SDP bounds on quantum codes: rational certificates

TL;DR

This paper develops rational infeasibility certificates for quantum codes using semidefinite programming, enabling rigorous bounds and improving known upper limits for certain quantum code sizes.

Contribution

It introduces a method to obtain exact rational certificates from SDP bounds, enhancing the rigor and reliability of quantum code size limitations.

Findings

Improved 18 upper bounds on quantum code sizes for 6 to 19 qubits.

Demonstrated scalability of SDP methods for quantum coding bounds.

Provided rigorous, algebraic infeasibility certificates for quantum codes.

Abstract

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.