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Hadamard’s Matrices, Grothendieck’s Constant, and Root Two

  • Dominique FortinEmail author
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 39)

Summary.

In this chapter, we start by a non-cooperative quantum game model for multiknapsack to give a flavor of quantum computing strength. Then, we show that many rank-deficient correlation matrices have Grothendieck’s constant that goes beyond \(\sqrt{2}\) for sufficiently large size. It suggests that cooperative quantum games relate powerset entanglement with Grothendieck’s constant.

Keywords:

non-cooperative quantum game multiknapsack entanglement grothendieck’s constant 

References

  1. 1.
    Adenier, G.: Refutation of Bell’s Theorem. In: Foundations of Probability and Physics (Vol. 13, pp. 29–38) QP–PQ: Quantum Probability and White Noise Analysis. World Science, Publishing, River Edge, NJ (2001)Google Scholar
  2. 2.
    Avis, D., Fukuda, K.: A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra. Discrete Comput. Geom. 8(3), 295–313 (1992) ACM Symposium on Computational Geometry (North Conway, NH, 1991)Google Scholar
  3. 3.
    Bannai, E., Sawano, M.: Symmetric designs attached to four-weight spin models. Des. Codes Cryptogr. 25 (1), 73–90 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bracken, C., McGuire, G., Ward, H.: New quasi-symmetric designs constructed using mutually orthogonal Latin squares and Hadamard matrices. Des. Codes Cryptogr. 41 (2), 195–198 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brassard, G., Hoyer, P., Mosca, M., Tapp, A.: Quantum amplitude amplification and estimation. In Quantum Computation and Information (Washington, DC, 2000) (Vol. 305, pp. 53–74), Contemporary Mathematics. Amer. Math. Soc., Providence, RI (2002)Google Scholar
  6. 6.
    Broughton, W., McGuire, G.: Some observations on quasi-3 designs and Hadamard matrices. Des. Codes Cryptogr. 18(1–3), 55–61 (1999) Designs and codes – A memorial tribute to Ed Assmus.Google Scholar
  7. 7.
    Childs, A.M., Landahl, A.J., Parrilo, P.A.: Improved quantum algorithms for the ordered search problem via semidefinite programming (2006)Google Scholar
  8. 8.
    Chu, P.C., Beasley, J.E.: A genetic algorithm for the multidimensional knapsack problem. J. Heuristics 4, 63–86 (1998)zbMATHCrossRefGoogle Scholar
  9. 9.
    Cirel’son, B.S.: Quantum generalizations of Bell’s inequality. Lett. Math. Phys. 4 (2), 93–100 (1980)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Collins, D., Gisin,N., Linden, N., Massar, S., Popescu, S.: Bell inequalities for arbitrarily high-dimensional systems. Phys. Rev. Lett. 88 (4), 040404, 4 (2002).MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cornuéjols, G., Guenin, B., Tunçel, L.: Lehman matrices (2006) http://integer.tepper.cmu.edu/webpub/Lehman-v06.pdf
  12. 12.
    Dasgupta, S., Gupta, A.: An elementary proof of a theorem of Johnson and Lindenstrauss. Random Struct. Algor. 22 (1), 60–65 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Du, J.F., Xu, X., Li, H., Zhou, X., Han, R.: Playing prisoner’s dilemma with quantum rules. Fluct. Noise Lett. 2 (4), R189–R203 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Dye, H.A.: Unitary solutions to the Yang-Baxter equation in dimension four. Quant. Inf. Process. 2 (1-2), 117–151 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Farebrother, R.W., Groß, J., Troschke, S.-O.: Matrix representation of quaternions. Linear Algebra Appl. 362, 251–255 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Fishburn, P.C., Reeds, J.A.: Bell inequalities, Grothendieck’s constant, and root two. SIAM J. Discrete Math. 7 (1), 48–56 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Fortin, D., Rudolf, R.: Weak monge arrays in higher dimensions. Discrete Math. 189 (1–3), 105–115 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Fukuda, K., Prodon, A.: Double Description Method Revisited. In: Combinatorics and Computer Science (Brest, 1995) (Vol. 1120, pp. 91–111) Lecture Notes in Comput. Sci., Springer: Berlin (1996)Google Scholar
