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

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Optimization and Optimal Control

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 39))

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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.

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References

  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. 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. Bannai, E., Sawano, M.: Symmetric designs attached to four-weight spin models. Des. Codes Cryptogr. 25 (1), 73–90 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. Childs, A.M., Landahl, A.J., Parrilo, P.A.: Improved quantum algorithms for the ordered search problem via semidefinite programming (2006)

    Google Scholar 

  8. Chu, P.C., Beasley, J.E.: A genetic algorithm for the multidimensional knapsack problem. J. Heuristics 4, 63–86 (1998)

    Article  MATH  Google Scholar 

  9. Cirel’son, B.S.: Quantum generalizations of Bell’s inequality. Lett. Math. Phys. 4 (2), 93–100 (1980)

    Article  MathSciNet  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  11. Cornuéjols, G., Guenin, B., Tunçel, L.: Lehman matrices (2006) http://integer.tepper.cmu.edu/webpub/Lehman-v06.pdf

  12. Dasgupta, S., Gupta, A.: An elementary proof of a theorem of Johnson and Lindenstrauss. Random Struct. Algor. 22 (1), 60–65 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  14. Dye, H.A.: Unitary solutions to the Yang-Baxter equation in dimension four. Quant. Inf. Process. 2 (1-2), 117–151 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Farebrother, R.W., Groß, J., Troschke, S.-O.: Matrix representation of quaternions. Linear Algebra Appl. 362, 251–255 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fishburn, P.C., Reeds, J.A.: Bell inequalities, Grothendieck’s constant, and root two. SIAM J. Discrete Math. 7 (1), 48–56 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fortin, D., Rudolf, R.: Weak monge arrays in higher dimensions. Discrete Math. 189 (1–3), 105–115 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  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. Glover, F., Rego, C.: Ejection chain and filter-and-fan methods in combinatorial optimization. 4OR 4 (4), 263–296 (2006)

    MathSciNet  MATH  Google Scholar 

  20. Grover, L., Patel, A., Tulsi, T.: A new algorithm for fixed point quantum search (2005)

    Google Scholar 

  21. Grover, L.K.: A Fast Quantum Mechanical Algorithm for Database Search, ACM, New York (1996)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  24. Jaeger, F.: On four-weight spin models and their gauge transformations. J. Algebraic Combin. 11 (3), 241–268 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kac, V., Cheung, P.: Quantum Calculus. Universitext, Springer, New York (2002)

    Book  Google Scholar 

  26. Kauffman, L.H., Lomonaco, S.J.: Entanglement criteria – Quantum and topological. New J. Phys. 4, 73.1–73.18 (electronic) (2002)

    Article  MathSciNet  Google Scholar 

  27. Kauffman, L.H., Lomonaco, S.J.: Braiding operators are universal quantum gates (2004)

    Google Scholar 

  28. Kauffman, L.H., Lomonaco, S.J.: Quantum knots (2004)

    Google Scholar 

  29. Khalfin, L.A., Tsirelson, B.S.: Quantum/classical correspondence in the light of Bell’s inequalities. Found. Phys. 22 (7), 879–948 (1992)

    Article  MathSciNet  Google Scholar 

  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. Kitaev, A.Y.: Quantum measurements and the abelian stabilizer problem (1995)

    Google Scholar 

  32. Li, D., Huang, H., Li, X.: The fixed-point quantum search for different phase shifts (2006)

    Google Scholar 

  33. Marinescu, D., Marinescu, G.D.: Approaching Quantum Computing. Prentice Hall (2004)

    Google Scholar 

  34. Martí, R., Laguna, M., Glover, F.: Principles of scatter search. Eur. J. Oper. Res. 169 (2), 359–372 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. Monge, G.: déblai et remblai. Mémoires de l’Académie des sciences, Paris (1781)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  37. Sica, L.: Bell’s inequalities i: An explanation for their experimental violation (2001)

    Google Scholar 

  38. Sica, L.: Correlations for a new Bell’s inequality experiment. Found. Phys. Lett. 15 (5), 473–486 (2002)

    Article  MathSciNet  Google Scholar 

  39. Vasquez, M., Hao, J.-K.: A hybrid approach for the 0–1 multidimensional knapsack problem (2001)

    Google Scholar 

  40. Werner, R.F., Wolf, M.M.: Bell inequalities and entanglement (2001)

    Google Scholar 

  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. Younes, A., Rowe, J., Miller, J.: Quantum search algorithm with more reliable behav0iour using partial diffusion (2003)

    Google Scholar 

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Authors and Affiliations

  1. Inria, Domaine de Voluceau, Rocquencourt, B.P. 105, 78153, Le Chesnay Cedex, France

    Dominique Fortin

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  1. Dominique Fortin
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Correspondence to Dominique Fortin .

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Editors and Affiliations

  1. South Kensington Campus, London, SW7 2AZ, United Kingdom

    Altannar Chinchuluun

  2. Dept. Industrial & Systems, Engineering, University of Florida, Weil Hall 303, Gainesville, 32611-6595, Florida, USA

    Panos M. Pardalos

  3. The School of Economic Studies, Dept.of Mathematics & Computer Science, National University of Mongolia, Ulaanbaatar, 210646, Mongolia

    Rentsen Enkhbat

  4. Labo. PRISM, Université Versailles, av. des Etats-Unis 45, Versailles CX, 78035, France

    Ider Tseveendorj

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Fortin, D. (2010). Hadamard’s Matrices, Grothendieck’s Constant, and Root Two. In: Chinchuluun, ., Pardalos, P., Enkhbat, R., Tseveendorj, I. (eds) Optimization and Optimal Control. Springer Optimization and Its Applications(), vol 39. Springer, New York, NY. https://doi.org/10.1007/978-0-387-89496-6_20

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