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Kempe–Pyber–Shalev minimal-degree conjecture for weak Fourier sampling
Kempe–Pyber–Shalev conjecture. If $H$ can be distinguished from the trivial subgroup by weak Fourier sampling, then the minimal degree of $H$ must be constant.
Conjecturesolvedfourier-sampling quantum-computing symmetric-groupsarXiv posted 2 days ago

Kempe–Pyber–Shalev minimal-degree conjecture for weak Fourier sampling