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Fault-Tolerant Quantum Error Correction for non-Abelian Anyons

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Abstract

While topological quantum computation is intrinsically fault-tolerant at zero temperature, it loses its topological protection at any finite temperature. We present a scheme to protect the information stored in a system supporting non-cyclic anyons against thermal and measurement errors. The correction procedure builds on the work of Gács (J Comput Syst Sci 32:15–78, 1986. doi:10.1145/800061.808730) and Harrington (Analysis of quantum error-correcting codes: symplectic lattice codes and toric code, 2004) and operates as a local cellular automaton. In contrast to previously studied schemes, our scheme is valid for both abelian and non-abelian anyons and accounts for measurement errors. We analytically prove the existence of a fault-tolerant threshold for a certain class of non-Abelian anyon models, and numerically simulate the procedure for the specific example of Ising anyons. The result of our simulations are consistent with a threshold between \({10^{-4}}\) and \({10^{-3}}\).

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References

  1. Gács P.: Reliable computation with cellular automata. J. Comput. Syst. Sci. 32, 15–78 (1986). doi:10.1145/800061.808730

    Article  MathSciNet  MATH  Google Scholar 

  2. Harrington, J.W.: Analysis of Quantum Error-Correcting Codes: Symplectic Lattice Codes and Toric Code. Ph.D. thesis, California Institude of Technology (2004). URL http://thesis.library.caltech.edu/1747/1/jimh_thesis.pdf

  3. Das Sarma S., Freedman M., Nayak C.: Majorana zero modes and topological quantum computation. NPJ Quantum Inf. 1, 15001 (2015). doi:10.1038/npjqi.2015.1

    Article  ADS  Google Scholar 

  4. Freedman M., Nayak C., Walker K.: Towards universal topological quantum computation in the \({\nu = \frac{5}{2}}\) fractional quantum Hall state. Phys. Rev. B 73, 245307 (2006). doi:10.1103/PhysRevB.73.245307

    Article  ADS  Google Scholar 

  5. Bonderson, P., Das Sarma, S., Freedman, M., Nayak, C.: A Blueprint for a Topologically Fault-tolerant Quantum Computer (2010). arXiv:1003.2856

  6. Kitaev A.Y.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003). doi:10.1016/S0003-4916(02)00018-0

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Nayak C., Simon S.H., Freedman M., Sarma S.: Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083 (2008). doi:10.1103/RevModPhys.80.1083

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Trebst S., Troyer M., Wang Z., Ludwig A.W.W.: A short introduction to Fibonacci anyon models. Prog. Theor. Phys. Suppl. 176, 384 (2008). doi:10.1143/PTPS.176.384

    Article  ADS  MATH  Google Scholar 

  9. Stern A.: Non-Abelian states of matter. Nat. Phys. 464, 187–193 (2010). doi:10.1038/nature08915

    Article  Google Scholar 

  10. Freedman M.H., Larsen M., Wang Z.: A modular functor which is universal for quantum computation. Commun. Math. Phys. 227, 605–622 (2002). doi:10.1007/s002200200645

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Freedman M.H., Kitaev A., Wang Z.: Simulation of topological field theories by quantum computer. Commun. Math. Phys. 227, 587–603 (2002). doi:10.1007/s002200200635

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Budich J.C., Walter S., Trauzettel B.: Failure of proctection of Majorana based qubits against decoherence. Phys. Rev. B 85, 121405 (2012). doi:10.1103/PhysRevB.85.121405

    Article  ADS  Google Scholar 

  13. Schmidt M.J., Rainis D., Loss D.: Decoherence of Majorana qubits by noisy gates. Phys. Rev. B 86, 085414 (2012). doi:10.1103/PhysRevB.86.085414

    Article  ADS  Google Scholar 

  14. Pedrocchi F.L., Bonesteel N.E., DiVincenzo D.P.: Monte Carlo studies of the properties of the Majorana quantum error correction code: is self-correction possible during braiding?. Phys. Rev. B 92, 115441 (2015). doi:10.1103/PhysRevB.92.115441

    Article  ADS  Google Scholar 

  15. Pedrocchi F.L., DiVincenzo D.P.: Majorana braiding with thermal noise. Phys. Rev. Lett. 115, 120402 (2015). doi:10.1103/PhysRevLett.115.120402

