NQP = co-C=P

Adleman, Demarrais, and Huang introduced the nondeterministic quantum polynomial-time complexity class NQP as an analogue of NP. It is known that, with restricted amplitudes, NQP is characterized in terms of the classical counting complexity class C=P. In this paper we prove that, with unrestricted amplitudes, NQP indeed coincides with the complement of C=P. As an immediate corollary, with unrestricted amplitudes BQP differs from NQP. key words: computational complexity, theory of computation

[1]  Lov K. Grover A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[2]  Umesh V. Vazirani,et al.  Quantum Complexity Theory , 1997, SIAM J. Comput..

[3]  Charles H. Bennett,et al.  Logical reversibility of computation , 1973 .

[4]  R. Feynman Quantum mechanical computers , 1986 .

[5]  Daniel R. Simon,et al.  On the power of quantum computation , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[6]  Stuart A. Kurtz,et al.  Gap-definable counting classes , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[7]  D. Deutsch,et al.  Rapid solution of problems by quantum computation , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[8]  D. Deutsch Quantum computational networks , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[9]  Leonard M. Adleman,et al.  Quantum Computability , 1997, SIAM J. Comput..

[10]  Jacobo Torán,et al.  Complexity classes defined by counting quantifiers , 1991, JACM.

[11]  Lance Fortnow,et al.  Complexity limitations on quantum computation , 1999, J. Comput. Syst. Sci..

[12]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[13]  Seinosuke Toda,et al.  PP is as Hard as the Polynomial-Time Hierarchy , 1991, SIAM J. Comput..

[14]  D. Deutsch Quantum theory, the Church–Turing principle and the universal quantum computer , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[15]  I PaulBenioff Quantum Mechanical Hamiltonian Models of Turing Machines , 1982 .

[16]  Stephen A. Fenner,et al.  Determining acceptance possibility for a quantum computation is hard for the polynomial hierarchy , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[17]  Stephen A. Fenner Quantum NP is Hard for PH , 1998 .

[18]  H. S. Allen The Quantum Theory , 1928, Nature.