Complexity of mixed-unitary detection under stronger distance promises

Determine the computational difficulty of deciding whether a quantum channel Φ: L(C^n) → L(C^n) is mixed-unitary under a stronger promise that either Φ is mixed-unitary or Φ lies at an inverse-logarithmic (or constant) distance from the boundary of the set of mixed-unitary channels, with distance measured using standard norms on Choi representations (e.g., trace norm, Frobenius norm, spectral norm) or the diamond norm on channels.

Background

The paper proves strong NP-hardness (via polynomial-time Turing reductions) of the mixed-unitary detection problem under the promise that the input channel is not within an inverse-polynomial distance of the boundary of the mixed-unitary channels. This ensures the hardness is not due to numerical precision issues.

The authors raise the question of how the complexity might change if the promise is strengthened to inverse-logarithmic or constant distance from the boundary. They note that, unlike the inverse-polynomial regime where various norms are polynomially equivalent, the choice of metric (trace norm, Frobenius norm, spectral norm on Choi matrices, or diamond norm) may affect complexity at logarithmic or constant distance scales.