  Amplitude Estimation (AE) is a critical subroutine in many quantum
algorithms, allowing for a quadratic speedup in various applications like those
involving estimating statistics of various functions as in financial Monte
Carlo simulations. Much work has gone into devising methods to efficiently
estimate the amplitude of a quantum state without expensive operations like the
Quantum Fourier Transform (QFT), which is especially prohibitive given the
constraints of current NISQ devices. Newer methods have reduced the number of
operations required on a quantum computer and are the most promising near-term
implementations of the AE subroutine. While it remains to be seen the exact
circuit requirements for a quantum advantage in applications relying on AE, it
is necessary to continue to benchmark the algorithm's performance on current
quantum computers and the circuit costs associated with such subroutines.
  Given these considerations, we expand on results from previous experiments in
using Maximum Likelihood Estimation (MLE) to approximate the amplitude of a
quantum state and provide empirical upper bounds on the current feasible
circuit depths for AE on a superconducting quantum computer. Our results show
that MLE using optimally-compiled circuits can currently outperform naive
sampling for up to 3 Grover Iterations with a circuit depth of 131, which is
higher than reported in other experimental results. This functional benchmark
is one of many that will be continually monitored against current quantum
hardware to measure the necessary progress towards quantum advantage.
