This study addresses the minor-embedding problem, which involves mapping the variables of an Ising model onto a quantum annealing processor. The primary motivation stems from the observed performance disparity of quantum annealers when solving problems suited to the processor's architecture versus those with non-hardware-native topologies. Our research has two main objectives: i) to analyze the impact of embedding quality on the performance of D-Wave Systems quantum annealers, and ii) to evaluate the quality of the embeddings generated by Minorminer, the standard minor-embedding technique in the quantum annealing literature, provided by D-Wave. Regarding the first objective, our experiments reveal a clear correlation between the average chain length of embeddings and the relative errors of the solutions sampled. This underscores the critical influence of embedding quality on quantum annealing performance. For the second objective, we evaluate Minorminer's embedding capabilities, the quality and robustness of its embeddings, and its execution-time performance on Erdös-Rényi graphs. We also compare its performance with Clique Embedding, another algorithm developed by D-Wave, which is deterministic and designed to embed fully connected Ising models into quantum annealing processors, serving as a worst-case scenario. The results demonstrate that there is significant room for improvement for Minorminer, suggesting that more effective embedding strategies could lead to meaningful gains in quantum annealing performance.

翻译：本研究针对小嵌入问题展开探讨，该问题涉及将伊辛模型变量映射至量子退火处理器。研究的主要动机源于量子退火机在解决适配处理器架构的问题与处理非硬件原生拓扑问题时表现出的性能差异。我们的研究包含两个核心目标：i) 分析嵌入质量对D-Wave Systems量子退火机性能的影响；ii) 评估由D-Wave提供的Minorminer（量子退火领域标准小嵌入技术）所生成嵌入的质量。关于第一个目标，实验结果表明嵌入的平均链长与采样解的相对误差之间存在明确关联，这凸显了嵌入质量对量子退火性能的关键影响。针对第二个目标，我们评估了Minorminer的嵌入能力、其嵌入质量与鲁棒性，以及在Erdös-Rényi图上的运行时性能。同时，我们将其性能与Clique Embedding（D-Wave开发的另一种确定性算法）进行对比，该算法专为将全连接伊辛模型嵌入量子退火处理器而设计，可作为最坏情况下的参照基准。实验结果表明Minorminer存在显著的改进空间，这预示着更有效的嵌入策略可能为量子退火性能带来实质性提升。