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On Metaplectic Modular Categories and Their Applications

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Abstract

For non-abelian simple objects in a unitary modular category, the density of their braid group representations, the #P-hard evaluation of their associated link invariants, and the BQP-completeness of their anyonic quantum computing models are closely related. We systematically study such properties of the non-abelian simple objects in the metaplectic modular categories SO(m)2 for an odd integer m ≥ 3. The simple objects with quantum dimensions \({\sqrt{m}}\) have finite image braid group representations, and their link invariants are classically efficient to evaluate. We also provide classically efficient simulations of their braid group representations. These simulations of the braid group representations can be regarded as qudit generalizations of the Knill–Gottesmann theorem for the qubit case. The simple objects of dimension 2 give us a surprising result: while their braid group representations have finite images and are efficiently simulable classically after a generalized localization, their link invariants are #P-hard to evaluate exactly. We sharpen the #P-hardness by showing that any sufficiently accurate approximation of their associated link invariants is already #P-hard.

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References

  1. Bruillard, P., Ng, S., Rowell, E., Wang, Z.: On modular categories. arXiv:1310.7050

  2. Bonderson, P.: Non-abelian anyons and interferometry. Caltech Ph.D. thesis (2007)

  3. Freedman, M., Larsen, M., Wang, Z.: A modular functor which is universal for quantum computation. Commun. Math. Phys. 227(3), 605–622 (2002). arXiv:quant-ph/0001108

    Google Scholar 

  4. Freedman, M., Larsen, M., Wang, Z.: The two-eigenvalue problem and density of Jones representation of braid groups. Commun. Math. Phys. 228, 1 (2002). arXiv:math.GT/0103200

    Google Scholar 

  5. Franko, J., Rowell, E., Wang, Z.: Extraspecial 2-groups and images of braid group representations. J. Knot Theory Ramifications 15(4) 413–427 (2006). arXiv:math.RT/0503435

    Google Scholar 

  6. Galindo, C., Hong, S., Rowell, E.C.: Generalized and quasi-localizations of braid group representations. Int. Math. Res. Not. 2013(3), 693–731 (2013). arXiv:1105.5048

  7. Goldschmidt David M., Jones V.: Metaplectic link invariants. Geometriae Dedicata 31(2), 165–191 (1989)

    MATH  MathSciNet  Google Scholar 

  8. Goldberg, L.A., Jerrum, M.: The complexity of computing the sign of the tutte polynomial (and consequent #P-hardness of approximation). Lect. Notes Comput. Sci. 73911, 399–410 (2012). arXiv:1202.0313, ICALP 2012

  9. Griess, R.L.: Automorphsims of extra-special groups and nonvanishing degree 2 cohomology. Pacific J. Math. 48(2), (1973)

  10. Hastings, M.B., Nayak, C., Wang, Z.: Metaplectic anyons, majorana zero modes, and their computational power. Phys. Rev. B 87, 165421 (2013). arXiv:1210.5477

  11. Hong, S.-M.: From ribbon categories to generalized Yang-Baxter operators and link invariants (after Kitaev and Wang). Int. J. Math. 24(1), 1250126, p. 12 (2013). arXiv:1205.4005

  12. Jones, V.: Braid groups, Hecke algebras and type II1 factors. in Geometric Methods in Operator Algebras (Kyoto, 1983), vol. 123. Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, pp. 242–273. Harlow (1986)

  13. Jones V.: Hecke algebra representations of braid groups and link polynomials. Ann. Math. (2) 126(2), 335–388 (1987)

    Article  MATH  Google Scholar 

  14. Jones V.: On a certain value of the Kauffman polynomial. Commun. Math. Phys. 125(3), 459–467 (1989)

    Article  ADS  MATH  Google Scholar 

  15. Kauffman, L., Lins, S.: Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds. Ann. Math. Stud., vol. 134. Princeton University Press, Princeton, NJ (1994)

  16. Krovi, H., Russell, A.: Quantum Fourier Transforms and the Complexity of Link Invariants for Quantum Doubles of Finite Groups. arXiv:1210.1550

  17. Kuperberg, G.: How hard is it to approximate the Jones polynomial? arXiv:0908.0512

  18. Kitaev, A., Wang, Z.: Solutions to generalized Yang–Baxter equations via ribbon fusion categories. Geometry Topol. Monogr. 18, 191–197 (2012). Proceedings of the FreedmanFest. arXiv:1203.1063

  19. Lickorish W.B.R., Millett K.C.: An evaluation of the F-polynomial of a link. Lect. Notes Math. 1350/1988, 104–108 (1988)

    Article  MathSciNet  Google Scholar 

  20. Moore G., Seiberg N.: Classical and quantum conformal field theory. Commun. Math. Phys. 123(2), 177–352 (1989)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. Nielsen M., Chuang I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  22. Naidu, D., Rowell, E.C.: A finiteness property for braided fusion categories. Algebr. Represent. Theor. 14(5), 837–855 (2011). arXiv:0903.4157

    Google Scholar 

  23. Read, N., Rezayi, E.: Beyond paired quantum Hall states: Parafermions and incompressible states in the first excited Landau level. Phys. Rev. B 59(12), 8084–8092 (1999). arXiv:cond-mat/9809384

    Google Scholar 

  24. Rowell, E., Wang, Z.: Localization of unitary braid group representations. Commun. Math. Phys. 311(3), 595–615 (2012). arXiv:1009.0241

    Google Scholar 

  25. Rowell, E., Wenzl, H.: SO(N)2 braid group representations are Gaussian. arXiv:1401.5329

  26. The fact that this problem is #P-hard is complete is a textbook exercise. See C. Moore and S. Mertens. The Nature of Computation, Ex. 13.11

  27. Turaev, V.: Quantum Invariants of Knots and 3-Manifolds. De Gruyter Studies in Mathematics, vol. 18. Walter de Gruyter & Co., Berlin (1994)

  28. Tuba I., Wenzl H.: On braided tensor categories of type BCD. J. Reine Angew. Math. 581, 31–69 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  29. Wang Z.: Topological quantum computation, CBMS monograph, vol. 112. American Mathematical Society, USA (2010)

    Google Scholar 

  30. Welsh, D.: Complexity: Knots, Colourings and Countings. London Mathematical Society Lecture Note Series, vol. 186. Cambridge University Press, Cambridge (1993)

  31. Winter D.: The automorphism group of an extraspecial p-group. Rocky Mountain J. Math. 2(2), 159–168 (1972)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Microsoft Station Q, University of California, Santa Barbara, CA, 93106, USA

    Matthew B. Hastings, Chetan Nayak & Zhenghan Wang

  2. Department of Physics, University of California, Santa Barbara, CA, 93106, USA

    Chetan Nayak

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  1. Matthew B. Hastings
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  2. Chetan Nayak
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  3. Zhenghan Wang
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Correspondence to Zhenghan Wang.

Additional information

Communicated by Y. Kawahigashi

M.B. Hastings was partially supported by a Simons Investigator award from the Simons Foundation. C. Nayak is partially supported by the DARPA QuEST program and the AFOSR under grant FA9550-10-1-0524. Z. Wang is partially supported by NSF DMS 1108736. We thank I. Arad for pointing out reference [GJ].

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Hastings, M.B., Nayak, C. & Wang, Z. On Metaplectic Modular Categories and Their Applications. Commun. Math. Phys. 330, 45–68 (2014). https://doi.org/10.1007/s00220-014-2044-7

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  • DOI: https://doi.org/10.1007/s00220-014-2044-7

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