Unfair Sampling of Quantum Annealing in Weighted Graph Bipartitioning Problems

TL;DR

This study investigates how increasing the penalty coefficient in quantum annealing improves sampling fairness in weighted graph bipartitioning problems, with implications for solution diversity and understanding of unfair sampling.

Contribution

It provides empirical evidence that higher penalty coefficients can enhance sampling fairness in constrained quantum annealing, supported by simulations and hardware experiments.

Findings

Increasing penalty coefficient reduces unfair sampling in a representative instance.

On the D-Wave system, higher penalties improve sampling fairness.

Over 70% of random instances show increased fairness with larger penalties.

Abstract

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.