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Communications in Mathematical Physics

, Volume 334, Issue 2, pp 743–777 | Cite as

Quantum Fourier Transforms and the Complexity of Link Invariants for Quantum Doubles of Finite Groups

  • Hari KroviEmail author
  • Alexander Russell
Article

Abstract

Knot and link invariants naturally arise from any braided Hopf algebra. We consider the computational complexity of the invariants arising from an elementary family of finite-dimensional Hopf algebras: quantum doubles of finite groups [denoted \({{\mathsf{D}(G)}}\), for a group G]. These induce a rich family of knot invariants and, additionally, are directly related to topological quantum computation.

Regarding algorithms for these invariants, we develop quantum circuits for the quantum Fourier transform over \({{\mathsf{D}(G)}}\); in general, we show that when one can uniformly and efficiently carry out the quantum Fourier transform over the centralizers Z(g) of the elements of G, one can efficiently carry out the quantum Fourier transform over \({{\mathsf{D}(G)}}\). We apply these results to the symmetric groups to yield efficient circuits for the quantum Fourier transform over \({{\mathsf{D}(S_n)}}\). With such a Fourier transform, it is straightforward to obtain additive approximation algorithms for the related link invariant.

As for hardness results, first we note that in contrast to those concerning the Jones polynomial—where the images of the braid group representations are dense in the unitary group—the images of the representations arising from \({{\mathsf{D}(G)}}\) are finite. This important difference appears to be directly reflected in the complexity of these invariants. While additively approximating “dense” invariants is \({{\mathsf{BQP}}}\)-complete and multiplicatively approximating them is \({{\#\mathsf{P}}}\)-complete, we show that certain \({{\mathsf{D}(G)}}\) invariants (such as \({{\mathsf{D}(A_n)}}\) invariants) are \({{\mathsf{BPP}}}\)-hard to additively approximate, \({{\mathsf{SBP}}}\)-hard to multiplicatively approximate, and \({{\#\mathsf{P}}}\)-hard to exactly evaluate. To show this, we prove that, for groups (such as A n ) which satisfy certain properties, the probability of success of any randomized computation can be approximated to within any \({\varepsilon}\) by the plat closure.

Finally, we make partial progress on the question of simulating anyonic computation in groups uniformly as a function of the group size. In this direction, we provide efficient quantum circuits for the Clebsch–Gordan transform over \({{\mathsf{D}(G)}}\) for “fluxon” irreps, i.e., irreps of \({{\mathsf{D}(G)}}\) characterized by a conjugacy class of G. For general irreps, i.e., those which are associated with a conjugacy class of G and an irrep of a centralizer, we present an efficient implementation under certain conditions, such as when there is an efficient Clebsch–Gordan transform over the centralizers (this could be a hard problem for some groups). We remark that this also provides a simulation of certain anyonic models of quantum computation, even in circumstances where the group may have size exponential in the size of the circuit.

Keywords

Irreducible Representation Conjugacy Class Hopf Algebra Quantum Computation Braid Group 
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.

