Uniform polynomial-time complexity in encoded input size

Establish a uniform polynomial bound in the encoded input size for the finite-precision computation of Gaussian-restricted entanglement quantities, beyond the paper’s polynomial dependence on the requested accuracy bits for each fixed rational input covariance matrix.

Background

The paper develops a finite-precision algorithm for optimizing Gaussian reference covariance matrices and proves that, for each fixed rational input covariance matrix, the computational cost is polynomial in the requested number of accuracy bits. However, the explicit runtime bound contains a prefactor depending on numerical covariance magnitudes and conditioning parameters, such as $M^2\Lambda^2$, rather than only on the binary encoding length of the input.

A uniform polynomial-time result in the encoded input size would require removing or controlling this unfavorable magnitude dependence. The paper explicitly identifies this as unresolved, while distinguishing it from the accuracy dependence that has already been established.