Skip to main content
SpringerLink
Log in

A Categorical Presentation of Quantum Computation with Anyons

  • Chapter
  • First Online:
New Structures for Physics

Part of the book series: Lecture Notes in Physics ((LNP,volume 813))

Abstract

In nature one observes that in three space dimensions particles are either symmetric under interchange (bosons) or antisymmetric (fermions). These phases give rise to the two possible “statistics” that one observes. In two dimensions, however, a whole continuum of phases is possible. “Anyon” is a term coined in by Frank Wilczek to describe particles in 2 dimensions that can acquire “any” phase when two or more of them are interchanged. The exchange of two such anyons can be expressed via representations of the braid group and hence, it permits one to encode information in topological features of a system composed of many anyons. Kitaev suggested the possibility that such topological excitations would be stable and could thus be used for robust quantum computation.

This paper aims to

  1. 1.

    give the categorical structure necessary to describe such a computing process;

  2. 2.

    illustrate this structure with a concrete example namely: Fibonacci anyons.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    To a mathematician this is part of the definition of Hilbert space. However, there have been proposals in physics to consider analogues of Hilbert spaces with an indefinite metric. In the physics literature these are sometimes also called “Hilbert spaces.”

  2. 2.

    Indeed, the braid group is not sufficient as anyons are extended objects. We need to have ribbons (or framed) strands to adequately represent all movements such as, for instance, a rotation of an anyon by 2π.

  3. 3.

    Being symmetric means that the braiding σ is such that \(\sigma_{B,A}\sigma_{A,B}=1_{A\otimes B}\) for all A and B.

  4. 4.

    Sometimes called a tortile category.

  5. 5.

    This definition is equivalent [23] to the standard definition of an abelian category [31].

  6. 6.

    Of course, this is an approximation in the effective field theory of these excitations.

  7. 7.

    Such a space is also called a fusion space however, as the initialisation of a state takes place via a splitting, we prefer our proposed terminology.

  8. 8.

    Strictly speaking, it is a biproduct of fusion spaces but as the later is simultaneously a product and a coproduct, it also makes sense to speak of tuples.

  9. 9.

    This algebra is defined as \(K(\textbf{C})\otimes_{\mathbb{Z}}\mathbb{K}\) where K(C) is the Grothendieck ring of C. See for instance [5] p. 32 and 53–54.

References

  1. Abramsky, S.: Temperley-lieb agebra: From knot theory to logic and quantum computation via quantum mechanics. In: Chen, G., Kauffman, L., Lomonaco, S. (eds.) Mathematics of Quantum Computing and Technology, pp. 515–558. Taylor and Francis, New York (2007)

    Chapter  Google Scholar 

  2. Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS’04), IEEE Computer Science Press (2004)

    Google Scholar 

  3. Aharonov, D., Jones, V., Landau, Z.: A polynomial quantum algorithm for approximating the jones polynomial. In: STOC ’06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pp. 427–436. ACM, New York (2006)

    Google Scholar 

  4. Bakalov, B., Kirilov, A. Jr.: On the Lego-teichmüller game. Transf. Groups 5, 7–44 (2000)

    Google Scholar 

  5. Bakalov, B., Kirillov, A.: Lectures on tensor categories and modular functors. American Mathematical Society in University Lecture Series (2001)

    Google Scholar 

  6. Balachandran, A.P., Daughton, A., Gu, Z., Sorkin, R.D., Marmo, G., Srivastava, A.M.: Spin-statistics theorems without relativity or field-theory. Int. J. Mod. Phys. A8, 2993–3044 (1993)

    MathSciNet  ADS  Google Scholar 

  7. Balachandran, A.P., McGlinn, W.D., O’Raifeartaigh, L., Sen, S., Sorkin, R.D., Srivastava, A.M.: Topological spin-statistics theorem for strings. Mod. Phys. Lett. A7, 1427–1442 (1992)

    MathSciNet  ADS  Google Scholar 

  8. Berry, M.V., Robbins, J.M.: Indistinguishability for quantum particles: spin, statistics and the geometric phase. Proc. Roy. Soc. A 453(1963), 1771–1790 (August 1997)

    MathSciNet  Google Scholar 

  9. Bonesteel, N.E., Hormozi, L., Zikos, G.: Braid topologies for quantum computation. Phys. Rev. Lett. 95, 140503 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  10. Bremner, M.J., Dawson, C.M., Dodd, J.L., Gilchrist, A., Harrow, A.W., Mortimer, D., Nielsen, M.A., Osborne, T.J.: Practical scheme for quantum computation with any two-qubit entangling gate. Phys. Rev. Lett. 89, 247902 (2002)

