Extracting GHZ states from linear cluster states

Quantum information processing architectures typically only allow for nearest-neighbor entanglement creation. In many cases, this prevents the direct generation of $\mathrm{GHZ}$ states, which are commonly used for many communication and computation tasks. Here, we show how to obtain $\mathrm{GHZ}$ states between nodes in a network that are connected in a straight line, naturally allowing them to initially share linear cluster states. We prove a strict upper bound of $\lfloor (n+3)/2\rfloor$ on the size of the set of nodes sharing a $\mathrm{GHZ}$ state that can be obtained from a linear cluster state of $n$ qubits, using local Clifford unitaries, local Pauli measurements, and classical communication. Furthermore, we completely characterize all selections of nodes below this threshold that can share a $\mathrm{GHZ}$ state obtained within this setting. Finally, we demonstrate these transformations on the IBMQ Montreal quantum device for linear cluster states of up to $n=19$ qubits.