Boosting Quantum Annealing Performance Using Evolution Strategies for Annealing Offsets Tuning

Abstract

In this paper we introduce a novel algorithm to iteratively tune annealing offsets for qubits in a D-Wave 2000Q quantum processing unit (QPU). Using a (1+1)-CMA-ES algorithm, we are able to improve the performance of the QPU by up to a factor of 12.4 in probability of obtaining ground states for small problems, and obtain previously inaccessible (i.e., better) solutions for larger problems. We also make efficient use of QPU samples as a resource, using 100 times less resources than existing tuning methods. The success of this approach demonstrates how quantum computing can benefit from classical algorithms, and opens the door to new hybrid methods of computing.

Appendices

A Evaluating the Fitness Function of $(1+1)$-CMA-ES using the QPU

Here we show an example of a single tuning run of $(1+1)$-CMA-ES for a 40 node graph, with the configuration of initial offsets set to all zeroes. As explained in Algorithm 1, we use the mean energy of 100 samples returned by the QPU as the evaluation for the fitness function of the CMA-ES. The sample size of 100 was determined empirically as being the minimum number of samples to determine the mean energy, and is consistent with previous results [4]. In Fig. 3 (left) we show the progression of the CMA-ES routine and the associated fitness function. The tuning shows a clear improvement in mean energy, as shown both in the fitness function and the cumulative minimum of the fitness function. Every time the objective function improves, the respective annealing offsets that were used in that sample set are recorded. The evolution of the annealing offsets for this 40 variable instance is shown in Fig. 3 (right). The final offsets after tuning were then used to test their performance.

Fig. 3.

Left: The fitness function evolution (mean energy of 100 samples) is shown as a function of the iteration number in CMA-ES. The red line represents the value of the fitness function at each iteration of CMA-ES, and the blue line is the cumulative minimum, representing the best solutions so far. Right: The evolution of the annealing offsets are shown as a function of the iteration number of CMA-ES (updated every time improvement is found by the CMA-ES). (Color figure online)

B Analysis of Tuned Annealing Offsets

Here we present the aggregated results of all the annealing offsets post tuning. Figures 4 (left and right) show the final offset values for all problem instances using the CMA-ES routine with initial offsets set to zero and uniform, respectively. We found that the final offset values were not correlated with chain length or success probability. However, we did see a systematic shift in the final offset values with respect to the degree of the node in the graph, and as a function of problem size. In both figures, we see divergent behavior in offsets for very small and very high degree nodes, with consistent stability in the mid-range. The main different between the two figures is the final value of the offsets in this middle region. In Fig. 4 (left), the average offset value rises from 0 at small problems, to roughly .02 for problems with 40 variables, then back down to 0 for the largest problems. There is also a slight increase in average offset value from degree 3 to degree 14, found consistently for all problem sizes. In contrast, Fig. 4 (right) shows that the final offset values were roughly .02 at all problem sizes, apart from the divergent behavior in the extrema of the degree axis. The difference between the two configurations could explain why initial offsets set to zero performed slightly better than the uniform initial offsets. Given the fixed resources of 10,000 samples for calibration per MIS instance, escaping from a local optimum (such as the null initial configuration) becomes increasingly difficult at larger problem sizes, thus degrading the uniform configuration’s performance. Other than the results shown here, we were not able to extract any meaningful information with respect to other interesting parameters.

Fig. 4.

Left: Final offset value as determined by (1+1)-CMA-ES with initial offsets set to zero, as a function of the degree of the logical node in the graph. Colors represent different problem sizes. Right: Same as in left, but for initial offsets set uniformly in their allowed range. (Color figure online)