Efficient Preparation of Graph States using the Quotient-Augmented Strong Split Tree

TL;DR

This paper introduces a scalable method using split decomposition and the quotient-augmented strong split tree to optimize the preparation of graph states, reducing resource costs in quantum computing.

Contribution

It presents a novel approach leveraging split decomposition and a new split-fuse construction to efficiently prepare graph states, especially for distance-hereditary graphs.

Findings

QASST characterizes LC orbits for DH graphs.

Split-fuse achieves linear scaling in entangling gates and time.

Methods outperform brute-force optimization on large graphs.

Abstract

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.