Graph minor embedding can affect sampling degenerate ground states using quantum annealing

Abstract

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 Schr\"odinger 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.