological subsystem codes. Phys. Rev. A, 81:032301, 2010.

    Article  MathSciNet  Google Scholar 

  7. H. Bombin. Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes. New J. Phys., 17:083002, 2015.

    Article  MathSciNet  Google Scholar 

  8. H. Bombin. Clifford gates by code deformation. New J. Physiscs, 13:43005, 2011.

    Article  MathSciNet  Google Scholar 

  9. H. Bombin, G. Duclos-Cianci, and D. Poulin. Universal topological phase of 2D stabilizer codes. New J. Phys., 14:73048, 2012.

    Article  Google Scholar 

  10. H. Bombin. Structure of 2D topological stabilizer codes. Comun. Math. Phys., 327:387–432, 2014.

    Article  MathSciNet  Google Scholar 

  11. S. Bravyi, and J. Haah. On the energy landscape of 3D spin hamiltonians with topological order. ar**v:1105.4159, 2011.

    Google Scholar 

  12. C. G. Brell, S.T. Flammia, S.D. Bartlet, and A. C. Doherty. Toric codes and quantum doubles from two-body Hamiltonians. New J. Phys., 13:053039, 2011.

    Article  Google Scholar 

  13. C.G. Brell. Generalized color codes supporting non-Abelian anyons. Phys. Rev. A, 91:042333, 2015.

    Article  Google Scholar 

  14. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill. Topological quantum memory. J. Math. Phys., 43: 4452, 2002.

    Article  MathSciNet  Google Scholar 

  15. J. Haah. Commuting Pauli Hamiltonians as maps between free modules. Commun. Math. Phys., 324:351, 2013.

    Article  MathSciNet  Google Scholar 

  16. A.Yu Kitaev. Fault-tolerant quantum computation by anyons. Ann. Phys., 303(1):2–30, 2003.

    Article  MathSciNet  Google Scholar 

  17. A. Kubica, B. Yoshida, and F. Pastawski. Unfolding the color code. New J. Physics, 17:083026, 2015.

    Article  MathSciNet  Google Scholar 

  18. A.J. Landahl, J.T. Anderson, and P.R. Rice. Fault-tolerant quantum computing with color codes. ar**v:1108.5738, 2011.

    Google Scholar 

  19. C. Nayak, S.H. Simon, A. Stern, M. Freedman, and S, das Sarma. Non-Abelian anyon and topological quantum computation. Rev. Mod. Phys., 80:1083, 2008.

    Google Scholar 

  20. J.K. Pachos. Introduction to Topological Quantum Computation, Cambridge University Press, 2012.

    Book  Google Scholar 

  21. A. Paetznick, and B.W. Reichardt. Universal fault-tolerant quantum computation with only transversal gates and error correction. Phys. Rev. Lett., 111:90505, 2013.

    Article  Google Scholar 

  22. J. Preskill, Reliable quantum computers. Proc. Roy. Soc. Lond., 454:385, 1998.

    Article  Google Scholar 

  23. P.W. Shor. Algorithms for quantum computation: Discrete logarithms and factoring. IEEE 35th Annual Symp. on Foundations of Computer Science, pp. 124–134, 1994.

    Google Scholar 

  24. P.W. Shor. Scheme for reducing decoherence in quantum memory. Phys. Rev. A, 52:2493, 1995.

    Article  Google Scholar 

  25. W.S. Soares Jr., and E.B. Silva. Hyperbolic quantum color codes. Quantum Inf. & Comp., 18:308–320, 2018.

    MathSciNet  Google Scholar 

  26. A.M. Steane. Error correcting codes in quantum theory. Phys. Rev. Letters, 77:793, 1996.

    Article  MathSciNet  Google Scholar 

  27. F. Wilczek. Quantum mechanics of fractional-spin particles. Phys. Rev. Lett., 49:957, 1982.

    Article  MathSciNet  Google Scholar 

  28. W. K. Wootters, and W.H. Zurek. A single quantum cannot be cloned. Nature, 299:802–803, 1982.

    Article  Google Scholar 

  29. B. Yoshida. Classification of quantum phases and topology of logical operators in an exactly solved model of quantum codes. Ann. Physics, 326(1):15–95, 2011.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Science and Technology Center, Federal University of Cariri, Juazeiro do Norte, Ceará, Brazil

    Clarice Dias de Albuquerque

  2. Mathematics Department, State University of Maringa, Maringá, Paraná, Brazil

    Eduardo Brandani da Silva

  3. Mathematics, Federal Technological University of Paraná, Pato Branco, Paraná, Brazil

    Waldir Silva Soares Jr.

Authors
  1. Clarice Dias de Albuquerque
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  2. Eduardo Brandani da Silva
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  3. Waldir Silva Soares Jr.
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Albuquerque, C.D.d., Silva, E.B.d., Soares Jr., W.S. (2022). The Interplay Between Color Codes and Toric Codes. In: Quantum Codes for Topological Quantum Computation. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-031-06833-1_5

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