Exploring Topologies in Quantum Annealing: A Hardware-Aware Perspective

Quantum Annealing (QA) offers a promising framework for solving NP-hard optimization problems, but its effectiveness is constrained by the topology of the underlying quantum hardware. Solving an optimization problem $P$ via QA involves a hardware-aware circuit compilation which requires representing $P$ as a graph $G_P$ and embedding it into the hardware connectivity graph $G_Q$ that defines how qubits connect to each other in a QA-based quantum processing unit (QPU). Minor Embedding (ME) is a possible operational form of this hardware-aware compilation. ME heuristically builds a map that associates each node of $G_P$ -- the logical variables of $P$ -- to a chain of adjacent nodes in $G_Q$ by means of one of its minors, so that the arcs of $G_P$ are preserved as physical connections among qubits in $G_Q$. The static topology of hardwired qubits can clearly lead to inefficient compilations because $G_Q$ cannot be a clique, currently. We propose a methodology and a set of criteria to evaluate how the hardware topology $G_Q$ can negatively affect the embedded problem, thus making the quantum optimization more sensible to noise. We evaluate the result of ME across two QPU topologies: Zephyr graphs (used in current D-Wave systems) and Havel-Hakimi graphs, which allow controlled variation of the average node degree. This enables us to study how the ratio `number of nodes/number of incident arcs per node' affects ME success rates to map $G_P$ into a minor of $G_Q$. Our findings, obtained through ME executed on classical, i.e. non-quantum, architectures, suggest that Havel-Hakimi-based topologies, on average, require shorter qubit chains in the minor of $G_P$, exhibiting smoother scaling of the largest embeddable $G_P$ as the QPU size increases. These characteristics indicate their potential as alternative designs for QA-based QPUs.