Initial D-Wave experiments were performed on the D-Wave 2X at NASA Ames Research Center and the D-Wave 2X online system at D-Wave Systems Inc. Later runs were made on the DW2Q at NASA Ames, once that machine came online. The 2X and 2000Q systems use a C12 (12 cells × 12 cells × 4 qubits) and C16 (16 × 16 × 4) Chimera, respectively. For all-to-all graphs, D-Wave 2X supports N ≤ 48 and 2000Q supports N ≤ 64 (the number is slightly smaller because of broken qubits). All N ≤ 48 runs were consistent across the three machines and with extrapolation of data in (42) from runs performed on a different set of instances on the earlier-generation machine D-Wave Two. All data reported in this paper came from the DW2Q.

Embeddings were precomputed for all problems (heuristic embeddings for sparse MAX-CUT and native clique embeddings for SK, dense MAX-CUT, and variable-density MAX-CUT) so that runs in different conditions (e.g., annealing times and constraint couplings) would use the same embeddings. For each problem type, the optimal annealing parameter Jc is found as a function of problem size N by sweeping Jc (section S1). The optimal Jc was found to be independent of the annealing time. The standard annealing schedule was used in all experiments, but the annealing time was tuned. Most instances were run 104 to 105 times in total depending on the observed success rate (the especially hard N ≥ 50 MAX-CUT instances were run up to 4 × 106 times). Five to 10 different embeddings were used per instance, and the success probability was averaged. Spin-reversal transformations were used to avoid spurious effects. After an anneal, each logical qubit value was determined by taking the majority vote of all qubits in the chain.

In all figures, the shaded regions give the [25th, 75th] percentile range [interquartile range (IQR)] for the data. Figures 2B, 3A, and 5A show individual instances as dots and the solid line gives the median. Figures 2C, 3 (B and C), and 5B are too crowded to show D-Wave instances; the dots give medians and the smooth lines give analytic fits. For CIM data, medians and IQR are shown in Figs. 2C and 3B, while Fig. 3C only shows medians and IQR due to crowding.

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