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QFT + NP = P Quantum Field Theory (QFT): A Possible Way of Solving NP-Complete Problems in Polynomial Time

  • Vladik Kreinovich
  • Luc Longpré
  • Adriana Beltran
Chapter
Part of the Studies in Big Data book series (SBD, volume 33)

Abstract

It has been recently theoretically shown that the dependency of some (potential observable) quantities in quantum field theory (QFT) on the parameters of this theory is discontinuous. This discovery leads to the theoretical possibility of checking whether the value of a given physical quantity is equal to 0 or different from 0 (here, theoretical means that this checking requires very precise measurements and because of that, this conclusion has not yet been verified by a direct experiment). This result from QFT enables us to do what we previously could not: check whether two computable real numbers are equal or not. In this paper, we show that we can use this ability to solve NP-complete (“computationally intractable”) problems in polynomial (“reasonable”) time. Specifically, we will introduce a new model of computation. This new model is based on solid mainstream physics (namely, on quantum field theory). It is capable of solving NP-complete problems in polynomial time.

Notes

Acknowledgements

This work was supported in part:

\(\bullet \) by the National Science Foundation grants

\(\quad \bullet \) HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and

\(\quad \bullet \) DUE-0926721, and

\(\bullet \) by an award “UTEP and Prudential Actuarial Science Academy and Pipeline Initiative” from

   Prudential Foundation.

We are thankful:

\(\bullet \) to Andrei A. Grib and Vladimir M. Mostepanenko who patiently explained their physical results

   to us,

\(\bullet \) to Michael G. Gelfond and Yuri Gurevich for valuable discussions of constructive real numbers,

   and

\(\bullet \) to James M. Salvador for valuable discussions of the corresponding chemical algorithms.

