  We introduce a new variant of Quantum Amplitude Estimation (QAE), called
Iterative QAE (IQAE), which does not rely on Quantum Phase Estimation (QPE) but
is only based on Grover's Algorithm, which reduces the required number of
qubits and gates. We provide a rigorous analysis of IQAE and prove that it
achieves a quadratic speedup up to a double-logarithmic factor compared to
classical Monte Carlo simulation. Furthermore, we show with an empirical study
that our algorithm outperforms other known QAE variants without QPE, some even
by orders of magnitude, i.e., our algorithm requires significantly fewer
samples to achieve the same estimation accuracy and confidence level.
