Transforming graph states via Bell state measurements

Graph states are key resources for measurement-based quantum computing which is particularly promising for photonic systems. Fusions are probabilistic Bell state measurements, measuring pairs of parity operators of two qubits. Fusions can be used to connect/entangle different graph states making them a powerful resource for measurement-based and the related fusion-based quantum computing. There are several different graph structures and types of Bell state measurements, yet the associated graph transformations have only been analyzed for a few specific cases. Here, we provide a full set of such graph transformation rules and we give an intuitive visualization based on Venn diagrams of local neighborhoods of graph nodes. We derive these graph transformations for all fusion types showing that there are five different types of fusion success cases. Finally, we give application examples of the derived graph transformation rules and show that they can be used for constructing graph codes or simulating fusion networks.