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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<\/mml:mi>=<\/mml:mo>5<\/mml:mn><\/mml:math> (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<\/mml:mi>=<\/mml:mo>8<\/mml:mn><\/mml:math> 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.<\/jats:p>","DOI":"10.22331\/q-2022-05-19-715","type":"journal-article","created":{"date-parts":[[2022,5,19]],"date-time":"2022-05-19T16:05:53Z","timestamp":1652976353000},"page":"715","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":22,"title":["Enumerating all bilocal Clifford distillation protocols through symmetry reduction"],"prefix":"10.22331","volume":"6","author":[{"given":"Sarah","family":"Jansen","sequence":"first","affiliation":[{"name":"Delft Institute of Applied Mathematics, Delft University of Technology, The Netherlands."},{"name":"Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kenneth","family":"Goodenough","sequence":"additional","affiliation":[{"name":"QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S\u00e9bastian","family":"de Bone","sequence":"additional","affiliation":[{"name":"QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands"},{"name":"QuSoft, CWI, Science Park 123, 1098 XG Amsterdam, The Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dion","family":"Gijswijt","sequence":"additional","affiliation":[{"name":"Delft Institute of Applied Mathematics, Delft University of Technology, The Netherlands."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David","family":"Elkouss","sequence":"additional","affiliation":[{"name":"QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands"},{"name":"Networked Quantum Devices Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2022,5,19]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Charles H. 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