- The paper demonstrates a practical heuristic that finds sparse graph minors for embedding optimization problems in quantum annealing systems.
- The algorithm employs dynamic priority-based updates in shortest path calculations to significantly reduce search complexity compared to exhaustive methods.
- The research offers insights for enhancing initialization strategies, improving minor-embedding performance for NP-hard optimization challenges in quantum computing.
An Overview of Heuristic Algorithms for Minor-Embedding in Sparse Graphs
The paper presents a heuristic algorithm aimed at efficiently finding graph minors, specifically when minor-embedding involves sparse graphs with input sizes stretching to hundreds of vertices. The focus is on practical applications, particularly in mapping optimization problems onto adiabatic quantum annealers, such as those developed by D-Wave Systems Inc.
The problem of determining whether a graph H is a minor of another graph G is pivotal, with numerous theoretical implications and practical applications. While Robertson-Seymour theory guarantees polynomial-time algorithms when H is fixed, the problem remains NP-hard when both graphs form part of the input. Therefore, heuristic methods have become essential, especially given the impracticality of known exact algorithms for large graphs due to their prohibitive constant factors and exponential time complexity.
The described algorithm operates under specific assumptions about the graphs: both G and H are sparse. It avoids exhaustive search, seeking instead to find minors with a probability rather than certainty, and does not verify minor-exclusion if it fails to find a minor. The authors justify its necessity due to its role in quantum computing; prominent applications involve translating quadratic boolean optimization problems onto D-Wave's adiabatic quantum annealing hardware, where the problem's graph of variable interactions must be a minor of the hardware's qubit interaction graph.
The algorithm's effectiveness relies significantly on the size relationship between H and G. It exploits scenarios where many distinct H-minors may reside in G. A heuristic is constructed by searching for optimal overlaps in representing vertices of H within G and is tested with various types of graph structures, including complete graphs and grid graphs, to establish reliability and performance. Notably, in computing shortest paths during iterations, the algorithm uses dynamic priority-based updates, achieving substantial efficiencies compared to exhaustive strategies.
While the algorithm is primarily heuristic-driven, experimental analysis reveals promising performance metrics, even achieving high success rates in embedding large graphs within the Chimera graph structure utilized by D-Wave.
The research implications are substantial, suggesting practical methods to enhance quantum computing's reach, given current technological constraints. A significant takeaway is that even failed attempts at minor-embedding can still produce G-decompositions, which may serve as useful approximations or starting points for solving specific classes of problems within adiabatic quantum frameworks.
Looking forward, potential improvements could target the algorithm's initialization strategies, potentially yielding more effective initial vertex-model placements. These advancements could further optimize quantum annealers’ utility, especially in scenarios where traditional computing paradigms struggle. Additionally, by extending this framework to consider different families of graphs and more complex real-world constraints, a broader class of problems can be addressed, contributing to expanding the computational capabilities of existing quantum systems.