  The quantum Fourier transform (QFT) is sometimes said to be the source of
various exponential quantum speed-ups. In this paper we introduce a class of
quantum circuits which cannot outperform classical computers even though the
QFT constitutes an essential component. More precisely, we consider normalizer
circuits. A normalizer circuit over a finite Abelian group is any quantum
circuit comprising the QFT over the group, gates which compute automorphisms
and gates which realize quadratic functions on the group. We prove that all
normalizer circuits have polynomial-time classical simulations. The proof uses
algorithms for linear diophantine equation solving and the monomial matrix
formalism introduced in our earlier work. We subsequently discuss several
aspects of normalizer circuits. First we show that our result generalizes the
Gottesman-Knill theorem. Furthermore we highlight connections to Shor's
factoring algorithm and to the Abelian hidden subgroup problem in general.
Finally we prove that quantum factoring cannot be realized as a normalizer
circuit owing to its modular exponentiation subroutine.
