Skip to main content
SpringerLink
Log in

Topological-Like Features in Diagrammatical Quantum Circuits

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

In this paper, we revisit topological-like features in the extended Temperley–Lieb diagrammatical representation for quantum circuits including the teleportation, dense coding and entanglement swapping. We describe these quantum circuits and derive characteristic equations for them with the help of topological-like operations. Furthermore, we comment on known diagrammatical approaches to quantum information phenomena from the perspectives of both tensor categories and topological quantum field theories. Moreover, we remark on the proposal for categorical quantum physics and information as described by dagger ribbon categories.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Nielsen and I. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 1999).

  2. S. Abramsky and B. Coecke, 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), Arxiv:quant-ph/0402130.

  3. B. Coecke, Kindergarten Quantum Mechanic–Lecture Notes, in Quantum Theory: Reconstructions of the Foundations III, edited by A. Khrennikov (American Institute of Physics Press), pp. 81–98. Arxiv: quant-ph/0510032.

  4. S. Abramsky, Temperley-Lieb Algebra: from Knot Theory to Logic and Computation via Quantum Mechanics, in G. Chen, L. Kauffman, and S. Lomonaco (eds.) Mathematics of Quantum Computation and Quantum Technology (to appear) (Taylor and Francis, 2007), pp. 523–566.

  5. Kauffman L.H. and Lomonaco S.J. (2004). New J. Phys. 6: 134. Arxiv: quant-ph/0401090

    Article  ADS  Google Scholar 

  6. Kauffman L.H. (2005). Opt. Spectrosc. 9: 227–232. Arxiv: quan-ph/0407224

    Article  ADS  Google Scholar 

  7. Griffiths R.B., Wu S., Yu L. and Cohen S.M. (2006). Phys. Rev. A 73: 052309. Arxiv: quant-ph/0507215

    Article  ADS  MathSciNet  Google Scholar 

  8. Zhang Y. (2006). J. Phys. A: Math. Gen. 39: 11599–11622. Arxiv: quant-ph/0610148

    Article  MATH  ADS  Google Scholar 

  9. Y. Zhang, Algebraic Structures Underlying Quantum Information Protocols. Arxiv: quant-ph/0601050.

  10. Temperley H.N.V. and Lieb E.H. (1971). Proc. Roy. Soc. A 322: 251–280

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. Zhang Y., Kauffman L.H. and Werner R.F. (2006). Int. J. Quant. Inform. 5(4): 469–507. Arxiv: quant-ph/0606005

    Article  Google Scholar 

  12. Y. Zhang, L. H. Kauffman, and M. L. Ge, Virtual Extension of Temperley–Lieb Algebra. Arxiv: math-ph/0610052.

  13. Brauer R. (1937). Ann. of Math. 38: 857–872

    Article  MathSciNet  Google Scholar 

  14. L. H. Kauffman and S. J. Lomonaco Jr., q-Deformed Spin Networks, Knot Polynomials and Anyonic Topological Quantum Computation. Arxiv: quant-ph/0606114.

  15. Bennett C.H., Brassard G., Crepeau C., Jozsa R., Peres A. and Wootters W.K. (1993). Phys. Rev. Lett. 70: 1895–1899

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. Vaidman L. (1994). Phys. Rev. A 49: 1473–1475

    Article  ADS  MathSciNet  Google Scholar 

  17. N. Erez, Teleportation from a Projection Operator Point of View. Arxiv: quant-ph/ 0510130.

  18. Braunstein S.L., D’Ariano G.M., Milburn G.J. and Sacchi M.F. (2000). Phys. Rev. Let. 84: 3486–3489

    Article  ADS  Google Scholar 

  19. Żukowski M., Zeilinger A., Horne M.A. and Ekert A.K. (1993). Phys. Rev. Let. 71: 4287–4290

    Article  ADS  Google Scholar 

  20. M. F. Atiyah, The Geometry and Physics of Knots (Cambridge University Press, 1990).

  21. Wu F.Y. (1992). Rev. Mod. Phys. 64: 1099–1131

    Article  ADS  Google Scholar 

  22. L. H. Kauffman, Knots and Physics (World Scientific Publishers, 2002).

  23. Jones V.F.R. (1987). Ann. of Math. 126: 335–388

    Article  MathSciNet  Google Scholar 

  24. Werner R.F. (2001). J. Phys. A 35: 7081–7094. Arxiv: quant-ph/0003070

    Article  ADS  Google Scholar 

  25. Joyal A. and Street R. (1991). Adv. in Math. 88: 55–112

    Article  MATH  MathSciNet  Google Scholar 

  26. Kelly G.M. (1972). Many-Variable Functional Calculus I. Spinger Lecture Notes in Mathematics 281: 66–105

    Google Scholar 

  27. C. Kassel, Quantum Groups (Spring-Verlag New York, 1995).

  28. V. G. Turaev, Quantum Invariants of Knots and 3-Manifolds (de Gruyter, 1994).

  29. Baez J. and Dolan J. (1995). JMP 36: 6703–6105. Arxiv: q-alg/9503002

    MathSciNet  Google Scholar 

  30. J. Baez, Quantum Quandaries: a Category-Theoretic Perspective, in S. French et al. (eds.) Structural Foundations of Quantum Gravity (Oxford University Press, Oxford, 2006), pp. 240-265. Arxiv: quant-ph/0404040

