Quantum Lower Bound for Approximate Counting Via Laurent Polynomials
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We consider the following problem: estimate the size of a nonempty set S⊆[ N] , given both quantum queries to a membership oracle for S, and a device that generates equal superpositions S〉 over S elements. We show that, if S is neither too large nor too small, then approximate counting with these resources is still quantumly hard. More precisely, any quantum algorithm needs either Ω( √(N/ S)) queries or else Ω( { S ^1/4,√(N/ S)}) copies of S〉 . This means that, in the black-box setting, quantum sampling does not imply approximate counting. The proof uses a novel generalization of the polynomial method of Beals et al. to Laurent polynomials, which can have negative exponents.
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