SDP bounds on quantum codes
<|reference_start|>SDP bounds on quantum codes: 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.<|reference_end|>
@article{munné2024sdp,
    title={SDP bounds on quantum codes},
    author={Gerard Angl`es Munn'e, Andrew Nemec, Felix Huber},
    journal={arXiv preprint arXiv:2408.10323},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.10323},
    primaryClass={quant-ph cs.IT math.IT math.OC}
}