Quantum Combing

  • Mario Rasetti


Leading idea of this note is to argue that quantum information manipulation tools may allow us to explore much wider fields than mere computation, reaching beyond its boundaries to touch the very roots of the universal structure of languages. The paper is mostly conjectural and touches just the few technical details necessary to pursue the general argument, because its main aim is simply to show how a complex blend of notions coming from formal language theory, finite group theory, and quantum computation theory can lead to new views. As working study-case the problem of combing finite groups will be dealt with, which bridges language theoretical issues with structural and algorithmic issues.


Word Problem Regular Expression Braid Group Regular Language Mapping Class 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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  1. 1.
    A. V. Aho: J. Assoc. Comp. Math 15, 647 (1968)Google Scholar
  2. 2.
    G. Baumslag, S. M. Gersten, M. Shapiro and H. Short: Journal of Pure and Applied Algebra 76, 229 (1991)CrossRefGoogle Scholar
  3. 3.
    D. Battaglia and M. Rasetti: Quantumlike Diffusion over Discrete Sets. Phys. Lett. A 313, 8 (2003)CrossRefGoogle Scholar
  4. 4.
    L. C. Biedenharn and J. D. Louck: The Racah-Wigner Algebra. In: Quantum Theory, Encyclopedia of Mathematics and its Applications textbf9 ed by G. C. Rota (Addison-Wesley Publ. Co., Reading 1981)Google Scholar
  5. 5.
    J. Birman: Braids, links, and mapping class groups, Annals of Math. Studies 8 (Princeton University Press, Princeton, 1975)Google Scholar
  6. 6.
    M. R. Bridson: Combings of semidirect products and 3-manifold groups. Geometric and Functional Analysis 3, 263 (1993)CrossRefGoogle Scholar
  7. 7.
    J. W. Cannon, W. J. Floyd and W. R. Parry: L’enseignement mathématique 42, 215 (1996)Google Scholar
  8. 8.
    N. Chomsky: Three models for the description of language. IRE Transactions on Information Theory 2, 113 (1956)CrossRefGoogle Scholar
  9. 9.
    N. Chomsky: On certain formal properties of grammars. Information and Control 2, 137 (1959)CrossRefGoogle Scholar
  10. 10.
    N. Chomsky: Aspects of the Theory of Syntax (MIT Press, Cambridge 1965)Google Scholar
  11. 11.
    N. Chomsky: Language and Mind (Harcourt, Brace & Jovanovich Inc., New York 1972)Google Scholar
  12. 12.
    N. Chomsky: Reflections on language (Pantheon Books, New York 1975)Google Scholar
  13. 13.
    N. Chomsky: Knowledge of Language (Praeger, New York 1986)Google Scholar
  14. 14.
    N. Chomsky: Language and the Problems of Knowledge. (MIT Press, Cambridge 1988)Google Scholar
  15. 15.
    N. Chomsky and M. P. Schötzenberger: The algebraic theory of context free languages. In: Computer Programming and Formal Languages, ed by P. Braffort and D. Hirschberg (North Holland Publ. Co., Amsterdam 1963) p 118CrossRefGoogle Scholar
  16. 16.
    J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson: Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups (Clarendon Press, Oxford 1985)Google Scholar
  17. 17.
    P. Dehornoy: Braids and Self-Distributivity, Progress in Math. 192 (Birkhäuser Verlag, Basel 2000)Google Scholar
  18. 18.
    P. Dehornoy: Geometric presentations for Thompson’s groups. Journal of Pure and Applied Algebra 203, 1 (2005)CrossRefGoogle Scholar
  19. 19.
    P. Dehornoy: The group of parenthesized braids. Advances in Mathematics 205, 354 (2006)Google Scholar
  20. 20.
    E. Dennis, A. Yu. Kitaev, A. Landahl and J. Preskill: Topological quantum memory. J. Math. Phys. 43, 4452 (2002)CrossRefGoogle Scholar
  21. 21.
    D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. Levy, M. S. Patterson and W. Thurston: Word processing in groups (Jones and Bartlett, Boston 1992)Google Scholar
  22. 22.
    R. P. Feynman: Simulating Physics with Computers. Int. J. of Theor. Phys. 21, 467 (1982)CrossRefGoogle Scholar
  23. 23.
    M. H. Freedman, A. Kitaev and Z. Wang: Simulation of topological field theories by quantum computers. Commun. Math. Phys. 227, 587 (2002)CrossRefGoogle Scholar