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References

  1. 1.
    Aaronson, S.: Quantum computing, postselection, and probabilistic polynomial-time. In: Proceedings of the Royal Society A, p. 0412187 (2005)Google Scholar
  2. 2.
    Aharonov, D., Arad, I.: The BQP-hardness of approximating the Jones polynomial. Technical Report quant-ph/0605181v1, Quant-ph e-print archive (2006). http://arxiv.org/abs/quant-ph/0605181
  3. 3.
    Aharonov, D., Jones, V., Landau, Z.: A polynomial quantum algorithm for approximating the Jones polynomial. In: Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, STOC ’06, pp. 427–436, New York, NY, USA (2006). ACM. doi: 10.1145/1132516.1132579
  4. 4.
    Alagic, G., Jordan, S.P., König, R., Reichardt, B.W.: Estimating Turaev–Viro three-manifold invariants is universal for quantum computation. Phys. Rev. A 82, 040302 (2010). doi: 10.1103/PhysRevA.82.040302
  5. 5.
    Alexander J.W.: A lemma on systems of knotted curves. Proc. Natl. Acad. Sci. (USA) 9, 93–95 (1923)ADSCrossRefGoogle Scholar
  6. 6.
    Mix Barrington D.A., Straubing H., Thérien D.: Non-uniform automata over groups. Inform. Comput. 89(2), 109–132 (1990). doi: 10.1016/0890-5401(90)90007-5 CrossRefzbMATHGoogle Scholar
  7. 7.
    Beals, R.: Quantum computation of Fourier transforms over symmetric groups. In: Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, STOC ’97, pp. 48–53, New York, NY, USA (1997). ACM. ISBN 0-89791-888-6. doi: 10.1145/258533.258548
  8. 8.
    Birman J.S.: On the stable equivalence of plat representations of knots and links. Can. J. Math. 28, 264–290 (1976). doi: 10.4153/CJM-1976-030-1 CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Birman, J.S., Menasco, W.W.: Studying links via closed braids. i. a finiteness theorem. Pacific J. Math. 154(1), 17–36 (1992). http://projecteuclid.org/euclid.pjm/1102635729
  10. 10.
    Böhler, E., Glaßer, C., Meister, D.: Error-bounded probabilistic computations between MA and AM. In: Rovan, B., Vojtás, P. (eds.) Mathematical Foundations of Computer Science 2003, 28th International Symposium (MFCS 2003), volume 2747 of Lecture Notes in Computer Science, pp. 249–258. Springer, Berlin (2003)Google Scholar
  11. 11.
    Bordewich M., Freedman M., Lovász L., Welsh D.: Approximate counting and quantum computation. Comb. Prob. Comput. 14(5–6), 737–754 (2005). doi: 10.1017/S0963548305007005 CrossRefzbMATHGoogle Scholar
  12. 12.
    Curtis C.W., Reiner I.: Representation Theory of Finite Groups and Associative Algebras. Ams Chelsea Publishing. AMS Chelsea Pub., New York (1962)zbMATHGoogle Scholar
  13. 13.
    Dijkgraaf, R., Pasquier, V., Roche, P.: Quasi hope algebras, group cohomology and orbifold models. Nuclear Phys. B Proc. Suppl. 18(2), 60–72 (1991). ISSN 0920-5632. doi: 10.1016/0920-5632(91)90123-V. http://www.sciencedirect.com/science/article/pii/092056329190123V
  14. 14.
    Etingof, P., Rowell, E., Witherspoon, S.: Braid group representations from twisted quantum doubles of finite groups. Pac. J. Math. 234(1), 33–41 (2007). http://arxiv.org/abs/math/0703274
  15. 15.
    Franko J.M., Rowell E.C., Wang Z.: Extraspecial 2-groups and images of braid group representations. J. Knot Theory Ramif. 15(413), 595–615 (2006). doi: 10.1142/S0218216506004580 ISSN 1793-6527MathSciNetGoogle Scholar
  16. 16.
    Freedman M.H., Kitaev A., Wang Z.: Simulation of topological field theories by quantum computers. Commun. Math. Phys. 227, 587 (2002). doi: 10.1007/s002200200635 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    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 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Freedman M.H., Kitaev A., Larsen M.J., Wang Z.: Topological quantum computation. Bull. Am. Math. Soc. (NS) 40(1), 31–38 (2003). doi: 10.1090/S0273-0979-02-00964-3 CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Garnerone S., Marzuoli A., Rasetti M.: Quantum geometry and quantum algorithms. J. Phys. A Math. Theor. 40(12), 3047 (2007). doi: 10.1088/1751-8113/40/12/S10 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Gould M.D.: Quantum double finite groups algebras and their representations. Bull. Austr. Math. Soc. 48, 275–301 (1993)CrossRefzbMATHGoogle Scholar
