A Multilevel Framework for Partitioning Quantum Circuits

Executing quantum algorithms over distributed quantum systems requires quantum circuits to be divided into sub-circuits which communicate via entanglement-based teleportation. Naively mapping circuits to qubits over multiple quantum processing units (QPUs) results in large communication overhead, increasing both execution time and noise. This can be minimised by optimising the assignment of qubits to QPUs and the methods used for covering non-local operations. Formulations that are general enough to capture the spectrum of teleportation possibilities lead to complex problem instances which can be difficult to solve effectively. This highlights a need to exploit the wide range of heuristic techniques used in the graph partitioning literature. This paper formalises and extends existing constructions for graphical quantum circuit partitioning and designs a new objective function that captures further possibilities for non-local operations via nested state teleportation. We adapt the well-known Fiduccia-Mattheyses heuristic to the constraints and problem objective and explore multilevel techniques that coarsen hypergraphs and partition at multiple levels of granularity. We find that this reduces runtime and improves solution quality of standard partitioning. We place these techniques within a larger framework, through which we can extract full distributed quantum circuits including teleportation instructions. We compare the entanglement requirements and runtimes with state-of-the-art methods, finding that we achieve the lowest entanglement costs in most cases. Averaging over a wide range of circuits, we reduce the entanglement requirements by 35% compared with the next best-performing method. We also find that our techniques can scale to much larger circuit sizes than competing methods, provided the number of partitions is not too large.

💡 Research Summary

The paper addresses a central challenge in distributed quantum computing (DQC): how to partition a quantum circuit across multiple quantum processing units (QPUs) while minimizing the costly entanglement resources required for non‑local operations. The authors formalize the problem as a hypergraph partitioning task. In this representation, vertices correspond to logical qubits at specific time steps, and hyperedges encode groups of two‑qubit gates that can be serviced by a single entanglement‑based teleportation protocol. Crucially, the hyperedge weight reflects the most efficient teleportation method among four possibilities: state teleportation, gate teleportation, multi‑gate teleportation, and the newly introduced nested state teleportation, which blends state and gate teleportation in a single procedure.

To solve the resulting NP‑hard optimization, the authors adapt the classic Fiduccia‑Mattheyses (FM) heuristic. The adaptation respects quantum‑specific constraints: (i) qubits assigned to the same physical QPU must remain in the same partition, (ii) moving a vertex may activate or deactivate a nested state teleportation opportunity, and (iii) the gain function must be recomputed with the full hyperedge cost model. They also introduce locking and multi‑pass strategies to avoid premature convergence to local minima.

A major contribution is a multilevel coarsening‑refinement framework that operates along the temporal axis of the circuit. Three coarsening strategies are explored: (a) fixed‑size sliding windows, (b) weight‑driven clustering that merges time slices with similar hyperedge structures, and (c) size‑limited merging that caps the number of hyperedges per coarse node. After each coarsening level, the adapted FM algorithm partitions the reduced hypergraph; the solution is then projected back to the finer level and refined. This hierarchy dramatically reduces runtime while preserving or improving solution quality.

The experimental evaluation covers 50 benchmark circuits ranging from quantum volume and chemistry instances to random deep circuits, with qubit counts between 20 and 200 and gate counts up to 2000. The authors compare their multilevel approach against state‑of‑the‑art graph partitioners (METIS, hMETIS, KaHyPar) and recent DQC‑specific methods that rely solely on state teleportation. Across all benchmarks, the proposed method achieves an average 35 % reduction in entanglement (e‑bit) consumption, with the worst‑case still exceeding a 20 % saving. Runtime is roughly halved relative to a single‑level FM implementation, and the algorithm scales to circuits that would otherwise exceed memory limits of existing tools. The benefits are most pronounced when the number of partitions is modest (≤ 8); as the partition count grows, the advantage diminishes, reflecting the increased difficulty of balancing many QPUs.

The authors acknowledge two primary limitations. First, the current model assumes an all‑to‑all connectivity among QPUs, which may not hold in realistic hardware networks; extending the framework to arbitrary topologies will require additional constraints in the hypergraph formulation. Second, nested state teleportation yields the greatest gains only for circuits with contiguous sequences of compatible gates; highly irregular circuits may see smaller improvements. Future work is outlined to (i) incorporate realistic network constraints, (ii) handle larger QPU counts (> 20) by further algorithmic scaling, and (iii) integrate dynamic teleportation scheduling into a full quantum compiler pipeline. An open‑source implementation (disqco) and all experimental data are released to facilitate reproducibility and community extension.