SDP bounds on quantum codes

TL;DR

This paper introduces a comprehensive semidefinite programming hierarchy to determine the existence of quantum codes with specific parameters, extending classical bounds and applying to general quantum codes beyond stabilizer types.

Contribution

It develops a complete SDP hierarchy based on state polynomial optimization for quantum codes, applicable to all code types, and derives quantum analogs of classical bounds.

Findings

Reproduces Lovász bound for quantum codes.

Recovers quantum Delsarte bound via symmetrization.

Proves non-existence of certain quantum codes such as (7,1,4), (8,9,3), and (10,5,4).

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 a $(\!(n, K, {\delta})\!)_2$ 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\'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.