Figure 2

Illustration of a counterexample where the heuristic of using the energy-optimal QAOA angles in RQAOA fails. Here, we show that for the weighted graph (9 vertices and 24 edges) depicted in (a), RQAOA makes a mistake even in its strongest regime, so at the very first iteration (i.e., \(n_{c} = 8\)). The two-correlation coefficients for each edge (at energy-optimal angles) are shown in the form of a horizontal bar plot in (b), where the edge \((0,2)\) has the maximal correlation coefficient. For the graph in (a), RQAOA with energy-optimal angles assigns a wrong edge-correlation (sign) to this edge which is precisely highlighted by a bold star in (c) and (d). Both (c) and (d) characterize the sets of good and bad QAOA angles where RQAOA makes a correct and a wrong choice, respectively. This example is counter-intuitive: as the edge \((0,2)\) has the highest weight in the graph, intuitively, the variables should be correlated (same sign) as to maximize the energy. However, this leads to a sub-optimal solution which RQAOA achieves with energy-optimal angles. Yet, for different settings of QAOA angles which do not maximize the overall energy, this edge will still have the largest magnitude of correlation, but in this case, anti-correlation, which leads to the true optimum (see sub-figure (c))