Systematic and Deterministic Graph-Minor Embedding for Cartesian Products of Graphs

TL;DR

This paper introduces a systematic and deterministic graph-minor embedding method tailored for Cartesian products of graphs, aiming to improve efficiency and scalability in quantum annealer problem embedding.

Contribution

The authors develop a new embedding technique exploiting graph structures, reducing heuristic reliance and enhancing scalability for quantum annealer architectures.

Findings

Embeddings are faster and more efficient than heuristic methods.

Produced embeddings scale well for larger quantum processors.

The method results in favorable qubit usage and chain length properties.

Abstract

The limited connectivity of current and next-generation quantum annealers motivates the need for efficient graph-minor embedding methods. These methods allow non-native problems to be adapted to the target annealer's architecture. The overhead of the widely used heuristic techniques is quickly proving to be a significant bottleneck for solving real-world applications. To alleviate this difficulty, we propose a systematic and deterministic embedding method, exploiting the structures of both the input graph of the specific problem and the quantum annealer. We focus on the specific case of the Cartesian product of two complete graphs, a regular structure that occurs in many problems. We divide the embedding problem by first embedding one of the factors of the Cartesian product in a repeatable pattern. The resulting simplified problem consists of the placement and connecting together of these copies to reach a valid solution. Aside from the obvious advantage of a systematic and deterministic approach with respect to speed and efficiency, the embeddings produced are easily scaled for larger processors and show desirable properties for the number of qubits used and the chain length distribution. To conclude, we briefly address the problem of circumventing inoperable qubits by presenting possible extensions of the method.