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_||^NP[2k]⊆EQP_||^NP[k] ; P_||^NP[2k+1-2]⊆EQP^NP[k]; FP_||^NP[2k=1-2]⊆FEQP^NP[2k]; FP _||^{NP⊆FEQP(NP[Olog n)]}. For sets A that are many-one complete for PSPACE or EXP we show that Fp^A⊆FEQP^A[1]. Sets A that are many-one complete for PP have the property that FP_||^A⊆FEQP^A[1]. In general we prove that for any set A there is a set X such that FP^A⊆FEQP^X[1], establishing that no set is superterse in the quantum setting