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.
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References
Aberth, O.: Precise Numerical Analysis. Wm. C. Brown Publishers, Dubuque, Iowa (1988)
Aihara, J., Hosoya, H.: Bull. Chem. Soc. Japan, 61, 2657–ff (1988)
Akhiezer, A.I., Berestetskii, V.B.: Quantum Electrodynamics. Pergamon Press, New York (1982)
Beeson, M.J.: Foundations of Constructive Mathematics. Springer-Verlag, New York (1985)
Beezer, R.A., Farrell, E.J.: The matching polynomial of a regular graph. Discrete Mathe. 137(1), 7–18 (1995)
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)
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)
Bernstein, E., Vazirani, U.: Quantum complexity theory. In: Proceedings of the 25th ACM Symposium on Theory of Computing, pp. 11–20 (1993)
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)
Bishop, E.: Foundations of Constructive Analysis, McGraw-Hill (1967)
Bishop, E., Bridges, D.S.: Constructive Analysis. Springer, New York (1985)
Bridges, D.S.: Constructive Functional Analysis. Pitman, London (1979)
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)
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)
Feynman, R.: Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982)
Finkelstein, D., Gibbs, J.M.: Quantum relativity. Int. J. Theor. Phys. 32, 1801 (1993)
Finkelstein, D.: Quantum Relativity. Springer-Verlag, Berlin, Heidelberg (1996)
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)
Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)
Grib, A.A., Mamayev, S.G., Mostepanenko, V.M.: Vacuum Quantum Effects in Strong Fields. Friedmann Laboratory Publishing, St.Petersburg (1994). (Chapter 11)
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)
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)
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)
Hosoya, H.: Comp. Math. Appls. 12B, 271–ff (1986)
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)
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)
Kushner, B.A.: Lectures on Constructive Mathematical Analysis, Translations of Mathematical Monographs, vol. 60. American Mathematical Society, Providence (1984)
Landau, L.D., Lifschitz, E.M.: Quantum Mechanics: Non-relativistic Theory. Pergamon Press, Oxford (1965)
Papadimitriou, C.H.: Computational Complexity. Addison Wesley, San Diego (1994)
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)
Rosenfeld, V.R., Gutman, I.: A novel approach to graph polynomials. Match, 24, 191–ff (1989)
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)
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)
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)
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)
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.
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Kreinovich, V., Longpré, L., Beltran, A. (2018). QFT + NP = P Quantum Field Theory (QFT): A Possible Way of Solving NP-Complete Problems in Polynomial Time. In: Hassanien, A., Elhoseny, M., Kacprzyk, J. (eds) Quantum Computing:An Environment for Intelligent Large Scale Real Application . Studies in Big Data, vol 33. Springer, Cham. https://doi.org/10.1007/978-3-319-63639-9_10
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