Finite-shot degradation of the magnitude-filter classification

Determine how the classification of coherent spectral filtering versus classical smoothing degrades at a finite shot budget, taking into account shot noise and the reduction in effective sampling rate caused by post-selection.

Background

The paper’s main classification is formulated in the large-sampling limit, where the total-variation gap between the coherently filtered distribution and the best classical convolutional smoothing can be evaluated without dominant shot noise. Magnitude filters also incur a post-selection cost, reducing the effective number of samples available for estimating the filtered distribution.

The unresolved issue is whether and how the dichotomy and associated gap-based conclusions persist when only finitely many shots are available. The analysis must account simultaneously for statistical resolution of the gap, shot noise, and the additional sampling overhead induced by post-selection.

References

Finally, the gap Eq.~eq:gap is a statement about distributions in the large-sampling limit. At a finite shot budget $\Phi_g$ must be resolved against shot noise, at a rate itself reduced by the post-selection cost $\, and how the classification degrades there we leave open.

— Classical Limits of Spectral Filtering in Quantum Generative Models  (2608.14169 - Roth, 14 Aug 2026) in Section 6, Discussion and Conclusion