  High-dimensional multipartite entanglement plays a crucial role in quantum
information science. However, existing schemes for generating such entanglement
become increasingly complex and costly as the dimension of quantum units
increases. In this work, we overcome the limitation by proposing a
significantly enhanced linear optical heralded scheme that generates the
$d$-level $N$-partite GHZ state with single-photon sources and their linear
operations. Our scheme requires $dN$ photons to generate the target state with
substantially improved success probability from previous schemes. It employs
linear optical logic gates compatible with any qudit encoding system and can
generate generalized GHZ states with installments of beamsplitters. With
efficient generations of high-dimensional resource states, our work opens
avenues for further exploration in high-dimensional quantum information
processing.
