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Topological Quantum Computation is Hyperbolic

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Abstract

We show that a topological quantum computer based on the evaluation of a Witten–Reshetikhin–Turaev TQFT invariant of knots can always be arranged so that the knot diagrams with which one computes are diagrams of hyperbolic knots. The diagrams can even be arranged to have additional nice properties, such as being alternating with minimal crossing number. Moreover, the reduction is polynomially uniform in the self-braiding exponent of the coloring object. Various complexity-theoretic hardness results regarding the calculation of quantum invariants of knots follow as corollaries. In particular, we argue that the hyperbolic geometry of knots is unlikely to be useful for topological quantum computation.

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Notes

  1. To be clear, we have in mind a strict notion of “topological quantum computation" along the lines of the original papers [FLW02a, FLW02b, FKW02] that means “approximate a TQFT invariant of a knot or link inside a closed 3-manifold." There are now more wide-ranging ideas of topological quantum computation that allow for invariants of colored, trivalent ribbon graphs (meaning projective measurements are performed during the computation), adaptive topological charge measurements (in which the amplitudes with which one computes are topologically protected but are not exactly topological invariants of 3-dimensional objects), or the braiding of Majorana zero modes. We shall make no attempt to address these broader paradigms.

  2. We note that all of our diagrams should be considered as diagrams on the 2-sphere \(S^2\), not as diagrams in \(\mathbb {R}^2\).

  3. Note that the first reference uses a slightly different (but equivalent) definition of distance. Our definition follows [Tom07] and [JM16].

References

  1. Aharonov, D., Arad, I.: The BQP-hardness of approximating the Jones polynomial. New J. Phys. 13, 035019 (2011). https://doi.org/10.1088/1367-2630/13/3/035019

    Article  MATH  ADS  Google Scholar 

  2. Anderson, G., Moore, G.: Rationality in conformal field theory. Comm. Math. Phys. 117(3), 441–450 (1988)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  3. Bachman, D., Schleimer, S.: Distance and bridge position. Pac. J. Math. 219(2), 221–235 (2005). https://doi.org/10.2140/pjm.2005.219.221

    Article  MathSciNet  MATH  Google Scholar 

  4. Burton, B.A., Maria, C., Spreer, J.: Algorithms and complexity for Turaev-Viro invariants. J. Appl. Comput. Topol. 2(1–2), 33–53 (2018). https://doi.org/10.1007/s41468-018-0016-2

    Article  MathSciNet  MATH  Google Scholar 

  5. Cui, S.X., Freedman, M.H., Wang, Z.: Complexity classes as mathematical axioms II. Quantum Topol. 7(1), 185–201 (2016). https://doi.org/10.4171/QT/75

    Article  MathSciNet  MATH  Google Scholar 

  6. Etingof, P.: On Vafa’s theorem for tensor categories, arXiv:math/0207007v1, (2002)

  7. Freedman, M.H.: Complexity classes as mathematical axioms. Ann. Math. (2) 170(2), 995–1002 (2009). https://doi.org/10.4007/annals.2009.170.995

  8. Freedman, M.H., Kitaev, A., Wang, Z.: Simulation of topological field theories by quantum computers. Comm. Math. Phys. 227(3), 587–603 (2002). https://doi.org/10.1007/s002200200635. arXiv:quant-ph/0001071

    Article  MathSciNet  MATH  ADS  Google Scholar 

  9. Freedman, M.H., Larsen, M., Wang, Z.: A modular functor which is universal for quantum computation. Comm. Math. Phys. 227(3), 605–622 (2002). https://doi.org/10.1007/s002200200645. arXiv:quant-ph/0001108

    Article  MathSciNet  MATH  ADS  Google Scholar 

  10. Freedman, M.H., Larsen, M.J., Wang, Z.: The two-eigenvalue problem and density of Jones representation of braid groups. Comm. Math. Phys. 228(1), 177–199 (2002). https://doi.org/10.1007/s002200200636. arXiv:math/0103200

    Article  MathSciNet  MATH  ADS  Google Scholar 

  11. Ham, S.L., Purcell, J.S.: Geometric triangulations and highly twisted links, arXiv:2005.11899.: To appear in Algebr. Geom, Topol (2021)

  12. Jaeger, F., Vertigan, D.L., Welsh, D.J.A.: On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Camb. Philos. Soc. 108(1), 35–53 (1990). https://doi.org/10.1017/S0305004100068936