  24. 24. M.H. Freedman, A. Kitaev, M. Larsen and Z. Wang: Topological quantum computation. Bull. Amer. Math. Soc. 40, 31 (2002)CrossRefGoogle Scholar
  25. 25.
    M.H. Freedman, M. Larsen and Z. Wang: A modular functor which is universal for quantum computation. Commun. Math. Phys. 227, 605 (2002)CrossRefGoogle Scholar
  26. 26.
    S. Garnerone, A. Marzuoli and M. Rasetti: Quantum geometry and quantum algorithms. J. Phys. A: Math. Theor. 40, 3047 (2007)CrossRefGoogle Scholar
  27. 27.
    S. Garnerone, A. Marzuoli and M. Rasetti: Quantum automata, braid group and link polynomials. Quantum Information & Computation 7, 479 (2007)Google Scholar
  28. 28.
    R. H. Gilman: Formal languages and infinite groups. Discrete Math. Theoret. Comput. Sci. 25, 27 (1996)Google Scholar
  29. 29.
    S. Ginsburg: The Mathematical Theory of Context-Free Languages (McGraw-Hill, Inc., New York 1966)Google Scholar
  30. 30.
    S. Ginsburg and M. Harrison: Bracketed Context-Free Languages. J. Comp. Sci. Soc. 1, 1 (1967)Google Scholar
  31. 31. N. Goodman: The Emperor’s New Ideas. In: Language and Philosophy, ed by S. Hook (New York University Press, New York 1969)Google Scholar
  32. 32.
    J. E. Hopcroft, R. Motwani and J. D. Ullman: Introduction to automata theory, languages and computation (Addison-Wesley, Boston 1979)Google Scholar
  33. 33.
    V. Jones: A Polynomial Invariant for Knots via von Neumann Algebras. Bull. Am. Math. Soc. 12, 103 (1985)CrossRefGoogle Scholar
  34. 34. A. Kitaev: Fault-tolerant quantum computation by anyons. Annals Phys. 303, 2 (2003)CrossRefGoogle Scholar
  35. 35.
    A. Marzuoli and M. Rasetti: Spin network quantum simulator. Phys. Lett. A306, 79 (2002)CrossRefGoogle Scholar
  36. 36.
    A. Marzuoli and M. Rasetti: Computing spin networks. Annals of Physics 318, 345 (2005)CrossRefGoogle Scholar
  37. 37.
    A. Marzuoli and M. Rasetti: Spin network setting of topological quantum computation. Int. J. Quantum Information 3, 65 (2005)CrossRefGoogle Scholar
  38. 38.
    R. McKenzie and R. J. Thompson: An elementary construction of unsolvable word problems in group theory. In: Word Problems, ed by W. W. Boon, F. B. Cannonito and R.C. Lyndon (North-Holland Publ. Co., Amsterdam 1973)Google Scholar
  39. 39.
    R. McNaughton: Parenthesis grammars. Journal of the ACM 14, 490 (1967)Google Scholar
  40. 40. C. Moore and J. P. Crutchfield: Quantum automata and quantum grammars. Theor. Comput. Sci. 37, 275 (2000)CrossRefGoogle Scholar
  41. 41. M. A. Nielsen and I. L. Chuang: Quantum Computation and Quantum Information, (Cambridge University Press, Cambridge 2000)Google Scholar
  42. 42. J. Pachos, P. Zanardi and M. Rasetti: Non-Abelian Berry connections for quantum computation. Phys. Rev. A 61, 010305(R) (2000)Google Scholar
  43. 43. R. Penrose: Angular Momentum: an approach to combinatorial space-time. In: Quantum Theory and Beyond, ed by T. Bastin (Cambridge Univ. Press, Cambridge 1971)Google Scholar
  44. 44. W. V. O. Quine: Methodological Reflections on Current Linguistic Theory. Sinthese 21, 386 (1970)CrossRefGoogle Scholar
  45. 45. R. J. Thompson: Embeddings into finitely generated simple groups which preserve the word problem. In: Word Problems II, ed by S. Adian, W. W. Boone, and G. Higman (North-Holland Publ. Co., Amsterdam 1980)Google Scholar
  46. 46. E. Witten: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351 (1989)CrossRefGoogle Scholar
  47. 47. A. P. Yutsis, I. B. Levinson and V. V. Vanagas: The Mathematical Apparatus of the Theory of Angular Momentum (Israel Program for Sci. Transl. Ltd., Jerusalem 1962)Google Scholar
  48. 48. P. Zanardi and M. Rasetti: Holonomic quantum computation. Phys. Lett. A 264, 94 (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia 2008

Authors and Affiliations

  • Mario Rasetti
    • 1
    • 2
  1. 1.Dipartimento di FisicaPolitecnico di TorinoTorinoItaly
  2. 2.Fondazione ISITorinoItaly

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