Modelling noise in global Mølmer-Sørensen interactions to quantum approximate optimization

Trapped ions with multi-qubit Mølmer-Sørensen (MS) interactions 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 (after experimental noise characterization). We validate the model in comparison with experiments that implement sequences of MS interactions on two and three ¹⁷¹Yb⁺ ions. The the model shows good agreement after several MS interactions as quantified by the reduced chi-squared statistic χ²< 2. As an application, we examine 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±0.02% and 0.92±0.06%, respectively, with disagreement in the latter value attributable to secondary noise sources beyond those considered in our analysis. We further compare experimental and simulated performance across heatmaps generated with varying algorithmic and experimental parameters, achieving χ²< 2 at six ions and a larger χ² with three ions. Our model predicts that realistic experimental improvements to reduce measurement error and radial trap frequency variations would achieve 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.