Optimal two-bucket algorithms across all distribution-estimation regimes

Determine whether local Chebyshev moment matching can produce a two-bucket algorithm for estimating an unknown d-element distribution up to permutation in total variation distance with optimal sample complexity across all accuracy regimes, including the intermediate range between ε≤1/d and ε≥1/d^{1−γ}.

Background

The classical analogue of quantum spectrum estimation is learning a distribution when the labels of its probabilities are irrelevant. Existing methods achieve optimal rates in the high-accuracy and constant-accuracy regimes, but the intermediate regime remains unresolved.

Because the paper’s quantum algorithm achieves its bound using only two buckets and local Chebyshev moment matching, the authors ask whether the same moment-matching approach can yield an optimal two-bucket classical algorithm throughout the full accuracy range.