  21. 21.
    Hastings M., Nayak C., Wang Z.: Metaplectic anyons, majorana zero modes and their applications. Phys. Rev. B 87, 165421 (2013). doi: 10.1103/PhysRevB.87.165421 ADSCrossRefGoogle Scholar
  22. 22.
    Hastings M., Nayak C., Wang Z.: On metaplectic modular categories and their applications. Commun. Math. Phys. 330, 45–68 (2014). doi: 10.1007/s0020-014-2044-7 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Johnson D.: Homomorphs of knot groups. Proc. Am. Math. Soc. 78(1), 135–138 (1980)CrossRefzbMATHGoogle Scholar
  24. 24.
    Kassel C.: Quantum Groups, Volume 155 of Graduate Texts in Mathematics. Springer, New York (1995)Google Scholar
  25. 25.
    Kitaev A.Y.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303(1), 2–30 (2003). doi: 10.1016/S0003-4916(02)00018-0 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Koenig, R., Kuperberg, G., Reichardt, B.: Quantum computation with Turaev–Viro codes. Ann. Phys. 325(12), 2707–2749 (2010). ISSN 0003-4916. doi: 10.1016/j.aop.2010.08.001. http://www.sciencedirect.com/science/article/pii/S0003491610001375
  27. 27.
    Kuperberg, G.: How hard is it to approximate the Jones polynomial? Technical Report http://arxiv.org/abs/0908.0512v1, Quant-ph e-print archive (2009). http://arxiv.org/abs/0908.0512
  28. 28.
    Majid, S.: A Quantum Groups Primer, Volume 292 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2002). doi: 10.1017/CBO9780511549892
  29. 29.
    Markov A.A.: Uber die freie Aquivalenz geschlosserner Zopfe. Recueil Mathematique Moscou 1, 73–78 (1935)Google Scholar
  30. 30.
    Maurer W.D., Rhodes John L.: A property of finite non-Abelian simple groups. Proc. Am. Math. Soc. 16, 552–554 (1965)CrossRefzbMATHGoogle Scholar
  31. 31.
    Mochon C.: Anyons from non-solvable finite groups are sufficient for universal quantum computation. Phys. Rev. A 67, 022315 (2003). doi: 10.1103/PhysRevA.67.022315 ADSCrossRefGoogle Scholar
  32. 32.
    Mochon C.: Anyon computers with smaller groups. Phys. Rev. A 69, 032306 (2004). doi: 10.1103/PhysRevA.69.032306 ADSCrossRefGoogle Scholar
  33. 33.
    Walter Ogburn, R., Preskill, J.: Topological quantum computation. In: QCQC, pp. 341–356 (1998). doi: 10.1007/3-540-49208-9_31
  34. 34.
    Pachos, J.K.: Introduction to Topological Quantum Computation. Cambridge University Press, UK (2012). ISBN 9781107005044. http://books.google.com/books?id=XDciVh6bAE0C
  35. 35.
    Preskill, J.: Topological quantum computation. Chapter 9 of Lecture Notes on Quantum Computation (2004). http://www.theory.caltech.edu/~preskill/ph219/
  36. 36.
    Rowell, E.: Two paradigms for topological quantum computation. Advances in Quantum Computation: Representation Theory, Quantum Field Theory, Category Theory, Mathematical Physics, and Quantum Information Theory, September 20–23, 2007, University of Texas at Tyler. In: Mahdavi, K., Koslover, D. (eds.) Contemporary mathematics—American Mathematical Society. American Mathematical Society, USA (2009)Google Scholar
  37. 37.
    Rowell E.C., Wang Z.: Localization of unitary braid group representations. Commun. Math. Phys. 311(3), 595–615 (2012). doi: 10.1007/s00220-011-1386-7 ISSN 0010-3616ADSCrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Serre J.P.: Linear representations of finite groups, volume 42 of Graduate texts in mathematics. Springer, Berlin (1977)CrossRefGoogle Scholar
  39. 39.
    Tsohantjis I., Gould M.D.: Quantum double finite group algebras and link polynomials. Bull. Austr. Math. Soc. 49, 177–204 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Wocjan P., Yard J.: The Jones polynomial: quantum algorithms and applications in quantum complexity theory. Quantum Inform. Comput. 8(1–2), 147–180 (2008)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Raytheon BBN TechnologiesCambridgeUSA
  2. 2.University of ConnecticutStorrsUSA

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