Determination of Chain Strength induced by Embedding in D-Wave Quantum Annealer

The D-wave quantum annealer requires embedding with ferromagnetic (FM) chains connected by several qubits, because it cannot capture exact long-range coupling between qubits, and retains the specific architecture that depends on the hardware type. Therefore, determination of the chain strength $J_c$ required to sustain FM order of qubits in the chains is crucial for the accuracy of quantum annealing. In this study, we devise combinatorial optimization problems with ordered and disordered qubits for various embeddings to predict appropriate $J_c$ values. We analyze the energy interval $\Delta_s$ and $\Delta_c$ between ground and first excited states in the combinatorial optimization problems without and with chains respectively, using the exact approach. We also measure the probability $p$ that the exact ground energy per site $E_g$ is observed in many simulated annealing shots. We demonstrate that the determination of $J_c$ is increasingly sensitive with growing disorder of qubits in the combinatorial optimization problems. In addition, the values of appropriate $J_c$, where the values of $p$ are at a maximum, increase with decreasing $\Delta_s$. Finally, the appropriate value of $J_c$ is shown to be observed at approximately $\Delta_c/\Delta_s=0.25$ and $2.1 E_g$ in the ordered and disordered qubits, respectively.