  We consider the use of arbitrary phases in quantum amplitude amplification
which is a generalization of quantum searching. We prove that the phase
condition in amplitude amplification is given by $\tan(\varphi/2) =
\tan(\phi/2)(1-2a)$, where $\phi$ and $\phi$ are the phases used and where $a$
is the success probability of the given algorithm. Thus the choice of phases
depends nontrivially and nonlinearly on the success probability. Utilizing this
condition, we give methods for constructing quantum algorithms that succeed
with certainty and for implementing arbitrary rotations. We also conclude that
phase errors of order up to $\frac{1}{\sqrt{a}}$ can be tolerated in amplitude
amplification.