    Article  ADS  Google Scholar 

  11. Burgoyne, N.: On the connection between spin and statistics. Nuovo Cimento VIII(4), 607–609 (1958)

    MathSciNet  Google Scholar 

  12. Coecke, B., ´Paquette, E.O.: Categories for the practising physicist. In: Coecke, B. (ed.) New Structures in Physics. Springer Lecture Notes in Physics, New York (2008)

    Google Scholar 

  13. Crane, L.: 2-d physics and 3-d topology. Comm. Math. Phys. 135, 615–640 (1991)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Das Sarma, S., Freedman, M., Nayak, C., Simon, S.H., Stern, A.: Non-abelian anyons and topological quantum computation. ArXiv e-prints, 707 (July 2007)

    Google Scholar 

  15. deWet, J.S.: On the connection between the spin and statistics of elementary particles. Phys. Rev. 57, 646–652 (1940). Ph.D. thesis, Princeton University 1939

    Article  MathSciNet  ADS  Google Scholar 

  16. Dowker, H.F., Sorkin, R.D.: A spin-statistics theorem for certain topological geons. Class. Quantum Grav. 15(5), 1153–1167 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Duck, I., Sudarshan, E.C.G.: Towards an understanding of the spin-statistics theorem. Am. J. Phys. 66(4), 284–303 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. Duck, I., Sudarshan, E.C.G., Arthur Wightman, S.: Pauli and the spin-statistics theorem. Am. J. Phys. 67(8), 742–746 (1999). Review by Wightman of the book by Duck and Sudarshan

    Article  ADS  Google Scholar 

  19. Fierz, M.: Uber die relativische Theorie kr¨aftfreir Teilchen mit beliebigem spin. Helv Phys. Acta 12(3), 3–37 (1939)

    Article  Google Scholar 

  20. Finkelstein, D., Rubinstein, J.: Connection between spin statistics and kinks. J. Math. Phys. 9(11), 1762–1779 (1968)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. Freedman, M.H., Kitaev, A., Wang, Z.: Simulation of topological field theories by quantum computers. Comm. Math. Phys. 227, 587–603 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Freedman, M.H., Larsen, M., Wang, Z.: A modular functor which is universal for quantum computation. Comm. Math. Phys. 227, 605–622 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Freyd, P.: Abelian Categories. Harper and Row (1964)

    Google Scholar 

  24. Hormozi, L., Zikos, G., Bonesteel, N.E.: Topological quantum compiling. Phys. Rev. B 75, 165310 (2007)

    Article  ADS  Google Scholar 

  25. Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102, 20–78 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kauffman, L.H., Lomonaco, S.J. Jr.: Braiding operators are universal quantum gates. New J. Phys. 6(134), (2004)

    Google Scholar 

  27. Khono, T.: Topological invariants for 3-manifolds using representations of the mapping class groups i. Topology 31, 203–230 (1992)

    Article  MathSciNet  Google Scholar 

  28. Kitaev, A.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303(1), 3–20 (2003)

    MathSciNet  ADS  Google Scholar 

  29. Kitaev, A.: Anyons in an exactly solved model and beyond (2006)

    Google Scholar 

  30. Klitzing, K.V., Dorda, G., Pepper, M.: New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance. Phys. Rev. Lett. 45(6), 494–497 (Aug 1980)

    Article  ADS  Google Scholar 

  31. Mac Lane, S.: Categories for the working mathematician. Springer graduate text in mathemathics, vol. 5. Springer, New-York (1998)

    Google Scholar 

  32. Laughlin, R.B.: Quantized hall conductivity in two dimensions. Phys. Rev. B 23(10), 5632–5633 (May 1981)

    Article  ADS  Google Scholar 

  33. Leinaas, J.M., Myrheim, J.: On the theory of identical particles. Il Nuovo Cimento 37B, 1–23 (1977)

    ADS  Google Scholar 

  34. Lüders, G., Zumino, B.: Connection between spin and statistics. Phys. Rev. 110, 1450–1453 (1958)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. Messiah, A.: Quantum Mechanics, vol. 1. Wiley, New York (1966)

    Google Scholar 

  36. Moore, G., Seiberg, N.: Polynomial equations for rational conformal field theories. Phys. Lett. B 212(4), 451–460 (1988)