  19. 19.
    Glover, F., Rego, C.: Ejection chain and filter-and-fan methods in combinatorial optimization. 4OR 4 (4), 263–296 (2006)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Grover, L., Patel, A., Tulsi, T.: A new algorithm for fixed point quantum search (2005)Google Scholar
  21. 21.
    Grover, L.K.: A Fast Quantum Mechanical Algorithm for Database Search, ACM, New York (1996)Google Scholar
  22. 22.
    Han, K.-H., Kim, J.-H.: Quantum-inspired evolutionary algorithm for a class of combinatorial optimization. IEEE Trans. Evol. Comput. 6 (6), 580–593 (2002)CrossRefGoogle Scholar
  23. 23.
    Han, K.-H., Kim, J.-H.: Quantum-inspired evolutionary algorithms with a new termination criterion, \(h_\epsilon\) gate, and two phase scheme. IEEE Trans. Evol. Comput. 8 (2), 580–593 (2004)CrossRefGoogle Scholar
  24. 24.
    Jaeger, F.: On four-weight spin models and their gauge transformations. J. Algebraic Combin. 11 (3), 241–268 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Kac, V., Cheung, P.: Quantum Calculus. Universitext, Springer, New York (2002)CrossRefGoogle Scholar
  26. 26.
    Kauffman, L.H., Lomonaco, S.J.: Entanglement criteria – Quantum and topological. New J. Phys. 4, 73.1–73.18 (electronic) (2002)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kauffman, L.H., Lomonaco, S.J.: Braiding operators are universal quantum gates (2004)Google Scholar
  28. 28.
    Kauffman, L.H., Lomonaco, S.J.: Quantum knots (2004)Google Scholar
  29. 29.
    Khalfin, L.A., Tsirelson, B.S.: Quantum/classical correspondence in the light of Bell’s inequalities. Found. Phys. 22 (7), 879–948 (1992)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Kitaev, A.Yu., Shen, A.H., Vyalyi, M.N.: Classical and Quantum Computation. Graduate Studies in Mathematics (Vol. 47). American Mathematical Society, Providence, RI (2002) Translated from the 1999 Russian original by Lester J. Senechal.Google Scholar
  31. 31.
    Kitaev, A.Y.: Quantum measurements and the abelian stabilizer problem (1995)Google Scholar
  32. 32.
    Li, D., Huang, H., Li, X.: The fixed-point quantum search for different phase shifts (2006)Google Scholar
  33. 33.
    Marinescu, D., Marinescu, G.D.: Approaching Quantum Computing. Prentice Hall (2004)Google Scholar
  34. 34.
    Martí, R., Laguna, M., Glover, F.: Principles of scatter search. Eur. J. Oper. Res. 169 (2), 359–372 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Monge, G.: déblai et remblai. Mémoires de l’Académie des sciences, Paris (1781)Google Scholar
  36. 36.
    Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev. 41 (2), 303–332 (electronic) (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Sica, L.: Bell’s inequalities i: An explanation for their experimental violation (2001)Google Scholar
  38. 38.
    Sica, L.: Correlations for a new Bell’s inequality experiment. Found. Phys. Lett. 15 (5), 473–486 (2002)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Vasquez, M., Hao, J.-K.: A hybrid approach for the 0–1 multidimensional knapsack problem (2001)Google Scholar
  40. 40.
    Werner, R.F., Wolf, M.M.: Bell inequalities and entanglement (2001)Google Scholar
  41. 41.
    Yost, D.: The Johnson-Lindenstrauss space. Extracta Math. 12(2), 185–192 1997. II Congress on Examples and Counterexamples in Banach Spaces (Badajoz, 1996)Google Scholar
  42. 42.
    Younes, A., Rowe, J., Miller, J.: Quantum search algorithm with more reliable behav0iour using partial diffusion (2003)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Inria, Domaine de Voluceau, RocquencourtLe Chesnay CedexFrance

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