    Article  ADS  Google Scholar 

  16. Goldstein G., Chamon C.: Decay rates for topological memories encoded with Majorana fermions. Phys. Rev. B 84, 205109 (2011). doi:10.1103/PhysRevB.84.205109

    Article  ADS  Google Scholar 

  17. Konschelle F., Hassler F.: Effects of nonequilibrium noise on a quantum memory encoded in Majorana zero modes. Phys. Rev. B 88, 075431 (2013). doi:10.1103/PhysRevB.88.075431

    Article  ADS  Google Scholar 

  18. Rainis D., Loss D.: Majorana qubit decoherence by quasiparticle poisoning. Phys. Rev. B 85, 174533 (2012). doi:10.1103/PhysRevB.85.174533

    Article  ADS  Google Scholar 

  19. Wootton J.R., Burri J., Iblisdir S., Loss D.: Error correction for non-Abelian topological quantum computation. Phys. Rev. X 4, 011051 (2014). doi:10.1103/PhysRevX.4.011051

    Google Scholar 

  20. Hamma A., Castelnovo C., Chamon C.: Toric-boson model: toward a topological quantum memory at finite temperature. Phys. Rev. B 79, 245122 (2009). doi:10.1103/PhysRevB.79.245122

    Article  ADS  Google Scholar 

  21. Chesi S., Röthlisberger B., Loss D.: Self-correcting quantum memory in a thermal environment. Phys. Rev. A 82, 022305 (2010). doi:10.1103/PhysRevA.82.022305

    Article  ADS  Google Scholar 

  22. Pedrocchi F.L., Hutter A., Wootton J.R., Loss D.: Enhanced thermal stability of the toric code through coupling to a bosonic bath. Phys. Rev. A 88, 062313 (2013). doi:10.1103/PhysRevA.88.062313

    Article  ADS  Google Scholar 

  23. Landon-Cardinal O., Poulin D.: Local topological order inhibits thermal stability in 2D. Phys. Rev. Lett. 110, 090502 (2013). doi:10.1103/PhysRevLett.110.090502

    Article  ADS  Google Scholar 

  24. Landon-Cardinal O., Yoshida B., Poulin D., Preskill J.: Perturbative instability of quantum memory based on effective long-range interactions. Phys. Rev. A 91(3), 032303 (2015). doi:10.1103/PhysRevA.91.032303

    Article  ADS  Google Scholar 

  25. Dennis E., Kitaev A., Landhal A., Presill J.: Topological quantum memory. J. Math. Phys. 43, 4452 (2002). doi:10.1063/1.1499754

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Duclos-Cianci G., Poulin D.: Fast decoders for topological quantum codes. Phys. Rev. Lett. 104, 050504 (2010). doi:10.1103/PhysRevLett.104.050504

    Article  ADS  Google Scholar 

  27. Duclos-Cianci, G., Poulin, D.: A renormalization group decoding algorithm for topological quantum codes. Inf. Theory Workshop 1–5 (2010). doi:10.1109/CIG.2010.5592866

  28. Wang, D.S., Fowler, A.G., Stephens, A.M., Hollenberg, L.C.L.: Threshold error rates for the toric and surface codes. Quantum Inf. Comput. 10, 456 (2010). URL https://arxiv.org/abs/0905.0531

  29. Wootton J.R., Loss D.: High threshold error correction for the surface code. Phys. Rev. Lett. 109, 160503 (2012). doi:10.1103/PhysRevLett.109.160503

    Article  ADS  Google Scholar 

  30. Bravyi S., Haah J.: Quantum self-correction in the 3D cubic code model. Phys. Rev. Lett. 111, 200501 (2013). doi:10.1103/PhysRevLett.111.200501

    Article  ADS  Google Scholar 

  31. Anwar H., Brown B.J., Campbell E.T., Browne D.E.: Efficient decoders for qudit topological codes. New J. Phys. 16, 063038 (2014). doi:10.1088/1367-2630/16/6/063038

    Article  ADS  MathSciNet  Google Scholar 

  32. Hutter A., Wootton J.R., Loss D.: Efficient Markov chain Monte Carlo algorithm for the surface code. Phys. Rev. A 89, 022326 (2014). doi:10.1103/PhysRevA.89.022326

