Addressing the Minor-Embedding Problem in Quantum Annealing and Evaluating State-of-the-Art Algorithm Performance

TL;DR

The paper investigates the minor-embedding problem for Ising models on quantum annealers, quantifying how embedding quality, measured by Average Chain Length ($ACL$), affects end-to-end QA performance. It evaluates Minorminer, the standard embedding algorithm, against Clique Embedding (CE) on Erdős–Rényi graphs embedded into D-Wave Pegasus topology, revealing that embedding quality dominates solution quality and that CE often outperforms Minorminer in non-sparse, larger instances. The findings show that higher $ACL$ leads to exponentially fewer valid configurations and higher median errors, and that Minorminer exhibits substantial variability and slower runtimes compared to CE, especially as problem size grows. The study suggests practical strategies for selecting embedding approaches based on problem density/degree and advocates future work combining embedding methods and exploring Layout Embedding to further improve end-to-end quantum annealing performance.

Abstract

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.

Figures (15)

• Figure 1: Flux diagram of the quantum annealing problem solving process in D-Wave Systems annealers. The image is an adapted version of the scheme originally presented by Goodrich et. al. goodrich2018optimizing.
• Figure 2: Section of a Pegasus graph. The size of the actual Pegasus employed in the processors follows the same shape but is 16 times larger in area. The graph has been generated with the D-Wave's python library dwave-networkx, which is an extension of Networkx hagberg2008exploring.
• Figure 3: The colored dots represent the median relative errors over the Number of Qubits of the embedded Ising Models for each solution sample. These samples are derived from solving randomly generated Ising models as described in Table \ref{['table:setup']}.
• Figure 4: Top row, first five panels: Median and minimum relative errors of the post‑processed solutions obtained from 100 samples, with fitted linear regressions shown as dashed lines. Each point corresponds to a complete sample generated using a distinct embedding, for five randomly selected problem instances (see Table \ref{['table:setup']}). Bottom row, first five panels: Median and minimum broken‑chain fractions plotted as a function of the ACL of the embeddings, with fitted linear regressions. Last column: Slopes of the linear regression models fitted for each chain‑strength prefactor. For every problem instance and embedding, five embedded Ising models were produced by varying the chain strength, controlled by the prefactor in the UTC function.
• Figure 5: Probability of success of Minorminer finding a valid embedding for ER graphs into D-Wave Advantage_system4.1's broken Pegasus, as a function of density and size of problem graph. The red line represents the embeddable limit for CE. HN corresponds to Hardware Native instances, taken as the lowest significant density value.

Theorems & Definitions (4)

• definition 1
• proposition 1
• proof
• proposition 2