  Multi-qubit photonic graph states are necessary for quantum communication and
computation. Preparing photonic graph states using probabilistic stitching of
single photons using linear optics results in a formidable resource requirement
due to the need of multiplexing. Quantum emitters present a viable solution to
prepare photonic graph states, as they enable controlled production of photons
entangled with the emitter qubit, and deterministic two-qubit interactions
among emitters. A handful of emitters often suffice to generate useful photonic
graph states that would otherwise require millions of single photon sources
using the linear-optics method. But, photon loss poses an impediment to this
method due to the large depth, i.e., age of the oldest photon, of the graph
state, given the typically large number of slow and noisy two-qubit CNOT gates
required on emitters. We propose an algorithm that can trade the number of
emitters with the graph-state depth, while minimizing the number of emitter
CNOTs. We apply our algorithm to generating a repeater graph state (RGS) for
all-photonic repeaters. We find that our scheme achieves a far superior
rate-vs.-distance performance than using the least number of emitters needed to
generate the RGS. Yet, our scheme is able to get the same performance as the
linear-optics method of generating the RGS where each emitter is used as a
single-photon source, but with orders of magnitude fewer emitters.
