We show that almost all n-bit Boolean functions have bounded-error quantum query complexity at least n/2, up to lower-order terms. This improves over an earlier n/4 lower bound of Ambainis, and shows that van Dams oracle interrogation is essentially optimal for almost all functions. Our proof uses the fact that the acceptance probability of a T-query algorithm can be written as the sum of squares of degree-T polynomials.
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