WoLLIC 2007: Logic, Language, Information and Computation pp 248-263 | Cite as
Spin Networks, Quantum Topology and Quantum Computation
Conference paper
Abstract
We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. The results are applied to the quantum computation of colored Jones polynomials.
Keywords
braiding knotting linking spin network Temperley – Lieb algebra unitary representationPreview
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References
- 1.Aharonov, D., Jones, V., Landau, Z.: A polynomial quantum algorithm for approximating the Jones polynomial, quant-ph/0511096.Google Scholar
- 2.Aharonov, D., Arad, I.: The BQP-hardness of approximating the Jones polynomial, quant-ph/0605181.Google Scholar
- 3.Freedman, M.: A magnetic model with a possible Chern-Simons phase, quant-ph/0110060v1 9 October 2001 (preprint)Google Scholar
- 4.Freedman, M.: Topological Views on Computational Complexity. Documenta Mathematica - Extra Volume ICM, 453–464 (1998)Google Scholar
- 5.Freedman, M., Larsen, M., Wang, Z.: A modular functor which is universal for quantum computation, quant-ph/0001108v2 (February 1, 2000)Google Scholar
- 6.Freedman, M.H., Kitaev, A., Wang, Z.: Simulation of topological field theories by quantum computers. Commun. Math. Phys. 227, 587–603 (2002) quant-ph/0001071.MATHCrossRefMathSciNetGoogle Scholar
- 7.Freedman, M.: Quantum computation and the localization of modular functors, quant-ph/0003128.Google Scholar
- 8.Jones, V.F.R.: A polynomial invariant for links via von Neumann algebras, Bull. Amer. Math. Soc. 129, 103–112 (1985)Google Scholar
- 9.Kauffman, L.H.: State models and the Jones polynomial. Topology 26, 395–407 (1987)MATHCrossRefMathSciNetGoogle Scholar
- 10.Kauffman, L.H.: An invariant of regular isotopy. Trans. Amer. Math. Soc. 318(2), 417–471 (1990)MATHCrossRefMathSciNetGoogle Scholar
- 11.Kauffman, L.H.: Temperley – Lieb Recoupling Theory and Invariants of Three-Manifolds, Princeton University Press, Annals Studies, 114 (1994)Google Scholar
- 12.Kauffman, L.H., Lomonaco Jr., S.J.: Braiding Operators are Universal Quantum Gates. New Journal of Physics 6(134), 1–39 (2004)MathSciNetGoogle Scholar
- 13.Kauffman, L.H., Lomonaoco Jr., S.J.: q - deformed spin networks, knot polynomials and anyonic topological computation, quant-ph/0606114 ( to appear in JKTR)Google Scholar
- 14.Kitaev, A.: Anyons in an exactly solved model and beyond, arXiv.cond-mat/0506438 v1 (17 June 2005)Google Scholar
- 15.Marzuoli, A., Rasetti, M.: Spin network quantum simulator. Physics Letters A 306, 79–87 (2002)MATHCrossRefMathSciNetGoogle Scholar
- 16.Penrose, R.: Angular momentum: An approach to Combinatorial Spacetime. In: Bastin, T. (ed.) Quantum Theory and Beyond, Cambridge University Press, Cambridge (1969)Google Scholar
- 17.Preskill, J.: Topological computing for beginners, (slide presentation), Lecture Notes for Chapter 9 - Physics 219 - Quantum Computation. http://www.iqi.caltech.edu/preskill/ph219
- 18.Wilczek, F.: Fractional Statistics and Anyon Superconductivity. World Scientific Publishing Company, Singapore (1990)Google Scholar
- 19.Witten, E.: Quantum field Theory and the Jones Polynomial. Commun. Math. Phys. 1989, 351–399 (1989)CrossRefMathSciNetGoogle Scholar
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