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 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.

Figures (4)

• Figure 1: (a) Original model, (b) Embedded model with the central chain (minimal embedding), and (c) Embedded model with the edge chains (redundant embedding). The solid and dashed lines represent ferromagnetic interactions $J_{ij}=+1$ and anti-ferromagnetic interactions $J_{ij}=-1$. The double line represents a chain $J_F$.
• Figure 2: Annealing time dependence on the ratio of probabilities in the original and minimally embedded model. The solid, dotted, and dashed lines represent $J_F \in \{0.5, 1.0, 1.5\}$ in the embedded model, and the dash-dot line is in the original model.
• Figure 3: Chain strength dependence on the probability of obtaining ground states in $\tau=1000$ by the perturbation theory (PT) and the Schrödinger equation (SE). Note that the lines (PT) and points (SE) for $\left|2\right\rangle$ and $\left|3\right\rangle$ overlap.
• Figure 4: Relationship between the probability of obtaining ground states and the relative flatness of their energy landscapes. Note that the lines $\left|2\right\rangle$ and $\left|3\right\rangle$ overlap.