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Quantum Computation Relative to Oracles

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Unconventional Models of Computation, UMC’2K

Part of the book series: Discrete Mathematics and Theoretical Computer Science ((DISCMATH))

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Abstract

The study of the power and limitations of quantum computation remains a major challenge in complexity theory. Key questions revolve around the quantum complexity classes EQP, BQP, NQP and their derivatives. This paper presents new relativized worlds in which (i) co-RP ⊈ NQE, (ii) P = BQP and UP = EXP, (iii) P = EQP and RP = EXP, and (iv) EQP ⊈ ∑ P2 ⋃ ∏ P2 . We also show a partial answer to the question of whether Almost-BQP = BQP.

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References

  1. L. Adleman, J. DeMarrais, and M. Huang, Quantum computability, SIAM J. Comput., 26 (1997), 1524–1540.

    Article  MathSciNet  MATH  Google Scholar 

  2. T. Baker, J. Gill, and R. Solovay, Relativizations ofthe P=?NP question, SIAM J. Comput., 4 (1975), 431–442.

    Google Scholar 

  3. R. Beals, H. Buhrman, R. Cleve, M. Mosca, and R. de Wolf, Quantum lower bounds by polynomials, in Proc. 39th Symposium on Foundations of Computer Science, pp.352–361,1998.

    Google Scholar 

  4. R. Beigel, Relativized counting classes: relations among threshold, parity, and mods, J. Comput. System Sci. 42 (1991), 76–96.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. Beigel, H. Buhrman, and L. Fortnow, NP might not be as easy as detecting unique solutions, in Proc. 30th IEEE Symposium on Theory of Computing, pp.203–208,1998.

    Google Scholar 

  6. P. Benioff, The computer as a physical system: a microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines, J. Statistical Physics, 22 (1980), 563–591.

    Article  MathSciNet  Google Scholar 

  7. A. Berthiaume and G. Brassard, Oracle quantum computing, Journal of Modern Optics, 41 (1994), 2521–2535.

    Article  MathSciNet  MATH  Google Scholar 

  8. E. Bernstein and U. Vazirani, Quantum complexity theory, SIAM J. Comput., 26 (1997),1411–1473.

    Article  MathSciNet  MATH  Google Scholar 

  9. C.H. Bennett and J. Gill, Relative to a random oracle A, PA ≠ NPA ≠cpNPA with probability 1, SIAM J. Comput., 10 (1981),96–113.

    Article  MathSciNet  MATH  Google Scholar 

  10. C.H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani, Strengths and weaknesses of quantum computing, SIAM J. Comput., 26 (1997),1510–1523.

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Brassard and P. Høyer, An exact quantum polynomial-time algorithm for Simon’s problem, in Proc. 5th Israeli Symposium on Theory of Computing and Systems, pp.12–23, 1997.

    Google Scholar 

  12. H. Buhrman and L. Fortnow, One-sided versus two-sided error in probabilistic computation, in Proc. 16th Symposium on TheoreticalAspects of Computer Science, Lecture Notes in Computer Science, Vol.1563, pp.100–109, 1999.

    MathSciNet  Google Scholar 

  13. D. Deutsch and R. Jozsa, Rapid solution of problems by quantum computation, Proc. Roy. Soc. London, Ser.A 439 (1992),553–558.

    Article  MathSciNet  MATH  Google Scholar 

  14. R. Feynman, Simulating Physics with computers, in Int. J. Theoretical Physics, 21 (1982),467–488.

    Article  MathSciNet  Google Scholar 

  15. S. Fenner, F. Green, S. Homer, and R. Pruim, Determining acceptance probability for a quantum computation is hard for PH, in Proc. 6th Italian Conference on Theoretical Computer Science, World-Scientific, Singapore, pp.241–252, 1998.

    Google Scholar 

  16. L. Fortnow and J. Rogers, Complexity limitations on quantum computation, in Proc. 13th Conference on Computational Complexity, pp.202–209, 1998.

    Google Scholar 

  17. M. Furst, J. Saxe, and M. Sipser, Parity, circuits and the polynomial-time hierarchy, Math. Syst. Theory, 17 (1984), 13–27.

    Article  MathSciNet  MATH  Google Scholar 

  18. L. Fortnow and T. Yamakami, Generic separations, J. Comput. System Sci., 52 (1996),191–197.

    Article  MathSciNet  MATH  Google Scholar 

  19. F. Green, Lower bounds for depth-three circuits with equals and mod-gates, in Proc. 12th Annual Symposium on Theoretical Aspect of Computer Science, Lecture Notes in Computer Science, Vol.900, pp.71–82, 1995.

