Optimization Tests and ResultsΒΆ

This section describes a set of tests and results that examine the quality of results returned by the optimization postprocessor. The goal is to describe the difference between postprocessed solutions from those obtained solely via the QPU.

Postprocessing for optimization is evaluated by generating 10 random problems on a D-Wave QPU, each with \(J\) values drawn uniformly at random from \(\{1, -1\}\). Each problem is evaluated based on a set of scaling factors. Problems are scaled to exaggerate the negative effects of analog noise on solution quality, so the optimization postprocessor can demonstrate that it can recover a high-quality solution from the QPU solution. Specifically, with small scaling factors, it is difficult to faithfully represent problems in the QPU because of the exaggeration of analog noise. This noise causes the solver to return lower-quality solutions, and provides a nice mechanism to evaluate the optimization postprocessor.

For each problem and scaling factor, 1000 samples were drawn with postprocessing on and off. As seen in Figure 60 and Figure 61, postprocessing for optimization can improve solutions significantly. Furthermore, the worse the non-postprocessed solutions are, the more postprocessing helps.

Graph showing the mean residual energies (that is, the mean energies above the ground-state energy) returned with and without optimization postprocessing. Along its horizontal axis is the scaling factor from 0.001 to 0, marked in exponential multiples of 10. Along its vertical axis is the mean residual energy from 10 to 10,000, marked in exponential multiples of 10. Two lines are plotted in the graph showing the residual energy with and without postprocessing. It shows that that optimization postprocessing does no harm, and helps more when scaling factors are smaller and the samples not as good. Error bars in the plot indicate 95% confidence intervals over input Hamiltonians.

Fig. 60 Line plot of mean residual energies (mean energies above ground-state energy) returned by the D-Wave system with and without optimization postprocessing. Observe that optimization postprocessing does no harm, and helps more when scaling factors are smaller and the non-postprocessed samples not as good. Error bars indicate 95% confidence intervals over input Hamiltonians.

Graph showing a scatter plot of mean residual energies (that is, the mean energies above the ground-state energy) returned with and without optimization postprocessing for various scaling factors. Along its horizontal axis are the energies with postprocessing off, from 0 to 1000, marked in increments of 200. Along its vertical axis are the energies with postprocessing on, from 0 to 1000, marked in increments of 200. Points plotted show 4 different scaling factors: 0.1, 0.2, 0.4, and 1, grouped in small clumps. The graph is annotated with a straight line running diagonally from 0,0 to 1000,1000, showing the imaginary line where postprocessing on and postprocessing off would yield identical results. Points pointed close to this line show little or no benefit to postprocessing. Those plotted below the line show the positive effect of postprocessing. The data here shows that optimization postprocessing does no harm and helps more for smaller scaling factors. For instance, problems scaled by a factor of 0.4 and 1.0 are close to the line, while those scaled by a factor of 0.2 and especially 0.1 are well below it. No points are plotted above the line, showing that optimization postprocessing does no harm to the returned results.

Fig. 61 Scatter plot of mean residual energies (mean energies above ground-state energy) returned by the D-Wave system with and without optimization postprocessing, with each point representing an input Hamiltonian. Observe that optimization postprocessing does no harm, and helps more when scaling factors are smaller and the non-postprocessed samples not as good.