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Noisy hardware forms one of the main hurdles to the realization of anear-term quantum internet. Distillation protocols allows one to overcome thisnoise at the cost of an increased overhead. We consider here an experimentallyrelevant class of distillation protocols, which distill $n$ to $k$ end-to-endentangled pairs using bilocal Clifford operations, a single round ofcommunication and a possible final local operation depending on the observedmeasurement outcomes. In the case of permutationally invariant depolarizingnoise on the input states, we find a correspondence between these distillationprotocols and graph codes. We leverage this correspondence to find provablyoptimal distillation protocols in this class for several tasks important forthe quantum internet. This correspondence allows us to investigate use casesfor so-called non-trivial measurement syndromes. Furthermore, we detail arecipe to construct the circuit used for the distillation protocol given agraph code. We use this to find circuits of short depth and small number oftwo-qubit gates. Additionally, we develop a black-box circuit optimizationalgorithm, and find that both approaches yield comparable circuits. Finally, weinvestigate the teleportation of encoded states and find protocols whichjointly improve the rate and fidelities with respect to prior art.

Near-term $n$ to $k$ distillation protocols using graph codes