Quantum networks are important for quantum communication and consist of
entangled states that are essential for many tasks such as quantum
teleportation, quantum key distribution, quantum sensing and quantum error
correction. Graph states are a specific class of multipartite entangled states
that can be represented by graphs. We propose a novel approach for distributing
graph states across a quantum network. We show that the distribution of graph
states can be characterised by a system of subgraph complementations, which we
also relate to the minimum rank of the underlying graph and the degree of
entanglement quantified by the Schmidt-rank of the quantum state. We analyse
resource usage for our algorithm and show it to match or be improved in the
number of qubits, bits for classical communication and EPR pairs utilised, as
compared to prior work. The number of local operations is efficient, and the
resource consumption for our approach scales linearly in the number of
vertices. This presents a quadratic improvement in completion time for several
classes of graph states represented by dense graphs, and implies a potential
for improved fidelity in the presence of noise. Common classes of graph states
are classified along with the optimal time for their distribution using
subgraph complementations. We also provide a framework to similarly find the
optimal sequence of operations to distribute an arbitrary graph state, and
prove upper bounds along with providing approximate greedy algorithms.Comment: Background section is condensed and requires further clarificatio