    Article  ADS  Google Scholar 

  33. Bravyi S., Suchara M., Vargo A.: Efficient algorithms for maximum likelihood decoding in the surface code. Phys. Rev. A 90, 032326 (2014). doi:10.1103/PhysRevA.90.032326

    Article  ADS  Google Scholar 

  34. Herold M., Campbell E.T., Eisert J., Kastoryano M.J.: Cellular-automaton decoders for topological quantum memories. NPJ Quantum Inf. 1, 15010 (2015). doi:10.1038/npjqi.2015.10

    Article  ADS  Google Scholar 

  35. Wootton J.R.: A simple decoder for topological codes. Entropy 17, 1946 (2015). doi:10.3390/e17041946

    Article  ADS  MathSciNet  Google Scholar 

  36. Andrist R.S., Wootton J.R., Katzgraber H.G.: Error thresholds for Abelian quantum double models: increasing the bit-flip stability of topological quantum memory. Phys. Rev. A 91, 042331 (2015). doi:10.1103/PhysRevA.91.042331

    Article  ADS  Google Scholar 

  37. Fowler A.G., Stephens A.M., Groszkowski P.: High-threshold universal quantum computation on the surface code. Phys. Rev. A 80, 052312 (2009). doi:10.1103/PhysRevA.80.052312

    Article  ADS  Google Scholar 

  38. Duclos-Cianci, G., Poulin, D.: Fault-tolerant renormalization group decoder for Abelian topological codes. Quantum Inf. Comput. 14(9–10), 0721–0740 (2014). URL http://epiq.physique.usherbrooke.ca/data/files/publications/DP13a1.pdf

  39. Watson F.H.E., Anwar H., Brown D.E.: A fast fault-tolerant decoder for qubit and qudit surface codes. Phys. Rev. A 92, 032309 (2015). doi:10.1103/PhysRevA.92.032309

    Article  ADS  Google Scholar 

  40. Fowler, A.G.: Minimum weight perfect matching of fault-tolerant topological quantum error correction in average O(1) parallel time. QIC 15, 0145–0158 (2015). URL http://arxiv.org/abs/1307.1740

  41. Herold, M., Kastoryano, M.J., Campbell, E.T., Eisert, J.: Fault tolerant dynamical decoders for topological quantum memories (2015). arXiv:1511.05579

  42. Brell C.G., Burton S., Dauphinais G., Flammia S.T., Poulin D.: Thermalization, error correction, and memory lifetime for Ising anyon systems. Phys. Rev. X 4, 031058 (2014). doi:10.1103/PhysRevX.4.031058

    Google Scholar 

  43. Burton S., Brell C.G., Flammia S.T.: Classical simulation of quantum error correction in a Fibonacci anyon code. Phys. Rev. A 95, 022309 (2017). doi:10.1103/PhysRevA.95.022309

    Article  ADS  Google Scholar 

  44. Wootton J.R., Hutter A.: Active error correction for Abelian and non-Abelian anyons. Phys. Rev. A 93, 022318 (2016). doi:10.1103/PhysRevA.93.022318

    Article  ADS  Google Scholar 

  45. Hutter A., Wootton J.R.: Continuous error correction for Ising anyons. Phys. Rev. A 93, 042327 (2016). doi:10.1103/PhysRevA.93.042327

    Article  ADS  Google Scholar 

  46. Gray, L.: A reader’s guide to P. Gács’ “positive rates” paper: “Reliable cellular automata with self-organization”. J. Stat. Phys. 103, 1–44 (2001). URL http://www.math.umn.edu/~grayx004/pdf/gacs2.pdf

  47. Kitaev A.: Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006). doi:10.1016/j.aop.2005.10.005

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. Bonderson, P.: Non-Abelian Anyons and Interferometry. Ph.D. thesis, California Institude of Technology (2007). URL http://thesis.library.caltech.edu/2447/2/thesis.pdf

  49. Mac Lane S.: Categories for the Working Mathematician. Springer, New York (1991)

    MATH  Google Scholar 

  50. Levaillant C., Bauer B., Freedman M., Wang Z., Bonderson P.: Universal gates via fusion and measurement operations on \({\rm{SU} (2)_{4}}\) anyons. Phys. Rev. A 92, 012301 (2015). doi:10.1103/PhysRevA.92.012301

