Analysis of randomized-noise Gowers-norm estimation

Analyze Gowers U^2-norm estimation when the additive shifts ω_i are independently sampled from an unknown distribution, and determine derandomization or symmetrization strategies that recover reliable norm estimation despite the resulting fluctuation of the Fourier peak.

Background

The paper proves robustness under fixed but arbitrary additive shifts, for which the Fourier peak is merely displaced and can still be used for norm estimation. It explicitly leaves unresolved the fully randomized setting, where independently varying unknown shifts spread the probability mass across Fourier outcomes and may degrade amplitude estimation.

References

We leave a full analysis of this random-noise model, and possible derandomization or symmetrization strategies, for future work.

— Quantum Algorithms for Gowers Norm Estimation, Polynomial Testing, and Arithmetic Progression Counting over Finite Abelian Groups  (2508.01231 - Kuo, 2 Aug 2025) in Section 7, Remark following the proof of the noisy U^2-estimation theorem