Almost-Everywhere Superiority for Quantum Polynomial Time

Simon as extended by Brassard and Hoyer shows that there are tasks on which polynomial-time quantum machines are exponentially faster than each classical machine infinitely often. The present paper shows that there are tasks on which polynomial-time quantum machines are exponentially faster than each classical machine almost everywhere.

[1]  Eric Allender,et al.  Almost-Everywhere Complexity Hierarchies for Nondeterministic Time , 1993, Theor. Comput. Sci..

[2]  Gilles Brassard,et al.  Oracle Quantum Computing , 1994 .

[3]  Zvi Galil,et al.  Proceedings of the 30th IEEE symposium on Foundations of computer science , 1994, FOCS 1994.

[4]  Lane A. Hemaspaandra,et al.  Almost-Everywhere Superiority for Quantum Polynomial-Time Languages , 2002 .

[5]  Ran Raz,et al.  Exponential separation of quantum and classical communication complexity , 1999, STOC '99.

[6]  Andris Ambainis,et al.  Average-Case Quantum Query Complexity , 2000, STACS.

[7]  Shao Chin Sung,et al.  A Quantum Polynomial Time Algorithm in Worst Case for Simon's Problem , 1998, ISAAC.

[8]  Alan L. Selman,et al.  A Hierarchy Theorem for Almost Everywhere Complex Sets With Application to Polynomial Complexity Degrees , 1987, Symposium on Theoretical Aspects of Computer Science.

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

[10]  Gilles Brassard,et al.  Tight bounds on quantum searching , 1996, quant-ph/9605034.

[11]  Ronald de Wolf,et al.  Quantum lower bounds by polynomials , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[12]  Gilles Brassard,et al.  An exact quantum polynomial-time algorithm for Simon's problem , 1997, Proceedings of the Fifth Israeli Symposium on Theory of Computing and Systems.

[13]  Zijian Diao,et al.  An Exponentially Fast Quantum Search Algorithm , 2000 .

[14]  John Gill,et al.  Relative to a Random Oracle A, PA != NPA != co-NPA with Probability 1 , 1981, SIAM J. Comput..

[15]  Joel I. Seiferas,et al.  A Note on Almost-Everywhere-Complex Sets and Separating Deterministic-Time-Complexity Classes , 1991, Inf. Comput..

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

[17]  Edith Hemaspaandra,et al.  Almost-Everywhere Superiority for Quantum Computing , 1999, ArXiv.

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

[19]  Ronald de Wolf,et al.  Quantum lower bounds by polynomials , 2001, JACM.