A Multilevel Framework for Partitioning Quantum Circuits

Abstract: 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 can achieve the lowest entanglement costs in most cases, while always being close to the best performing method. We achieve an average improvement of 33% over the next best performing method across a wide range of circuits. We also find that our techniques can scale to much larger circuit sizes than state-of-the-art methods, provided the number of partitions is not too large.

• The paper introduces a hypergraph-based multilevel framework that reduces communication overhead in distributed quantum circuits by up to 35%.
• It adapts the Fiduccia-Mattheyses algorithm for temporal-domain hypergraphs, using constant-time gain updates for efficient local optimization.
• The framework employs multilevel coarsening, with recursive strategies that scale to large circuits and enable scheduleable circuit extraction.

Multilevel Framework for Partitioning Quantum Circuits

Introduction

The paper presents a multilevel framework aimed at optimizing the partitioning of quantum circuits for distributed quantum computing (DQC). Quantum circuits, which need to be executed across multiple quantum processing units (QPUs), face the challenge of minimizing communication overhead. The paper introduces a generalized hypergraph-based approach to capture teleportation possibilities and employs a modified Fiduccia-Mattheyses (FM) algorithm within a multilevel coarsening-uncoarsening paradigm. The multilevel FM (MLFM) framework is proposed to efficiently find lower-cost partitions and to scale across large circuit sizes, with potential improvements in entanglement requirements of up to 35% compared to existing methods.

Quantum Circuit Partitioning

The approach formalizes quantum circuit partitioning as a hypergraph problem. Each qubit interaction is represented as either a state edge or a gate edge, which are then grouped into teleportation-compatible hyper-edges. The partitioning objective minimizes the entanglement cost, determined by the number of e-bits required corresponding to cut edges and hyper-edges after partitioning.

Fiduccia-Mattheyses Adaptation

The FM algorithm is adapted for temporal-domain hypergraphs, with a custom cost function reflecting entanglement costs. Gain update calculations are executed in constant time by leveraging the pre-computation and caching of possible edge configurations. This adaptation allows for efficient local search for minimizing the cut (entanglement cost).

Multilevel Coarsening and Partitioning

The multilevel methodology involves coarsening the temporal axis of the quantum circuit hypergraph, contracting nodes to reduce problem size, and then iteratively refining the partitioning across coarse-to-fine levels. Three strategies are considered: window-based, block-based, and recursive coarsening. The recursive approach was shown to be particularly effective in balancing scalability with solution quality.

Performance Evaluation

The approach was tested against benchmark circuits such as quantum Fourier transforms (QFT), quantum volume (QV) circuits, QAOA, and others. Results show significant reductions in entanglement requirements across different circuit types, with recursive MLFM achieving the lowest costs and scaling efficiently to larger circuit sizes.

Circuit Extraction

The framework includes methods for converting partitioned temporal hypergraphs back into schedule-able quantum circuits, taking into account communication qubit constraints and applying nested state teleportations where beneficial.

Conclusion

The proposed multilevel framework efficiently reduces communication overhead and optimizes qubit assignments across various QPUs, making it a valuable addition to the tools available for compiling quantum circuits in DQC environments. Extensions to address communication qubit constraints and integrate new protocols like runtime embedding and state merging are suggested as future work.