    Article  Google Scholar 

  20. J.T. Håstad, Computational Limitations for Small-Depth Circuits, The MIT Press, 1987.

    Google Scholar 

  21. E. Hemaspaandra, L.A. Hemaspaandra, and M. Zimand, Almost-everywhere superiority for quantum polynomial time. Technical Report TR-CS-99-720, University of Rochester. See also ph-quantl9910033.

    Google Scholar 

  22. K. Ko and H. Friedman, Computational complexity of real functions, Theor. Comput. Sci., 20 (1982), 323–352.

    Article  MathSciNet  MATH  Google Scholar 

  23. K. Ko, Some observations on the probabilistic algorithms and NP-hard problems, Inform. Process. Lett., 14 (1982), 39–43.

    Article  MathSciNet  MATH  Google Scholar 

  24. K. Ko, Separating and collapsing results on the relativized probabilistic polynomial-time hierarchy, J. Assoc. Comput. Mach., 37 (1990), 415–438.

    Article  MathSciNet  MATH  Google Scholar 

  25. S.A. Kurtz, A note on randomized polynomial time, SIAM J. Comput., 16 (1987), 852–853.

    Article  MathSciNet  MATH  Google Scholar 

  26. H. Nishimura and M. Ozawa, Computational complexity of uniform quantum circuit families and quantum Turing machines, manuscript, 1999. See also LANL quant-ph/9906095.

    Google Scholar 

  27. N. Nisan and M. Szegedy, On the degree of Boolean functions as real polynomials, Computational Complexity, 4 (1994),301–313.

    Article  MathSciNet  MATH  Google Scholar 

  28. N. Nisan and A. Wigderson, Hardness and randomness, J. Comput. System Sci., 49 (1994), 149–167.

    Article  MathSciNet  MATH  Google Scholar 

  29. C.H. Papadimitriou, Computational Complexity, Addison-Wesley, 1994.

    Google Scholar 

  30. D. Simon, On the power of quantum computation, SIAM J. Comput., 26 (1997), 1340–1349.

    Google Scholar 

  31. P.W. Shor, Polynomial-time algorithms for integer factorization and discrete logarithms on a quantum computer, SIAM J. Comput., 26 (1997), 1484–1509.

    Article  MathSciNet  MATH  Google Scholar 

  32. T. Yamakami, A foundation of programming a multi-tape quantum Turing machine, in Proc. 24th International Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science, Vol.1672, pp.430–441, 1999. See also LANL quant-ph/990684.

    Article  Google Scholar 

  33. T. Yamakami, Analysis of quantum functions, in Proc. 19th International Conference on Foundations of Software Technology and Theoretical Computer Science, Lecture Notes in Computer Science, Vol.1738, pp.407–419, 1999. See also LANL quant-ph/9909012.

    Article  MathSciNet  Google Scholar 

  34. T. Yamakami and A.C. Yao, NQPc = co-C=P. Inform. Process. Lett., 71 (1999),63–69.

    Google Scholar 

Additional References

  1. L. Fortnow, Personal communication, October 2000.

    Google Scholar 

  2. F. Green and R. Pruim, Relativized separation of EQP from PNP, manuscript, 2000.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Clarkson University, Potsdam, New York, USA

    Christino Tamon

  2. School of Information Technology and Engineering, University of Ottawa, Ottawa, Ontario, Canada

    Tomoyuki Yamakami

Authors
  1. Christino Tamon
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  2. Tomoyuki Yamakami
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Editor information

Editors and Affiliations

  1. Solvay Institutes, Free University of Brussels, Belgium

    I. Antoniou

  2. Department of Computer Science, University of Auckland, Auckland, New Zealand

    C. S. Calude  & M. J. Dinneen  & 

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© 2001 Springer-Verlag London

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Tamon, C., Yamakami, T. (2001). Quantum Computation Relative to Oracles. In: Antoniou, I., Calude, C.S., Dinneen, M.J. (eds) Unconventional Models of Computation, UMC’2K. Discrete Mathematics and Theoretical Computer Science. Springer, London. https://doi.org/10.1007/978-1-4471-0313-4_20

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  • DOI: https://doi.org/10.1007/978-1-4471-0313-4_20

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-85233-415-4

  • Online ISBN: 978-1-4471-0313-4

  • eBook Packages: Springer Book Archive

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