Multipartite Entanglement in Quantum Networks Using Subgraph Complementations

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 $\boldsymbol{a}$ 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.