SDP bounds on quantum codes: rational certificates
This provides more reliable non-existence proofs for quantum codes, which is important for quantum error correction researchers, though it is incremental as it builds on existing semidefinite programming approaches.
The authors tackled the problem of determining rigorous upper bounds on quantum code sizes by addressing floating-point inaccuracies in semidefinite programming methods, resulting in improved bounds for 18 cases of n-qubit codes with 6 ≤ n ≤ 19.
A fundamental problem in quantum coding theory is to determine the maximum size of quantum codes of given block length and distance. A recent work introduced bounds based on semidefinite programming, strengthening the well-known quantum linear programming bounds. However, floating-point inaccuracies prevent the extraction of rigorous non-existence proofs from the numerical methods. Here, we address this by providing rational infeasibility certificates for a range of quantum codes. Using a clustered low-rank solver with heuristic rounding to algebraic expressions, we can improve upon $18$ upper bounds on the maximum size of $n$-qubit codes with $6 \leq n \leq 19$. Our work highlights the practicality and scalability of semidefinite programming for quantum coding bounds.