  In this work, we provide the first QFT-free algorithm for Quantum Amplitude
Estimation (QAE) that is asymptotically optimal while maintaining the leading
numerical performance. QAE algorithms appear as a subroutine in many
applications for quantum computers. The optimal query complexity achievable by
a quantum algorithm for QAE is $O\left(\frac{1}{\epsilon}\log
\frac{1}{\alpha}\right)$ queries, providing a speedup of a factor of
$1/\epsilon$ over any other classical algorithm for the same problem. The
original algorithm for QAE utilizes the quantum Fourier transform (QFT) which
is expected to be a challenge for near-term quantum hardware. To solve this
problem, there has been interest in designing a QAE algorithm that avoids using
QFT. Recently, the iterative QAE algorithm (IQAE) was introduced by Grinko et
al. with a near-optimal $O\left(\frac{1}{\epsilon}\log \left(\frac{1}{\alpha}
\log \frac{1}{\epsilon}\right)\right)$ query complexity and small constant
factors. In this work, we combine ideas from the preceding line of work to
introduce a QFT-free QAE algorithm that achieves a query complexity upper bound
of $\frac{62}{\epsilon}\ln\frac{6}{\alpha}$ which matches the optimal
$O\left(\frac{1}{\epsilon}\log \frac{1}{\alpha}\right)$ query complexity. We
supplement our analysis with numerical experiments comparing our performance
with IQAE where we find that our modifications retain the high performance, and
in some cases even improve the numerical results.
