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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: o P_||^NP[2k] is included in EQP_||^NP[k] o P_||^NP[2^(k+1)-2] is included in EQP^NP[k] o FP_||^NP[2^(k+1)-2] is included in FEQP^NP[2k] o FP_||^NP is included in FEQP^NP[O(log n)] For sets A that are many-one complete for PSPACE or EXP we show that FP^A is included in FEQP^A[1]. Sets A that are many-one complete for PP have the property that FP_||^A is included in FEQP^A[1]. In general we prove that for any set A there is a set X such that FP^A is included in FEQP^X[1], establishing that no set is superterse in the quantum setting.
We study the quantum query complexity of finding a certificate for a d-regular, k-level balanced NAND formula. Up to logarithmic factors, we show that the query complexity is Theta(d^{(k+1)/2}) for 0-certificates, and Theta(d^{k/2}) for 1-certificate
We present three new quantum algorithms in the quantum query model for textsc{graph-collision} problem: begin{itemize} item an algorithm based on tree decomposition that uses $Oleft(sqrt{n}t^{sfrac{1}{6}}right)$ queries where $t$ is the treewidth of
We study quantum algorithms that learn properties of a matrix using queries that return its action on an input vector. We show that for various problems, including computing the trace, determinant, or rank of a matrix or solving a linear system that
I offer a case that quantum query complexity still has loads of enticing and fundamental open problems -- from relativized QMA versus QCMA and BQP versus IP, to time/space tradeoffs for collision and element distinctness, to polynomial degree versus
We study the query complexity of computing a function f:{0,1}^n-->R_+ in expectation. This requires the algorithm on input x to output a nonnegative random variable whose expectation equals f(x), using as few queries to the input x as possible. We ex