Classes of domains with low-degree indicator approximations (naïve completion sufficiency)

Classify the classes of domains X⊆{0,1}^n for which the indicator function 1_X admits a low-degree polynomial approximation, thereby determining precisely when the naïve completion (setting values outside the domain uniformly) suffices to rule out superpolynomial quantum speedups for partial Boolean functions f defined on X.

Background

The paper shows that the naïve completion F (assigning a constant value outside the promise) has low approximate degree whenever membership in the domain can be efficiently decided (via a low-degree approximator for 1_Dom(f)).

A full characterization of domains whose indicators have low approximate degree would pinpoint when simple completions already preclude superpolynomial quantum advantages.

References

We list some open problems that would further the usefulness of our techniques. For what classes of functions is a low degree approximation of the indicator function achievable? An exact characterization would quantify precisely when the na"ive completion is sufficient to prove no superpolynomial speedup.

— From Promises to Totality: A Framework for Ruling Out Quantum Speedups  (2603.29256 - Huffstutler et al., 31 Mar 2026) in Discussion and further work (Introduction)