Fair sampling of weighted ground state configurations in quantum annealing

Asher Lantz, Chris M. Herdman

Brian Hu Zhang, Gene Wagenbreth, Víctor Martín-Mayor +1 more

Shunta Ide, Shu Tanaka

Quantum annealing (QA) is a promising approach for solving combinatorial optimization problems; however, it is known to exhibit unfair sampling, in which degenerate ground states are not sampled with equal probability even for sufficiently long annealing times. Fair sampling is important in applications such as solution diversity assessment and combinatorial counting, yet the mechanism of unfair sampling remains poorly understood, particularly in constrained combinatorial optimization problems. In this work, we investigate unfair sampling of QA in weighted graph bipartitioning problems (GBP), a representative constrained optimization problem. We study how the penalty coefficient in the penalty method affects sampling fairness. Through numerical simulations and experiments on the D-Wave Advantage2 system, we show that increasing the penalty coefficient reduces unfair sampling in a representative single instance, and that this qualitative behavior is also observed on actual hardware. A scaling analysis over randomly generated instances with up to 12 spins reveals that, while this trend does not hold universally, more than 70% of instances exhibit monotonically increasing sampling fairness as the penalty coefficient increases, even at the largest system size studied. These results show that increasing the penalty coefficient improves sampling fairness, though at the cost of ground-state probability under practical annealing conditions, and call for a deeper theoretical understanding of unfair sampling in constrained optimization problems.

E. Arvidsson

Masayuki Yamamoto, Masayuki Ohzeki, Kazuyuki Tanaka

Conventional quantum annealing does not sample all ground states fairly. We demonstrate that fair sampling can be achieved by performing simulated annealing on a quantum annealer. We discuss the problems that occur when implementing this method and propose an alternative way to overcome them. We numerically verify the fair sampling ability of our method in a small-scale toy model.

Naoki Maruyama, Masayuki Ohzeki

In applications where multiple optimal solutions are needed, transverse-field quantum annealing (QA) is known to sample degenerate ground states in a strongly biased manner. Despite extensive empirical observations, it remains unclear which features of degenerate ground states are preferentially sampled and why by QA. Here we analyze the final states using degenerate perturbation theory to characterize the preference among them. In this analysis, the adjacency matrix of the graph composed by the ground states naturally emerges, and we can predict the eigenvector centralities (one of the node centralities) are related to the probabilities of these states. We verify this prediction on toy models where degeneracy is lifted at first and second order, and we show that second-order weights encode local barrier information, relating sampling fairness to the flatness of the local energy landscape. Finally, this perspective suggests two practical routes toward fair sampling -- promoting connectivity of the graph and reducing heterogeneity of centralities -- and we illustrate consistency with higher-order drivers and minor-embedding transformations.