SDP bounds on quantum codes

• URL: http://arxiv.org/abs/2408.10323v1
• Date: Mon, 19 Aug 2024 18:00:07 GMT
• Title: SDP bounds on quantum codes
• Authors: Gerard Anglès Munné, Andrew Nemec, Felix Huber,
• Abstract summary: This paper provides a semidefinite programming hierarchy based on state optimization to determine the existence of quantum codes. The hierarchy is complete, in the sense that if a $(!(n,K,delta)!)$ code does not exist then a level of the hierarchy is infeasible. While it is formally-free, we restrict it to qubit codes through quasi-Clifford algebras.
• Score: 6.417777780911225
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper 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 if a $(\!(n,K,\delta)\!)_2$ code does not exist then a level of the hierarchy is infeasible. It is not limited to stabilizer codes and thus applicable generally. While it 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\'asz bound for self-dual quantum codes is recovered. Second, a symmetrization of a minor variation of this Lov\'asz bound recovers the quantum Delsarte bound. Third, a symmetry reduction using the Terwilliger algebra leads to semidefinite programming bounds of size $O(n^4)$. With this we give an alternative proof that there is no $(\!(7,1,4)\!)_2$ quantum code, and show that $(\!(8,9,3)\!)_2$ and $(\!(10,5,4)\!)_2$ codes do not exist.

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