  A graph picture of linear quantum systems (LQSs) is introduced in Quantum,
5:611 (2021) and arXiv:2211.04042 (2022) to provide systematic methods for
generating multipartite genuine entanglement with and without postselection. An
intriguing and pertinent question emerges from this approach: Can we find a
shared structure between LQS graphs and qubit graphs, commonly referred to as
graph states? If such a structure can be identified, it can be exploited to
generate graph states with LQSs. Our work presents a partial but noteworthy
answer to this question. Within this research, \emph{we suggest a directed
graph structure, which enables the generation of arbitrary caterpillar graph
states employing heralded schemes in LQSs}. The caterpillar graph states
encompass various useful graph states for one-way quantum computing such as
linear graphs, star graphs, and networks of star graphs. The states generated
through this approach will serve as valuable resource states for arbitrary
graph states with fusion gates.

Quantum Physics