  We propose a graph method for systematically searching for schemes that can
generate multipartite entanglement in linear bosonic systems with heralding.
While heralded entanglement generation offers more tolerable schemes for
quantum tasks than postselected ones, it is generally more challenging to find
appropriate circuits for multipartite systems. We show that our graph mapping
from boson subtractions provides handy tactics to overcome the limitations in
circuit designs. We present a practical strategy to mitigate the limitation
through the implementation of our graph technique. Our physical setup is based
on the sculpting protocol, which utilizes an $ N$ spatially overlapped
subtractions of single bosons to convert Fock states of evenly distributed
bosons into entanglement. We have identified general schemes for qubit
N-partite GHZ and W states, which are significantly more efficient than
previous schemes. In addition, our scheme for generating the superposition of
$N=3$ GHZ and W entangled states illustrates that our approach can be extended
to derive more generalized forms of entangled states. Furthermore, we have
found an N-partite GHZ state generation scheme for qudits, which requires
substantially fewer particles than previous proposals. These results
demonstrate the power of our approach in discovering optimized solutions for
the generation of intricate heralded entangled states. As a proof of concept,
we propose a linear optical scheme for the generation of the Bell state by
heralding detections. We expect our method to serve as a promising tool in
generating diverse entanglement.
