Higher-order asymptotics of the optimized rank-one gap

Determine the higher-order corrections to the optimized rank-one encoder gap for the constructed quantum-noise family, beyond its established quadratic asymptotic coefficient, and thereby obtain a closed-form expression for the optimized gap or the corresponding optimized fidelity.

Background

For the explicitly constructed noise family, the paper derives the exact leading quadratic behavior of the optimized gap between unrestricted encoders and rank-one encoders as the noise parameter approaches zero. The analysis establishes the coefficient of the quadratic term but does not resolve the subsequent terms in the asymptotic expansion. Consequently, the full dependence of the optimized rank-one fidelity and the optimized gap on the noise parameter remains unknown.

References

This analysis does not determine higher-order corrections to the optimized gap. Fig.~\ref{fig:gap}(b) provides numerical estimates of $$ at finite $p$.

— High-Rank Encoding Can Improve Approximate Quantum Error Correction  (2609.00778 - Li et al., 1 Sep 2026) in Supplemental Material, Section S3, subsection “The exact quadratic gap coefficient”