Connectivity-based obstruction to quantum speedups on partial domains

Develop and formalize a measure of connectedness for subsets of the Boolean hypercube (viewed as subgraphs under Hamming adjacency) and prove that on highly connected domains any partial Boolean function f cannot exhibit superpolynomial quantum speedup over deterministic query complexity.

Background

Using promise-aware measures, the paper shows that when certain collapses occur, superpolynomial quantum speedups are impossible; moreover, large-domain promises already preclude speedups in some cases.

The authors ask whether a graph-theoretic connectedness property of the domain itself can yield a general no-speedup theorem, capturing the intuition that well-connected promises limit the potential for quantum advantage.

References

We list some open problems that would further the usefulness of our techniques. Theorem~\ref{thm:measures_no_speedup} directly implies that partial functions on domains with a large promise (i.e., low number of undefined inputs) do not attain superpolynomial speedup. Can one define a measure of connectedness on the Boolean hypercube and show that on highly-connected domains one cannot obtain 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)