Advertisement

Algorithmica

, Volume 55, Issue 3, pp 395–421 | Cite as

A Polynomial Quantum Algorithm for Approximating the Jones Polynomial

  • Dorit Aharonov
  • Vaughan Jones
  • Zeph Landau
Article

Abstract

The Jones polynomial, discovered in 1984, is an important knot invariant in topology. Among its many connections to various mathematical and physical areas, it is known (due to Witten) to be intimately connected to Topological Quantum Field Theory ( \({\sf{TQFT}}\) ). The works of Freedman, Kitaev, Larsen and Wang provide an efficient simulation of \({\sf{TQFT}}\) by a quantum computer, and vice versa. These results implicitly imply the existence of an efficient (namely, polynomial) quantum algorithm that provides a certain additive approximation of the Jones polynomial at the fifth root of unity, e 2π i/5, and moreover, that this problem is \({\sf{BQP}}\) -complete. Unfortunately, this important algorithm was never explicitly formulated. Moreover, the results of Freedman et al. are heavily based on \({\sf{TQFT}}\) , which makes the algorithm essentially inaccessible to computer scientists.

We provide an explicit and simple polynomial quantum algorithm to approximate the Jones polynomial of an n strands braid with m crossings at any primitive root of unity e 2π i/k , where the running time of the algorithm is polynomial in m, n and k. Our algorithm is based, rather than on \({\sf{TQFT}}\) , on well known mathematical results (specifically, the path model representation of the braid group and the uniqueness of the Markov trace for the Temperley-Lieb algebra). By the results of Freedman et al., our algorithm solves a \({\sf{BQP}}\) complete problem.

Our algorithm works by encoding the local structure of the problem into the local unitary gates which are applied by the circuit. This structure is significantly different from previous quantum algorithms, which are mostly based on the Quantum Fourier transform. Since the results of the current paper were presented in their preliminary form, these ideas have been extended and generalized in several interesting directions. Most notably, Aharonov, Arad, Eban and Landau give a simplification and extension of these results that provides additive approximations for all points of the Tutte polynomial, including the Jones polynomial at any point, and the Potts model partition function at any temperature and any set of coupling strengths. We hope and believe that the ideas presented in this work will have other extensions and generalizations.

