  Many-qubit M{\o}lmer-S{\o}rensen (MS) interactions applied to trapped ions
offer unique capabilities for quantum information processing, with applications
including quantum simulation and the quantum approximate optimization algorithm
(QAOA). Here, we develop a physical model to describe many-qubit MS
interactions under four sources of experimental noise: vibrational mode
frequency fluctuations, laser power fluctuations, thermal initial vibrational
states, and state preparation and measurement errors. The model parameterizes
these errors from simple experimental measurements, without free parameters. We
validate the model in comparison with experiments that implement sequences of
MS interactions on two $^{171}$Yb$^+$ ions. The model shows reasonable
agreement after several MS interactions as quantified by the reduced
chi-squared statistic $\chi^2_\mathrm{red} \approx 2$. As an application we
examine MaxCut QAOA experiments on three and six ions. The experimental
performance is quantified by approximation ratios that are $91\%$ and $83\%$ of
the optimal theoretical values. Our model predicts $0.93^{+0.03}_{-0.02}$ and
$0.95^{+0.04}_{-0.03}$, respectively, with disagreement in the latter value
attributable to secondary noise sources beyond those considered in our
analysis. With realistic experimental improvements to reduce measurement error
and radial trap frequency variations the model achieves approximation ratios
that are 99$\%$ of the optimal. Incorporating these improvements into future
experiments is expected to reveal new aspects of noise for future modeling and
experimental improvements.
