Completion-based approximate-degree analogue on dense domains

Prove that if a partial Boolean function f has approximate degree d and its promise domain has size at least a constant fraction of the Boolean hypercube, then there exists a completion F extending f that matches f on a (1−δ)-fraction of the domain and satisfies \tilde{\deg}(F)=poly(d,1/ε,1/δ).

Background

Motivated by the Aaronson–Ambainis conjecture, the paper formulates an approximate-degree analogue tailored to the completion framework: construct a low-degree completion that agrees with f on most of its domain when the domain is dense.

Showing such completions exist would connect dense promises with extendability in approximate degree, strengthening the completion method’s reach.

References

In our completion framework the equivalent approximate degree question is as follows. Conjecture Let $f: {f} \rightarrow {-1,1}$ be a partial function with approximate degree $d$ where $|{f}|\geq0.12{n}$. Then there exists a completion $F$\footnote{Technically, the completion is from a subfunction of $f$ and not the original $f$ itself.} of $f$ where $F(x)=f(x)$ on $(1-\delta)|{f}|$ many inputs $x\in {f}$ and set arbitrarily otherwise, with $(F)=(d,1/0.1,1/\delta)$.

— From Promises to Totality: A Framework for Ruling Out Quantum Speedups  (2603.29256 - Huffstutler et al., 31 Mar 2026) in Subsubsection: Perturbing approximate polynomials (Towards general partial function completions)