Sharp Quantum vs. Classical Query Complexity Separations

de Beaudrap, J. Niel; Cleve, Richard; Watrous, John

Sharp Quantum vs. Classical Query Complexity Separations

Authors

de Beaudrap, J. Niel

Cleve, Richard

Watrous, John

Description

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.
13 pages, change in title, improvements in presentation, and minor corrections. To appear in Algorithmica