THE QUANTUM INTELLIGENCER

Evaluating Quantum Hardware Topologies: Havel-Hakimi Graphs Offer Improved Embedding Efficiency Over Zephyr in Quantum Annealing In Plain English: Quantum computers that solve optimization problems have to map those problems onto their physical hardware, but the way their parts are connected can make this process inefficient. This study looks at different ways to arrange the connections between quantum bits (qubits) and finds that one design—called Havel-Hakimi—lets problems fit better and use fewer extra connections than the current design used by D-Wave systems. This matters because better arrangements mean the computer is less likely to make errors and can solve bigger, more complex problems in the real world, bringing us closer to practical quantum computing. Summary: This paper presents a hardware-aware analysis of quantum annealing (QA) systems, focusing on how the physical connectivity of qubits—defined by the hardware graph $G_Q$—impacts the ability to embed problem graphs $G_P$ via minor embedding (ME). ME is a crucial compilation step that maps logical variables to chains of physical qubits while preserving adjacency. The fixed, non-clique nature of current QA hardware limits embedding efficiency, often requiring long qubit chains that increase susceptibility to noise and reduce success rates. The authors evaluate two topologies: Zephyr, used in D-Wave’s current quantum processing units (QPUs), and Havel-Hakimi graphs, which allow precise control over average node degree. Using classical simulations of ME, they analyze how the ratio of nodes to incident arcs (i.e., connectivity density) affects the success and quality of embeddings. Results indicate that Havel-Hakimi topologies, on average, require shorter qubit chains and support smoother scaling of the largest embeddable problem size as the QPU grows. This suggests improved scalability and hardware efficiency. The study proposes a methodology for assessing hardware topologies based on their embedding performance and argues that tunable, high-connectivity graphs like Havel-Hakimi could serve as superior alternatives to current fixed architectures. The findings are particularly relevant for the design of next-generation QA hardware, where minimizing chain length and maximizing embeddability are critical for noise resilience and computational performance. Key Points: - Quantum annealing requires mapping problem graphs onto fixed hardware connectivity via minor embedding. - Minor embedding often creates long qubit chains, which degrade performance due to noise and decoherence. - The hardware’s topology (how qubits are connected) significantly impacts embedding efficiency. - Zephyr is the current topology used in D-Wave systems - Havel-Hakimi allows controlled variation in node connectivity. - Havel-Hakimi topologies require shorter qubit chains and scale better with increasing QPU size. - Higher average node degree improves the success rate of embedding larger problem graphs. - The study proposes evaluation criteria for hardware topologies based on embedding performance. - Findings suggest Havel-Hakimi could be a promising alternative for future quantum annealing hardware design. Notable Quotes: - "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 $G_Q$..." - "Our findings... 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." Data Points: - The study compares two quantum hardware topologies: Zephyr (used in D-Wave systems) and Havel-Hakimi graphs. - Evaluation is based on minor embedding success rates and qubit chain lengths. - Findings are derived from simulations on classical (non-quantum) architectures. - The ratio of 'number of nodes / number of incident arcs per node' is used as a key metric in topology evaluation. - Havel-Hakimi topologies show smoother scaling of the largest embeddable problem size with increasing QPU scale. Controversial Claims: - The suggestion that Havel-Hakimi graphs could outperform Zephyr topologies assumes that such synthetic, highly tunable graphs can be physically realized in actual quantum hardware, which may not be feasible with current fabrication techniques. - The claim that shorter chains lead directly to better noise resilience, while plausible, is inferred rather than experimentally validated in this study, as all tests were run on classical simulators. - The paper implies a general superiority of higher average node degree, but does not address potential trade-offs such as increased crosstalk or control complexity in denser topologies. Technical Terms: - Quantum Annealing (QA), Optimization Problems, NP-hard, Hardware Topology, Quantum Processing Unit (QPU), Minor Embedding (ME), Graph Embedding, Logical Variables, Physical Qubits, Chain Length, Connectivity Graph ($G_Q$), Problem Graph ($G_P$), Zephyr Topology, Havel-Hakimi Graphs, Average Node Degree, Node Connectivity, Hardware-Aware Compilation, Adiabatic Quantum Computing, Noise Susceptibility, Embeddability, Graph Minors —Ada H. Pemberley Dispatch from The Prepared E0