Quantum bounded query complexity

We combine the classical notions and techniques for bounded query classes with those developed in quantum computing. We give strong evidence that quantum queries to an oracle in the class NP does indeed reduce the query, complexity of decision problems. Under traditional complexity assumptions, we obtain an exponential speedup between the quantum and the classical query complexity of function classes. For decision problems and function classes we obtain the following results: P/sub /spl par///sup NP[2k]//spl sube/EQP/sub /spl par///sup NP[k]/; P/sub /spl par///sup NP[2k+1-2]//spl sube/EQP/sup NP[k]/; FP/sub /spl par///sup NP[2k=1-2]//spl sube/FEQP/sup NP[2k]/; FP/sub /spl par///sup NP/spl sube/FEQP(NP[Olog n)]/. For sets A that are many-one complete for PSPACE or EXP we show that Fp/sup A//spl sube/FEQP/sup A[1]/. Sets A that are many-one complete for PP have the property that FP/sub /spl par///sup A//spl sube/FEQP/sup A[1]/. In general we prove that for any set A there is a set X such that FP/sup A//spl sube/FEQP/sup X[1]/, establishing that no set is superterse in the quantum setting.

[1]  R. Cleve,et al.  Quantum algorithms revisited , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[2]  Klaus W. Wagner,et al.  Bounded Query Classes , 1990, SIAM J. Comput..

[3]  M. Sipser,et al.  A Limit on the Speed of Quantum Computation for Insertion into an Ordered List , 1998 .

[4]  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.

[5]  John Gill,et al.  Terse, Superterse, and Verbose Sets , 1993, Inf. Comput..

[6]  M. Sipser,et al.  Invariant quantum algorithms for insertion into an ordered list , 1999, quant-ph/9901059.

[7]  R. Beigel.  Query-limited reducibilities , 1988 .

[8]  R. Beigel,et al.  Bounded Queries to SAT and the Boolean Hierarchy , 1991, Theor. Comput. Sci..

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

[10]  Vikraman Arvind,et al.  Quasi-Linear Truth-Table Reductions to p-Selective Sets , 1996, Theor. Comput. Sci..

[11]  Jim Kadin.  The Polynomial Time Hierarchy Collapses if the Boolean Hierarchy Collapses , 1988, SIAM J. Comput..

[12]  Wim van Dam.  Two Classical Queries versus One Quantum Query , 1998, ArXiv.

[13]  Andris Ambainis,et al.  A better lower bound for quantum algorithms searching an ordered list , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[14]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[15]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[16]  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.

[17]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[18]  Lance Fortnow,et al.  Two Queries , 1999, J. Comput. Syst. Sci..

[19]  Frank Stephan,et al.  Approximable sets , 1994, Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory.

[20]  R. Beigel.  NP-hard Sets are P-Superterse Unless R = NP , 1988 .

[21]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

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

[23]  Amihood Amir,et al.  Some connections between bounded query classes and nonuniform complexity , 1990, Proceedings Fifth Annual Structure in Complexity Theory Conference.

[24]  A. Berthiaume.  Quantum computation , 1998 .

[25]  José L. Balcázar,et al.  Structural Complexity II , 2012, EATCS.

[26]  Mitsunori Ogihara.  Polynomial-Time Membership Comparable Sets , 1995, SIAM J. Comput..

[27]  William I. Gasarch,et al.  On the Complexity of Finding the Chromatic Number of a Recursive Graph I: The Bounded Case , 1989, Ann. Pure Appl. Log..

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

[29]  Ronald de Wolf,et al.  Lower Bounds for Quantum Search and Derandomization , 1998, ArXiv.

[30]  Samuel R. Buss,et al.  On Truth-Table Reducibility to SAT , 1991, Inf. Comput..

[31]  Gilles Brassard,et al.  Strengths and Weaknesses of Quantum Computing , 1997, SIAM J. Comput..

[32]  Edith Hemaspaandra,et al.  A Downward Translation in the Polynomial Hierarchy , 1997, STACS.

[33]  Alan L. Selman,et al.  A Taxonomy of Complexity Classes of Functions , 1994, J. Comput. Syst. Sci..

[34]  Amihood Amir,et al.  Polynomial Terse Sets , 1988, Inf. Comput..