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To perform amplitude estimation with precision 1/M, it suffices to perform about M rounds of amplitude amplification on the state and measure it in the {good, bad} basis a constant number of times.
To perform amplitude estimation with precision 1/M, it suffices to perform about M rounds of amplitude amplification on the state and measure it in the {good, bad} basis M times. (Though this is uninteresting because the total number of calls to the state preparation algorithm is then M^2, identical to the complexity of Monte Carlo).
Which of the following variant of QFT-free amplitude estimation would you choose if you wanted an algorithm running practically fast and with a theoretical guarantee of correctness?