Enumerating all bilocal Clifford distillation protocols through symmetry reduction
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
Published: | 2022-05-19, volume 6, page 715 |
Eprint: | arXiv:2103.03669v5 |
Doi: | https://doi.org/10.22331/q-2022-05-19-715 |
Citation: | Quantum 6, 715 (2022). |
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Abstract
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.

Featured image: Probabilities that describe the state of a system consisting of 3 qubit pairs. The light blue cylinders highlight the probabilities that correspond to the 'pillars'. The sum of these probabilities yields the success probability of a distillation protocol. The darker circles highlight the probabilities that correspond to the 'base'. These probabilities correspond to the fidelity of the output state of the protocol. Note that we have only labelled the coefficients that are on the front, right and top face.
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