  We obtain the strongest separation between quantum and classical query
complexity known to date -- specifically, we define a black-box problem that
requires exponentially many queries in the classical bounded-error case, but
can be solved exactly in the quantum case with a single query (and a polynomial
number of auxiliary operations). The problem is simple to define and the
quantum algorithm solving it is also simple when described in terms of certain
quantum Fourier transforms (QFTs) that have natural properties with respect to
the algebraic structures of finite fields. These QFTs may be of independent
interest, and we also investigate generalizations of them to noncommutative
finite rings.
