Minimizing matter qubits and entangling gates in photonic resource state generation schemes

Fusion-based quantum computing (FBQC) relies on a set of small, typically photonic, resource states that are fused together through Bell state measurements. The main bottleneck of FBQC is the probabilistic nature and low rate of resource state generation. We employ our previously developed algorithm that can find a photon generation protocol within polynomial time using a set of emitter qubits. However, finding the minimal number of emitters or CNOTs for different emission orderings is an NP-Hard problem. We analyze various emission orderings for photonic resource states and find efficient deterministic protocols for their generation.