    Article  MathSciNet  ADS  Google Scholar 

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

    MATH  Google Scholar 

  38. Pauli, W.: The connection between spin and statistics. Phys. Rev. 58, 716–722 (1940)

    Article  ADS  Google Scholar 

  39. Pauli, W.: On the connection between spin and statistics. Prog. Theor. Phys. 5(4), 526–543 (1950)

    Article  MathSciNet  ADS  Google Scholar 

  40. Preskill, J.: Lectures notes physics 219/computer science 219 quantum computation (2004)

    Google Scholar 

  41. Reshetikhin, Y.N., Tuarev, V.G.: Ribbon graphs and their invariants derived from quantum groups. Comm. Math. Phys. 127, 1–26 (1990)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. Reshetikhin, Y.N., Tuarev, V.G.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103, 547–597 (1991)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  43. Das Sarma, S., Freedman, M., Nayak, C.: Topologically-protected qubits from a possible non-abelian fractional quantum hall state. Phys. Rev. Lett. 94, 166802 (2005)

    Article  ADS  Google Scholar 

  44. Schwinger, J.: Spin, statistics and the TCP theorem and Addendum. Proc. Natl. Acad. Sci. USA 44, 223–228, 617–619 (1958)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  45. Sorkin, R.D.: Particle-statistics in three dimensions. Phys. Rev. D 27, 1787–1797 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  46. Störmer, H.: Nobel Lecture: Physics 1996–2000, Chapter The Fractional Quantum Hall Effect, pp. 295–326. World Scientific, Singapore (2002)

    Google Scholar 

  47. Streater, R.F., Wightman, A.S.: PCT, Spin, Statistics and All That. Benjamin, New York (1964)

    Google Scholar 

  48. Tuarev, V.G.: Quantum Invariant of Knots and 3-Manifolds. W. de Gruyter, Berlin (1994)

    Google Scholar 

  49. Verlinde, E.: Fusion rules and modular transformations in 2d conformal field theory. Nuc. Phys. B. 300, 360–376 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  50. Wightman, A.S.: Quantum field theory in terms of vacuum expectation values. Phys. Rev. 101(2), 860–866 (January 1956)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  51. Wightman, A.S.: The spin-statistics connection: Some pedagogical remarks in response to neuenschwander’s question. Electro. J. Diff. Eq. 207–213 (2000). Conference on Mathematical Physics and Quantum Field Theory

    Google Scholar 

  52. Wilczek, F.: Magnetic flux, angular momentum, and statistics. Phys. Rev. Lett. 48(17), 1144–1146 (April 1982)

    Article  ADS  Google Scholar 

  53. Wilczek, F.: Quantum mechanics of fractional-spin particles. Phys. Rev. Lett. 49(14), 957–959 (Oct 1982)

    Article  MathSciNet  ADS  Google Scholar 

  54. Wilczek, F. (ed.): Fractional Statistics and Anyon Superconductivity. World Scientific, New Jersey (1990)

    Google Scholar 

Download references

Acknowledgement

The authors would like to thank Samson Abramsky, Michel Boyer, Ross Duncan, Rafael Sorkin, Colin Stephen, Benoît Valiron and Jamie Vicary for useful discussions.

Author information

Authors and Affiliations

    Authors
    1. P. Panangaden
      View author publications

      You can also search for this author in PubMed Google Scholar

    2. É.O. Paquette
      View author publications

      You can also search for this author in PubMed Google Scholar

    Corresponding authors

    Correspondence to P. Panangaden or É.O. Paquette .

    Editor information

    Editors and Affiliations

    1. Parks Road, Oxford, OX1 3QD, United Kingdom

      Bob Coecke

    Rights and permissions

    Reprints and permissions

    Copyright information

    © 2010 Springer-VerlagBerlin Heidelberg

    About this chapter

    Cite this chapter

    Panangaden, P., Paquette, É. (2010). A Categorical Presentation of Quantum Computation with Anyons. In: Coecke, B. (eds) New Structures for Physics. Lecture Notes in Physics, vol 813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12821-9_15

    Download citation

    • DOI: https://doi.org/10.1007/978-3-642-12821-9_15

    • Published:

    • Publisher Name: Springer, Berlin, Heidelberg

    • Print ISBN: 978-3-642-12820-2

    • Online ISBN: 978-3-642-12821-9

    • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

    Publish with us

    Policies and ethics

    Search

    Navigation