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.
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.
Quantum annealing, as currently implemented in hardware, cannot fairly sample all ground states. Graph minor embedding, which maps a problem to the hardware graph of quantum annealers, affects sampling all states. In this study, we demonstrate the influence of graph minor embedding on fair sampling of degenerate ground states. For two embedded models that introduce auxiliary variables, numerical simulations of Schrödinger evolution revealed that fairness varies significantly depending on the embedding, and the chain strength is related to ground-state fairness. Using perturbation theory, we found that chain strength determines the energy landscape around ground states, with flatter landscapes having higher probabilities of being obtained.
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.
Recently, it was demonstrated both theoretically and experimentally on the D-Wave quantum annealer that transverse-field quantum annealing does not find all ground states with equal probability. In particular, it was proposed that more complex driver Hamiltonians beyond transverse fields might mitigate this shortcoming. Here, we investigate the mechanisms of (un)fair sampling in quantum annealing. While higher-order terms can improve the sampling for selected small problems, we present multiple counterexamples where driver Hamiltonians that go beyond transverse fields do not remove the sampling bias. Using perturbation theory we explain why this is the case. In addition, we present large-scale quantum Monte Carlo simulations for spin glasses with known degeneracy in two space dimensions and demonstrate that the fair-sampling performance of quadratic driver terms is comparable to standard transverse-field drivers. Our results suggest that quantum annealing machines are not well suited for sampling applications, unless postprocessing techniques to improve the sampling are applied.
Quantum annealing generally samples degenerate ground states with nonuniform probabilities. This unfair sampling depends on how the driver Hamiltonian couples these states near the end of annealing. For combinatorial optimization problems with a cardinality constraint, the penalty method and constrained quantum annealing (CQA) differ in these couplings. The transverse-field driver of the penalty method does not directly couple feasible ground states, whereas the fully connected XY driver of CQA couples pairs of them related by a single spin exchange. We investigate how this difference affects ground-state sampling, using closed-system simulations of random graph bipartitioning instances together with degenerate perturbation theory. For the instances studied, the conditional ground-state distribution of penalty-based quantum annealing approaches that of CQA at sufficiently large penalties and long annealing times. This convergence can be explained by the effective Hamiltonian of penalty-based quantum annealing, in which virtual single-spin flips to infeasible states generate a second-order interaction proportional to the CQA driver projected onto the feasible ground-state manifold, which is the leading term of the effective Hamiltonian of CQA. In the numerical simulations, the Jensen--Shannon divergence between the conditional ground-state distributions of penalty-based quantum annealing and CQA decreases approximately as the inverse square of the penalty-to-objective energy ratio, consistent with the perturbative analysis. This correspondence between the effective Hamiltonians extends to equal-weight fixed-magnetization constraints under the same perturbative conditions, providing a common mechanism for sampling bias in the two implementations.
Sampling all ground states of a Hamiltonian with equal probability is a desired feature of a sampling algorithm, but recent studies indicate that common variants of transverse field quantum annealing sample the ground state subspace unfairly. In this note, we present perturbation theory arguments suggesting that this deficiency can be corrected by employing reverse annealing-inspired paths. We confirm that this conclusion holds in simulations of previously studied small instances with degeneracy, as well as larger instances on quantum annealing hardware.
Reaching a degenerate ground-state manifold and sampling it uniformly are distinct properties of a quantum annealer. We quantify this distinction using an Ising ring with additional couplings between maximally separated sites. For antiferromagnetic benchmark couplings and N≥5, we derive the exact ground-state energy, degeneracy, and defect structure for each congruence class of N modulo four. This classification separates ground-state hit probability from conditional sampling bias. We embed instances with up to 162 logical spins on a D-Wave Advantage processor using length-two chains and compare its output with simulated annealing on the embedded and logical problems. For a frustrated N=83 instance, the processor visits 164 of the 166 exact ground states, yet the conditional distributions over states and defect sectors remain nonuniform. For N=160, the processor has a larger ground-state hit probability than logical simulated annealing (0.869 versus 0.519), but also a larger total-variation distance from uniformity (0.317 versus 0.136). A pause during the anneal raises the ground-state hit probability from 0.42 to 0.86 without restoring uniform sampling. These results identify a relaxation window and demonstrate that optimization success and sampling quality need not improve together, providing an exactly characterized benchmark for open-system quantum annealing.