Template-based Minor Embedding for Adiabatic Quantum Optimization

TL;DR

This paper introduces a new integer linear programming method for embedding problem graphs into quantum annealing hardware, improving scalability and success rate over existing OCT-based approaches.

Contribution

It presents a novel template-based embedding approach using ILP, with exact formulations and superior performance compared to prior OCT-based methods.

Findings

Embeds more graphs than OCT-based methods.

Scales better with hardware size.

Runs significantly faster on test sets.

Abstract

Quantum Annealing (QA) can be used to quickly obtain near-optimal solutions for Quadratic Unconstrained Binary Optimization (QUBO) problems. In QA hardware, each decision variable of a QUBO should be mapped to one or more adjacent qubits in such a way that pairs of variables defining a quadratic term in the objective function are mapped to some pair of adjacent qubits. However, qubits have limited connectivity in existing QA hardware. This has spurred work on preprocessing algorithms for embedding the graph representing problem variables with quadratic terms into the hardware graph representing qubits adjacencies, such as the Chimera graph in hardware produced by D-Wave Systems. In this paper, we use integer linear programming to search for an embedding of the problem graph into certain classes of minors of the Chimera graph, which we call template embeddings. One of these classes corresponds to complete bipartite graphs, for which we show the limitation of the existing approach based on minimum Odd Cycle Transversals (OCTs). One of the formulations presented is exact, and thus can be used to certify the absence of a minor embedding using that template. On an extensive test set consisting of random graphs from five different classes of varying size and sparsity, we can embed more graphs than a state-of-the-art OCT-based approach, our approach scales better with the hardware size, and the runtime is generally orders of magnitude smaller.