    Article  ADS  Google Scholar 

  51. Bonderson P., Freedman M., Nayak C.: Measurement-only topological quantum computation. Phys. Rev. Lett. 101, 010501 (2008). doi:10.1103/PhysRevLett.101.010501

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. Bonderson P., Freedman M., Nayak C.: Measurement-only topological quantum computation via Anyonic Interferometry. Ann. Phys. 324, 787–826 (2009). doi:10.1016/j.aop.2008.09.009

    Article  ADS  MathSciNet  MATH  Google Scholar 

  53. Verlinde E.: Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys. B 300, 360–376 (1988). doi:10.1016/0550-3213(88)90603-7

    Article  ADS  MATH  Google Scholar 

  54. Rowell E., Stong R., Wang Z.: On classification of modular tensor categories. Commun. Math. Phys. 292, 343–389 (2009). doi:10.1007/s00220-009-0908-z

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. Bais F.A., van Driel P., de Wild Propitius M.: Quantum symmetries in discrete gauge theories. Phys. Lett. B 280, 63–67 (1992). doi:10.1016/0370-2693(92)90773-W

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. de Wild Propitius, M., Bais, F.A.: Discrete gauge theories (1995). arXiv:hep-th/9511201

  57. Xu C., Ludwig A.W.W.: Topological quantum liquids with quaternion non-Abelian statistics. Phys. Rev. Lett. 108, 047202 (2012). doi:10.1103/PhysRevLett.108.047202

    Article  ADS  Google Scholar 

  58. Tambara D., Yamagami S.: Tensor categories with fusion rules of self-duality for finite Abelian groups. J. Algebra 209, 692–707 (1998). doi:10.1006/jabr.1998.7558

    Article  MathSciNet  MATH  Google Scholar 

  59. Borel E.: Les probabilités dénombrables et leurs applications arithmétiques. Rend. Cric. Mat. Palermo (2) 27, 247–271 (1909)

    Article  MATH  Google Scholar 

  60. Cantelli, F.P.: Sulla probabilitá come limite della frequenza. Atti Accad. Naz. Lincei 26, 39–45 (1917)

  61. Gács, P.: Self-correcting two-dimensoinal array. Adv. Comput. Res. 5, 223–326 (1989). URL http://www.cs.bu.edu/~gacs/papers/self-correcting-2d.pdf

  62. Bravyi S.: Universal quantum computation with the \({\nu = \frac{5}{2}}\) fractional quantum Hall state. Phys. Rev. A 73, 042313 (2006). doi:10.1103/PhysRevA.73.042313

    Article  ADS  Google Scholar 

  63. Nayak C., Wilczek F.: 2n-quasiholes states realize \({2^{n-1}}\)-dimensional spinor braiding statistics in paired quantum Hall states. Nucl. Phys. B 479, 529 (1996). doi:10.1016/0550-3213(96)00430-0

    Article  ADS  MathSciNet  MATH  Google Scholar 

  64. Aaronson S., Gottesman D.: Improved simulation of stabilizer circuits. Phys. Rev. A 70, 052328 (2004). doi:10.1103/PhysRevA.70.052328

    Article  ADS  Google Scholar 

  65. Kolmogorov V., Blossom V.: A new implementation of a minimum cost perfect matching algorithm. Math. Program. Comput. 1, 43 (2009). doi:10.1007/s12532-009-0002-8

    Article  MathSciNet  MATH  Google Scholar 

  66. Beverland, A., Buerschaper, König, R., Pastawski, F., Preskill, J., Sijhner, S.: Protected gates for topological quantum field theories. J. Math. Phys. 57, 022201 (2016). doi:10.1063/1.4939783

  67. Freedman M.H., Meyer D.A.: Projective plane and planar quantum codes. Found. Comput. Math. 1, 325–332 (2001). doi:10.1007/s102080010013

    Article  MathSciNet  MATH  Google Scholar 

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  1. Institut quantique & Département de physique, Université de Sherbrooke, 2500 boulevard de l’Université, Sherbrooke, J1K 2R1, Canada

    Guillaume Dauphinais & David Poulin

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  1. Guillaume Dauphinais
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Dauphinais, G., Poulin, D. Fault-Tolerant Quantum Error Correction for non-Abelian Anyons. Commun. Math. Phys. 355, 519–560 (2017). https://doi.org/10.1007/s00220-017-2923-9

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