Aaronson–Ambainis conjecture (dense-domain query upper bound)

Establish that for any function f defined on a dense subset S⊆{−1,1}^N with |S|≥c·2^N, there exists a deterministic classical decision tree making poly(\tilde{\deg}(f), 1/α, 1/c) queries that computes f(X) on at least a 1−α fraction of inputs X∈S.

Background

The paper recalls a well-known conjecture connecting approximate degree to randomized/deterministic query algorithms on dense domains, positing that low approximate degree implies low-query algorithms that are correct on most inputs.

This conjecture, if resolved, would further link structural polynomial approximations to classical algorithmic performance in partial-domain settings.