  Amplitude estimation is a fundamental quantum algorithmic primitive that
enables quantum computers to achieve quadratic speedups for a large class of
statistical estimation problems, including Monte Carlo methods. The main
drawback from the perspective of near term hardware implementations is that the
amplitude estimation algorithm requires very deep quantum circuits. Recent
works have succeeded in somewhat reducing the necessary resources for such
algorithms, by trading off some of the speedup for lower depth circuits, but
high quality qubits are still needed for demonstrating such algorithms.
  Here, we report the results of an experimental demonstration of amplitude
estimation on a state-of-the-art trapped ion quantum computer. The amplitude
estimation algorithms were used to estimate the inner product of randomly
chosen four-dimensional unit vectors, and were based on the maximum likelihood
estimation (MLE) and the Chinese remainder theorem (CRT) techniques.
Significant improvements in accuracy were observed for the MLE based approach
when deeper quantum circuits were taken into account, including circuits with
more than ninety two-qubit gates and depth sixty, achieving a mean additive
estimation error on the order of $10^{-2}$. The CRT based approach was found to
provide accurate estimates for many of the data points but was less robust
against noise on average. Last, we analyze two more amplitude estimation
algorithms that take into account the specifics of the hardware noise to
further improve the results.
