  Future quantum internet applications will derive their power from the ability
to share quantum information across the network. Quantum teleportation allows
for the reliable transfer of quantum information between distant nodes, even in
the presence of highly lossy network connections. While many experimental
demonstrations have been performed on different quantum network platforms,
moving beyond directly connected nodes has so far been hindered by the
demanding requirements on the pre-shared remote entanglement, joint qubit
readout and coherence times. Here we realize quantum teleportation between
remote, non-neighboring nodes in a quantum network. The network employs three
optically connected nodes based on solid-state spin qubits. The teleporter is
prepared by establishing remote entanglement on the two links, followed by
entanglement swapping on the middle node and storage in a memory qubit. We
demonstrate that once successful preparation of the teleporter is heralded,
arbitrary qubit states can be teleported with fidelity above the classical
bound, even with unit efficiency. These results are enabled by key innovations
in the qubit readout procedure, active memory qubit protection during
entanglement generation and tailored heralding that reduces remote entanglement
infidelities. Our work demonstrates a prime building block for future quantum
networks and opens the door to exploring teleportation-based multi-node
protocols and applications.
