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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. 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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. 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