Keywords

Unitary Representation Braid Group Quantum Algorithm Jones Polynomial Tutte Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aharonov, D., Arad, I.: On the \(\mathsf{BQP}\) -hardness of approximating the Jones polynomial. Preprint (2006) Google Scholar
  2. 2.
    Aharonov, D., Arad, I., Eban, E., Landau, Z.: Polynomial Quantum algorithms for additive approximations of the Potts model and the Tutte polynomial. quant-ph/0702008 (2007) Google Scholar
  3. 3.
    Aharonov, D., Jones, V., Landau, Z.: Preliminary version of this paper. In: Proceedings of STOC (2006). quant-ph/0511096 Google Scholar
  4. 4.
    Artin, E.: Theory of braids. Ann. Math. 48, 101–126 (1947) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Bisch, D., Jones, V.: Algebras associated to intermediate subfactors. Invent. Math. 128, 89–157 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bordewich, M., Freedman, M., Lovasz, L., Welsh, D.: Approximate counting and quantum computation. Comb. Probab. Comput. 14(5–6), 737–754 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Childs, A., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.: Exponential algorithmic speedup by quantum walk. In: STOC (2003) Google Scholar
  8. 8.
    Conway, J.H.: An enumeration of knots and links, and some of their algebraic properties. In: Proceeding of the Conference on Computational Problems in Abstract Algebra, Oxford, 1967, pp. 329–358 (1970) Google Scholar
  9. 9.
    van Dam, W., Hallgren, S.: Efficient quantum algorithms for shifted quadratic character problems. quant-ph/0011067 Google Scholar
  10. 10.
    Freedman, M.: P/NP and the quantum field computer. Proc. Natl. Acad. Sci., USA 95, 98–101 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Freedman, M., Kitaev, A., Larsen, M., Wang, Z.: Topological quantum computation. Mathematical challenges of the 21st century. Bull. Am. Math. Soc. (N.S.) 40(1), 31–38 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Freedman, M.H., Kitaev, A., Wang, Z.: Simulation of topological field theories by quantum computers. Commun. Math. Phys. 227, 587–603 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Freedman, M.H., Larsen, M., Wang, Z.: A modular functor which is universal for quantum computation. Commun. Math. Phys. 227(3), 605–622 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Garnerone, S., Marzuoli, A., Rasetti, M.: quant-ph/0601169 Google Scholar
  15. 15.
    Garnerone, S., Marzuoli, A., Rasetti, M.: quant-ph/0606167 Google Scholar
  16. 16.
    Garnerone, S., Marzuoli, A., Rasetti, M.: quant-ph/0607203 Google Scholar
  17. 17.
    Garnerone, S., Marzuoli, A., Rasetti, M.: quant-ph/0703037v1 Google Scholar
  18. 18.
    Goodman, F.M., de la Harpe, P., Jones, V.F.R.: Coxeter Graphs and Towers of Algebras. Springer, Berlin (1989) zbMATHGoogle Scholar
  19. 19.
    Hallgren, S.: Polynomial-time quantum algorithms for Pell’s Equation and the principal ideal problem. In: STOC, pp. 653–658 (2002) Google Scholar
  20. 20.
    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) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries. In: STOC, pp. 712–721 (2001) Google Scholar
  22. 22.
    Jones, V.F.R.: A polynomial invariant for knots via von Neumann algebras. Bull. Am. Math. Soc. 12(1), 103–111 (1985) zbMATHCrossRefGoogle Scholar
  23. 23.
    Jones, V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Jones, V.F.R.: Braid groups, Hecke algebras and type II factors. In: Geometric Methods in Operator Algebras. Pitman Research Notes in Math., vol. 123, pp. 242–273. Longman, Harlow (1986) Google Scholar
  25. 25.
    Kauffman, L.: State models and the Jones polynomial. Topology 26, 395–407 (1987) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Kauffman, L.H., Lomonaco, S.J. Jr.: q-deformed spin networks, knot polynomials and anyonic topological computation. J. Knot Theory Ramif. 16(3), 267–332 (2007). quant-ph/0606114 v3 zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Kauffman, L.H.: Quantum computing and the Jones polynomial. AMS Contemp. Math. Ser. 305, 101–137 (2002) MathSciNetGoogle Scholar
  28. 28.
    Kauffman, L.H., Lomonaco, S.J.: A three-stranded quantum algorithm for the Jones polynomial. In: Donkor, E.J., Pirich, A.R., Brandt, H.E. (eds.) Quantum Information and Quantum Computation V. Proceedings of Spie, April 2007, pp. 1–16. Int. Soc. Opt. Eng. arXiv:0706.0020 Google Scholar
  29. 29.
    Kuperberg, G.: A subexponential-time quantum algorithm for the dihedral hidden subgroup problem. arXiv:quant-ph/0302112 Google Scholar
  30. 30.
    Lomonaco, S.J., Kauffman, L.H.: Topological quantum computing and the Jones polynomial. arXiv:quant-ph/0605004v1 Google Scholar
  31. 31.
    Neilsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge Univ. Press, Cambridge (2000) Google Scholar
  32. 32.
    Podtelezhnikov, A., Cozzarelli, N., Vologodskii, A.: Equilibrium distributions of topological states in circular DNA: interplay of supercoiling and knotting. Proc. Natl. Acad. Sci. USA 96(23), 12974–12979 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Preskill, J.: Topological quantum computation. In: Lecture Notes for Caltech Course # 219 in Physics. http://www.theory.caltech.edu/~preskill/ph229/#lecture
  34. 34.
    Subramaniam, V., Ramadevi, P.: Quantum computation of Jones’ polynomials. quant-ph/0210095 Google Scholar
  35. 35.
    Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Shor, P.W., Jordan, S.P.: Estimating Jones polynomials is a complete problem for one clean qubit. arXiv:quant-ph0707.2831v1 Google Scholar
  37. 37.
    Vogel, P.: Representation of links by braids: a new algorithm. Comment. Math. Helv. 65(1), 104–113 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Watrous, J.: Quantum algorithms for solvable groups. In: STOC, pp. 60–67 (2001) Google Scholar
  39. 39.
    Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121(3), 351–399 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Wocjan, P., Yard, J.: The Jones polynomial: quantum algorithms and applications in quantum complexity theory. quant-ph/0603069 Google Scholar
  41. 41.
    Wu, F.Y.: Knot theory and statistical mechanics, Rev. Mod. Phys. 64(4) (1992) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.School of Computer Science and EngineeringThe Hebrew UniversityJerusalemIsrael
  2. 2.Department of MathematicsU.C. BerkeleyBerkeleyUSA
  3. 3.Department of MathematicsThe City College of New YorkNew YorkUSA

Personalised recommendations