Exact finite-sample worst-case error under product noise

Determine whether the finite-sample worst-case spectrum-estimation error for weak Schur sampling under per-copy trace-distance noise is, up to constants, the maximum of the noise floor and the noiseless statistical error, or whether the two contributions genuinely add, and determine whether an adversary can prevent cancellation among oppositely directed per-register perturbations.

Background

The main theorem gives an upper bound consisting of the additive noise floor plus the noiseless error, while the lower bounds separately establish the necessity of each term. Thus the paper does not determine the exact finite-n combination of the two effects.

The authors also note that the telescoping proof uses absolute values for each register and may discard cancellation. An unresolved issue is whether admissible perturbations can be arranged to avoid such cancellation, beyond the already understood case of identical perturbations on every register. Computing the exact Lipschitz constant of the relevant outcome observables is identified as a related refinement.

References

First, is the noisy error ever larger than \max(\epsilon,\eta(n,d)) up to constants, i.e. do the two terms in \cref{eq:mainbound} genuinely add, or does the true worst case behave like their maximum?

— Robustness of quantum spectrum estimation: weak Schur sampling under noisy inputs  (2610.03582 - Visnevskyi et al., 2 Oct 2026) in Section 8, subsection “Open questions,” item 4