Enumerating all bilocal Clifford distillation protocols through symmetry reduction

Sarah Jansen1,2, Kenneth Goodenough3, Sébastian de Bone3,4, Dion Gijswijt1, and David Elkouss3,5

1Delft Institute of Applied Mathematics, Delft University of Technology, The Netherlands.
2Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands
3QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
4QuSoft, CWI, Science Park 123, 1098 XG Amsterdam, The Netherlands
5Networked Quantum Devices Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa, Japan

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Entanglement distillation is an essential building block in quantum communication protocols. Here, we study the class of near-term implementable distillation protocols that use bilocal Clifford operations followed by a single round of communication. We introduce tools to enumerate and optimise over all protocols for up to $n=5$ (not necessarily equal) Bell-diagonal states using a commodity desktop computer. Furthermore, by exploiting the symmetries of the input states, we find all protocols for up to $n=8$ copies of a Werner state. For the latter case, we present circuits that achieve the highest fidelity with perfect operations and no decoherence. These circuits have modest depth and number of two-qubit gates. Our results are based on a correspondence between distillation protocols and double cosets of the symplectic group, and improve on previously known protocols.

Entanglement forms a key component for quantum networks, but is usually left imperfect by noise in experimental setups. By performing distillation on the noisy entanglement, one can compensate for this. Distillation trades off quantity with quality — it transforms noisy entanglement into a smaller amount of entanglement, but which is less noisy. Understanding how to perform such distillation is crucial for future networks. We use group theoretic tools to exponentially reduce the number of protocols, allowing us to find new and improved such protocols, that can be implemented in the near-term.

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[3] Michelle Victora, Spyros Tserkis, Stefan Krastanov, Alexander Sanchez de la Cerda, Steven Willis, and Prineha Narang, "Entanglement purification on quantum networks", Physical Review Research 5 3, 033171 (2023).

[4] Benjamin Desef and Martin B. Plenio, "Optimizing quantum codes with an application to the loss channel with partial erasure information", Quantum 6, 667 (2022).

[5] Stefan Krastanov, Alexander Sanchez de la Cerda, and Prineha Narang, "Heterogeneous multipartite entanglement purification for size-constrained quantum devices", Physical Review Research 3 3, 033164 (2021).

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