References

  1. 1.
    Aberth, O.: Precise Numerical Analysis. Wm. C. Brown Publishers, Dubuque, Iowa (1988)MATHGoogle Scholar
  2. 2.
    Aihara, J., Hosoya, H.: Bull. Chem. Soc. Japan, 61, 2657–ff (1988)Google Scholar
  3. 3.
    Akhiezer, A.I., Berestetskii, V.B.: Quantum Electrodynamics. Pergamon Press, New York (1982)Google Scholar
  4. 4.
    Beeson, M.J.: Foundations of Constructive Mathematics. Springer-Verlag, New York (1985)CrossRefMATHGoogle Scholar
  5. 5.
    Beezer, R.A., Farrell, E.J.: The matching polynomial of a regular graph. Discrete Mathe. 137(1), 7–18 (1995)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Beezer, R.A., Farrell, E.J., Riegsecker, J., Smith, B.: Graphs with a minimum number of pairs of independent edges I: Matching polynomials. Bulletin of the Institute of Combinatorics and Its Applications, 1996 (to appear)Google Scholar
  7. 7.
    Beltran, A., Salvador, J.M.: The Ulam index. Abstracts of the Second SC-COSMIC Conference in Computational Sciences, October 25–27, El Paso, TX, Rice University Center for Research on Parallel Computations and University of Texas at El Paso, p. 6 (1996)Google Scholar
  8. 8.
    Bernstein, E., Vazirani, U.: Quantum complexity theory. In: Proceedings of the 25th ACM Symposium on Theory of Computing, pp. 11–20 (1993)Google Scholar
  9. 9.
    Berthiaume, A., Brassard, G.: The quantum challenge to structural complexity theory. In: Proceedings of the 7th IEEE Conference on Structure in Complexity Theory, pp. 132–137 (1992)Google Scholar
  10. 10.
    Bishop, E.: Foundations of Constructive Analysis, McGraw-Hill (1967)Google Scholar
  11. 11.
    Bishop, E., Bridges, D.S.: Constructive Analysis. Springer, New York (1985)CrossRefMATHGoogle Scholar
  12. 12.
    Bridges, D.S.: Constructive Functional Analysis. Pitman, London (1979)MATHGoogle Scholar
  13. 13.
    Deutsch, D.: Quantum theory, the Church-Turing principle, and the universal quantum computer. In: Proceedings of the Royal Society of London, Ser. A, Vol. 400, pp. 96–117 (1985)Google Scholar
  14. 14.
    Deutsch, D., Jozsa, R.: Rapid solution of problem by quantum computation. In: Proceedings of the Royal Society of London, Ser. A, Vol. 439, pp. 553–558 (1992)Google Scholar
  15. 15.
    Feynman, R.: Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Finkelstein, D., Gibbs, J.M.: Quantum relativity. Int. J. Theor. Phys. 32, 1801 (1993)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Finkelstein, D.: Quantum Relativity. Springer-Verlag, Berlin, Heidelberg (1996)CrossRefMATHGoogle Scholar
  18. 18.
    Frolov, V.M., Grib, A.A., Mostepanenko, V.M.: Conformal symmetry breaking and quantization in curved space-time. Phys. Lett. A 65, 282–284 (1978)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)MATHGoogle Scholar
  20. 20.
    Grib, A.A., Mamayev, S.G., Mostepanenko, V.M.: Vacuum Quantum Effects in Strong Fields. Friedmann Laboratory Publishing, St.Petersburg (1994). (Chapter 11)Google Scholar
  21. 21.
    Grib, A.A., Mostepanenko, V.M.: Spontaneous breaking of gauge symmetry in a homogeneous isotropic universe if the open type. Sov. Phys.-JETP Lett., 25, 277–279 (1977)Google Scholar
  22. 22.
    Grib, A.A., Mostepanenko, V.M., Frolov, V.M.: Spontaneous breaking of gauge symmetry in a nonstationary isotropic metric. Theor. Math. Phys. 33, 869–876 (1977)CrossRefGoogle Scholar
  23. 23.
    Grib, A.A., Mostepanenko, V.M., Frolov, V.M.: Spontaneous breaking of CP symmetry in a nonstationary isotropic metric. Theor. Math. Phys. 37(2), 975–983 (1978)CrossRefGoogle Scholar
  24. 24.
    Hosoya, H.: Comp. Math. Appls. 12B, 271–ff (1986)Google Scholar
  25. 25.
    Hosoya, H., Balasubramanian, K.: Computational algorithms for matching polynomials of graphs from the characteristic polynomials of edge-weighted graphs. J. Comput. Chem. 10(5), 698–710 (1989)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)CrossRefGoogle Scholar
  27. 27.
    Kushner, B.A.: Lectures on Constructive Mathematical Analysis, Translations of Mathematical Monographs, vol. 60. American Mathematical Society, Providence (1984)MATHGoogle Scholar
  28. 28.
    Landau, L.D., Lifschitz, E.M.: Quantum Mechanics: Non-relativistic Theory. Pergamon Press, Oxford (1965)Google Scholar
  29. 29.
    Papadimitriou, C.H.: Computational Complexity. Addison Wesley, San Diego (1994)MATHGoogle Scholar
  30. 30.
    Randic, M., Hosoya, H., Polansky, O.E.: On the construction of the matching polynomial for unbranched catacondensed benzenoids. J. Comput. Chemi. 10(5), 683–697 (1989)CrossRefGoogle Scholar
  31. 31.
    Rosenfeld, V.R., Gutman, I.: A novel approach to graph polynomials. Match, 24, 191–ff (1989)Google Scholar
  32. 32.
    Salvador, J.M.: Topological indices and polynomials: the partial derivatives. In: Abstracts of the 5th International Conference on Mathematical and Computational Chemistry, May 17–21, Kansas City, Missouri, p. 154 (1993)Google Scholar
  33. 33.
    Shor, P.W.: Algorithms for quantum computations: discrete logarithms and factoring. In: Proceedings of the 35th Annual Symposium on Fundamentals of Computer Science (FOCS), pp. 124–134 (1994)Google Scholar
  34. 34.
    Simon, P.: On the power of quantum computation. In: Proceedings of the 35th Annual Symposium on Fundamentals of Computer Science (FOCS), pp. 116–123 (1994)Google Scholar
  35. 35.
    Turing, A.M.: On computable numbers, with an application to the em Entscheidungsproblem. In: Proceedings London Mathematical Society, vol. 42, pp. 230–265 (1936); see also Proceedings London Mathematics Society, vol. 43, pp. 544–546 (1937)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Vladik Kreinovich
    • 1
  • Luc Longpré
    • 1
  • Adriana Beltran
    • 1
  1. 1.Department of Computer ScienceUniversity of Texas at El PasoEl PasoUSA

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