  Quantum teleportation is not merely a fascinating corollary of quantum
entanglement, it also finds utility in quantum processing and circuit
compilation. In this paper, we measure and track the entanglement and fidelity
of two-qubit states prepared on a 127-qubit IBM Quantum device, as one of the
qubits is teleported across 19 qubits. We design, evaluate and compare two
distinct approaches to teleportation: post-selected measurement categorisation
and dynamic circuit corrections based on mid-circuit measurements, and compare
with direct state transportation using SWAP gates. By optimally choosing the
teleportation path which exhibits the highest total negativity entanglement
measure across nearest-neighbour pairs, we show the entanglement of a two-qubit
graph state is sustained after at least 19 hops in teleportation using the
post-selection approach and 17 hops using the dynamic circuit approach. We
observe a higher level of teleported entanglement in paths determined from
two-qubit negativities compared to those obtained from gate errors,
demonstrating an advantage in using the negativity map over the gate error map
for compiling quantum circuits.
