Graph states are a key resource for measurement-based quantum computation and quantum networking, but state-preparation costs limit their practical use. Graph states related by local complement (LC) operations are equivalent up to single-qubit Clifford gates; one may reduce entangling resources by preparing a favorable LC-equivalent representative. However, exhaustive optimization over the LC orbit is not scalable. We address this problem using the split decomposition and its quotient-augmented strong split tree (QASST). For several families of distance-hereditary (DH) graphs, we use the QASST to characterize LC orbits and identify representatives with reduced controlled-Z count or preparation circuit depth. We also introduce a split-fuse construction for arbitrary DH graph states, achieving linear scaling with respect to entangling gates, time steps, and auxiliary qubits. Beyond the DH setting, we discuss a generalized divide-and-conquer split-fuse strategy and a simple greedy heuristic for generic graphs based on triangle enumeration. Together, these methods outperform direct implementations on sufficiently large graphs, providing a scalable alternative to brute-force optimization.

翻译：图态是测量型量子计算和量子网络的关键资源，但其制备成本限制了实际应用。通过局域互补（LC）操作相关联的图态在单量子比特克利福德门作用下等价，因此可通过制备有利的LC等价代表态来减少纠缠资源。然而，对LC轨道进行穷举优化并不具有可扩展性。我们利用分裂分解及其商增强强分裂树（QASST）来解决该问题。对于多个距离遗传（DH）图族，我们使用QASST刻画LC轨道并识别出具有降低受控-Z计数或制备电路深度的代表态。我们还引入了针对任意DH图态的分裂-融合构造方法，在纠缠门数、时间步长和辅助量子比特方面实现了线性缩放。在DH情形之外，我们讨论了基于三角形枚举的广义分治-分裂-融合策略和简单贪心启发式算法。这些方法在足够大的图上优于直接实现方案，为暴力优化提供了可扩展的替代方案。