  The most efficient automated way to construct a large class of quantum
photonic experiments is via abstract representation of graphs with certain
properties. While new directions were explored using Artificial intelligence
and SAT solvers to find such graphs, it becomes computationally infeasible to
do so as the size of the graph increases. So, we take an analytical approach
and introduce the technique of local sparsification on experiment graphs, using
which we answer a crucial open question in experimental quantum optics, namely
whether certain complex entangled quantum states can be constructed. This
provides us with more insights into quantum resource theory, the limitation of
specific quantum photonic systems and initiates the use of graph-theoretic
techniques for designing quantum physics experiments.
