1. SDP bounds on quantum codes
- Author
-
Munné, Gerard Anglès, Nemec, Andrew, and Huber, Felix
- Subjects
Quantum Physics ,Computer Science - Information Theory ,Mathematics - Optimization and Control - 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., Comment: 51 pages
- Published
- 2024