Bennett's Protocol for Entanglement Purification and Fast Routing with Local Operations and Classical Communication

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@Krastanov, I am unfamiliar with the already implemented protocols and content here, so here goes nothing. I didn't see the paper: Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels in the Bibliography or references.bib. My assumption is that this protocol is not present, or if it is present, then at least a quick search does not provide documentation about it. If it is present or there is a similar protocols that provides the same functionality, this is more of a tutorial rather than a feature and vice versa, otherwise.

I think the entanglement purification protocol and Teleportation routing (Greedy Routing) in the context of LOCC presented in this paper Toward a 2D Local Implementation of Quantum LDPC Codes are relevant here. Although most of the paper is about Clifford Simulation, there is some portion on Teleportation Routing and Entanglement Purification so I think it was appropriate to share this here as well. I hope this is useful.

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The following steps:

1. Initial Setup: A circuit is set up to generate two noisy long-range Bell pairs, similar to the configuration in Fig. 2(a) of the reference.
   
   Fig 2 from https://arxiv.org/pdf/2404.17676
2. Entanglement Purification: Bennett et al.'s entanglement purification protocol is applied to these pairs in order to improve their fidelity.
3. No Failure Retries: This simplified version of the protocol does not include reattempts for failures and utilizes only a single donor Bell pair.
4. Simulation Execution: By running the simulation multiple times, it is possible to estimate the success probability of the purification process and the resulting fidelity of the purified Bell pairs.

P.S. More details are presented in IV. Section B about the details of this procedure.

These steps are summed up in the following figure from the aforementioned paper:

Fig 4 from https://arxiv.org/pdf/2404.17676

There is also some discussion and algorithm presented for Greedy Routing (Page 5)

Additional context

Miscellaneous: The Clifford portion of this paper is in progress (if that is relevant w.r.t current context of this issue, although I don't think it will be that relevant here as I think the protocol's functionality might be what's appropriate): The codes and Toric Layout mentioned in this paper are completed in QuantumClifford waiting scrutiny, and 2D Embedding is under progress. Only the Detector part is left as the authors relate 'Detectors with errors in the circuit by constructing a bipartite graph'.

Details about the specific of Detector are presented in FIG 5, IV Section C in space-time decoder which I think is a discussion mostly related to Clifford simulation. I think the Detector specifics (Page 8 complete) in the aforementioned might be interesting as I think you might have some plans about Detector as you were considering the capabilities of the Detector (based on user requirements) in a recent discussion.