The paper introduces the concept of amplitude amplification, a technique that enhances the success probability of a quantum algorithm by boosting the amplitude of the desired outcome. This technique generalizes Grover's search algorithm, which uses a Fourier transform to find a single solution with a quadratic speedup over classical algorithms. The authors extend this idea to scenarios where the initial success probability \(a\) is not known, showing that a quadratic speedup can still be achieved with an expected number of applications proportional to \(1/\sqrt{a}\). They also present methods to de-randomize the algorithm when the success probability is known, achieving certainty in the worst case. Additionally, the paper discusses the application of amplitude amplification to classical heuristics, demonstrating how quantum computers can leverage these heuristics to solve problems more efficiently. Finally, the authors introduce amplitude estimation, a method to estimate the success probability \(a\) of a quantum algorithm, which is crucial for problems like approximate counting. The paper provides algorithms and theoretical guarantees for these techniques, showing their effectiveness in various settings.The paper introduces the concept of amplitude amplification, a technique that enhances the success probability of a quantum algorithm by boosting the amplitude of the desired outcome. This technique generalizes Grover's search algorithm, which uses a Fourier transform to find a single solution with a quadratic speedup over classical algorithms. The authors extend this idea to scenarios where the initial success probability \(a\) is not known, showing that a quadratic speedup can still be achieved with an expected number of applications proportional to \(1/\sqrt{a}\). They also present methods to de-randomize the algorithm when the success probability is known, achieving certainty in the worst case. Additionally, the paper discusses the application of amplitude amplification to classical heuristics, demonstrating how quantum computers can leverage these heuristics to solve problems more efficiently. Finally, the authors introduce amplitude estimation, a method to estimate the success probability \(a\) of a quantum algorithm, which is crucial for problems like approximate counting. The paper provides algorithms and theoretical guarantees for these techniques, showing their effectiveness in various settings.