    Article  MathSciNet  MATH  Google Scholar 

  13. Johnson, J., Moriah, Y.: Bridge distance and plat projections. Algebr. Geom. Topol. 16(6), 3361–3384 (2016). https://doi.org/10.2140/agt.2016.16.3361

    Article  MathSciNet  MATH  Google Scholar 

  14. Kashaev, R.M.: A link invariant from quantum dilogarithm. Mod. Phys. Lett. A 10(19), 1409–1418 (1995). https://doi.org/10.1142/S0217732395001526

    Article  MathSciNet  MATH  ADS  Google Scholar 

  15. Kauffman, L.H.: State models and the Jones polynomial. Topology 26(3), 395–407 (1987). https://doi.org/10.1016/0040-9383(87)90009-7

    Article  MathSciNet  MATH  Google Scholar 

  16. Kuperberg, G.: How hard is it to approximate the Jones polynomial? Theory Comput. 11, 183–219 (2015). https://doi.org/10.4086/toc.2015.v011a006. arXiv:0908.0512

    Article  MathSciNet  MATH  Google Scholar 

  17. Kuperberg, G., Samperton, E.: Coloring invariants of knots and links are often intractable. Algebr. Geom. Topol. 21(3), 1479–1510 (2021). https://doi.org/10.2140/agt.2021.21.1479. arXiv:1907.05981

  18. Lackenby, M.: with an appendix by Ian Agol and Dylan Thurston. The volume of hyperbolic alternating link complements. Proc. London Math. Soc. (3) 88(1), 204–224 (2004). https://doi.org/10.1112/S0024611503014291

  19. Lackenby, M.: Private communication. (2023)

  20. Menasco, W.: Closed incompressible surfaces in alternating knot and link complements. Topology 23(1), 37–44 (1984). https://doi.org/10.1016/0040-9383(84)90023-5

    Article  MathSciNet  MATH  Google Scholar 

  21. Menasco, W., Thistlethwaite, M.: The classification of alternating links. Ann. Math. (2) 138(1), 113–171 (1993). (10.2307/2946636)

  22. Murakami, H., Murakami, J.: The colored Jones polynomials and the simplicial volume of a knot. Acta Math. 186(1), 85–104 (2001). https://doi.org/10.1007/BF02392716

    Article  MathSciNet  MATH  Google Scholar 

  23. Murasugi, K.: Jones polynomials and classical conjectures in knot theory. Topology 26(2), 187–194 (1987). https://doi.org/10.1016/0040-9383(87)90058-9

    Article  MathSciNet  MATH  Google Scholar 

  24. Murasugi, K.: Jones polynomials and classical conjectures in knot theory. II. Math. Proc. Camb. Philos. Soc. 102(2), 317–318 (1987). https://doi.org/10.1016/0040-9383(87)90058-9

    Article  MathSciNet  MATH  ADS  Google Scholar 

  25. Rankin, S., Flint, O., Schermann, J.: Enumerating the prime alternating knots. Part II. J. Knot Theory Ramif. 13(1), 101–149 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Thistlethwaite, M.B.: A spanning tree expansion of the Jones polynomial. Topology 26(3), 297–309 (1987). https://doi.org/10.1016/0040-9383(87)90003-6

    Article  MathSciNet  MATH  Google Scholar 

  27. Thistlethwaite, M.B.: Kauffman’s polynomial and alternating links. Topology 27(3), 311–318 (1988). https://doi.org/10.1016/0040-9383(87)90003-6

    Article  MathSciNet  MATH  Google Scholar 

  28. Tomova, M.: Multiple bridge surfaces restrict knot distance. Algebr. Geom. Topol. 7, 957–1006 (2007). https://doi.org/10.2140/agt.2007.7.957

    Article  MathSciNet  MATH  Google Scholar 

  29. Vafa, C.: Toward classification of conformal theories. Phys. Lett. B 206(3), 421–426 (1988). https://doi.org/10.1016/0370-2693(88)91603-6

    Article  MathSciNet  ADS  Google Scholar 

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Acknowledgements

We thank Chris Leininger for many helpful conversations about curve complexes. We also thank Colleen Delaney for very helpful feedback on early drafts of this work, as well as Nathan Dunfield.

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  1. Departments of Mathematics and Computer Science, Purdue University, 150 N University St, West Lafayette, IN, 47907, USA

    Eric Samperton

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Samperton, E. Topological Quantum Computation is Hyperbolic. Commun. Math. Phys. 402, 79–96 (2023). https://doi.org/10.1007/s00220-023-04713-w

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