This work provides a semidefinite programming hierarchy based on state polynomial optimization to determine the existence of quantum codes with given parameters. The hierarchy is complete, in the sense that a ((n, K, \u03b4))\u2082 code exists if and only if every level of the hierarchy is feasible. It is not limited to stabilizer codes and thus is applicable generally. While the machinery is formally dimension-free, we restrict it to qubit codes through quasi-Clifford algebras. We derive the quantum analog of a range of classical results: first, from an intermediate level a Lov\u00e1sz bound for self-dual quan- tum codes is recovered. Second, a symmetrization of a minor variation of this Lov\u00e1sz bound recovers the quantum Delsarte bound. Third, a symmetry reduction using the Terwilliger algebra leads to semidefinite programming bounds of size O(n\u2074). With this we give an alternative proof that there is no ((7, 1, 4))\u2082 quantum code, and show that ((8, 9, 3))\u2082 and ((10, 5, 4))\u2082 codes do not exist.<\/p>\n\n\n\n
<\/p>\n","protected":false},"excerpt":{"rendered":"
This work provides a semidefinite programming hierarchy based on state polynomial optimization to determine the existence of quantum codes with given parameters. The hierarchy is complete, in the sense that a ((n, K, \u03b4))\u2082 code exists if and only if every level of the hierarchy is feasible. It is not limited to stabilizer codes and […]<\/p>\n","protected":false},"author":2605,"featured_media":0,"template":"","class_list":["post-2478","seminar","type-seminar","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/webs.uab.cat\/giq\/wp-json\/wp\/v2\/seminar\/2478","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/webs.uab.cat\/giq\/wp-json\/wp\/v2\/seminar"}],"about":[{"href":"https:\/\/webs.uab.cat\/giq\/wp-json\/wp\/v2\/types\/seminar"}],"author":[{"embeddable":true,"href":"https:\/\/webs.uab.cat\/giq\/wp-json\/wp\/v2\/users\/2605"}],"wp:attachment":[{"href":"https:\/\/webs.uab.cat\/giq\/wp-json\/wp\/v2\/media?parent=2478"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}