  31. P. Selinger, Dagger Compact Closed Categories and Completely Positive Maps. Electronic Notes in Theoretical Computer Science (special issue: Proceedings of the 3rd International Workshop on Quantum Programming Languages).

  32. Levin M. and Wen X.-G. (2005). Phys. Rev. B 71: 045110

    Article  ADS  Google Scholar 

  33. Zhang Y., Kauffman L.H. and Ge M.L. (2005). Int. J. Quant. Inform. 3(4): 669–678. Arxiv: quant-ph/0412095

    Article  MATH  Google Scholar 

  34. Zhang Y., Kauffman L.H. and Ge M.L. (2005). Quant. Inf. Proc. 4: 159–197. Arxiv: quant-ph/0502015

    Article  MATH  MathSciNet  Google Scholar 

  35. Witten E. (1989). Commun. Math. Phys. 121: 351–399

    Article  MATH  ADS  MathSciNet  Google Scholar 

  36. Freedman M., Larsen M. and Wang Z. (2002). Commun. Math. Phys. 227(3): 605–622. Arxiv: quant-ph/0001108

    Article  MATH  ADS  MathSciNet  Google Scholar 

  37. Freedman M.H., Kitaev A. and Wang Z. (2002). Commun. Math. Phys. 227: 587–603. Arxiv: quant-ph/0001071

    Article  MATH  ADS  MathSciNet  Google Scholar 

  38. G. Segal, Two-dimensional Conformal Field Theories and Modular Functors, IXth. (International Congress on Mathematical Physics, Swansea, 1988) 22C37.

  39. Rovelli C. and Smolin L. (1995). Phys. Rev. D 52: 5743–5759

    Article  ADS  MathSciNet  Google Scholar 

  40. Kelly G.M. and Laplaza M.L. (1980). J. Pure Appl. Algebra 19: 193–213

    Article  MATH  MathSciNet  Google Scholar 

  41. E. O. Paquette, A Categorical Semantics for Topological Quantum Computation, Master’s thesis, University of Ottawa (2004).

  42. Rowell E.C. (2005). Math. Z. 250(4): 745–774

    Article  MATH  MathSciNet  Google Scholar 

  43. E. C. Rowell, From Quantum Groups to Unitary Modular Tensor Categories, to appear in Contemp. Math. (Conference Proceedings).

  44. Y. Zhang, Categorical Quantum Physics and Information (in preparation).

Download references

Author information

Authors and Affiliations

  1. Department of Physics, University of Utah, 115 S, 1400 E, Room 201, Salt Lake City, UT, 84112-0830, USA

    Yong Zhang

  2. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL, 60607-7045, USA

    Louis H. Kauffman

Authors
  1. Yong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Louis H. Kauffman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong Zhang.

Additional information

The main results in this paper had been presented at the workshop “Cats, Kets and Cloisters” (Computing Laboratory, Oxford University, July 17–23, 2006). Categorical quantum physics and information is proposed as described by dagger ribbon categories. The functor between Abramsky and Coecke’s categorical quantum mechanics and the extended Temperley–Lieb categorical approach is recognized as the same type as those defining topological quantum field theories. On the one hand, fundamental objects in the physical world may be string-like (even brane-like) and satisfy the braid statistics. On the other hand, fundamental particles at the Planck energy scale (or quasi-particles of many-body systems) may obey the braid statistics and have an effective (or a new internal) degree of freedom called the “twist spin”. The name “categorical quantum physics and information” hereby refers to quantum physics and information which can be recast in terms of the language of categories. This is a simple and intuitional generalization of the name “categorical quantum mechanics”. The latter does not yet recognize conformal field theories, topological quantum field theories, quantum gravity and string theories which have been already described in the categorical framework by different research groups. Besides, the proposal categorical quantum physics and information is strongly motivated by the present study in quantum information phenomena and theory. It is aimed at setting up a theoretical platform on which both categorical quantum mechanics and topological quantum computing by Freedman, Larsen and Wang are allowed to stand.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, Y., Kauffman, L.H. Topological-Like Features in Diagrammatical Quantum Circuits. Quantum Inf Process 7, 3–32 (2008). https://doi.org/10.1007/s11128-007-0067-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11128-007-0067-0

Keywords

PACS

Search

Navigation