Close the existential–algorithmic gap for quantum OPI

Determine the largest satisfaction ratio s^*_{\mathrm{alg}} for which a solution to every Optimal Polynomial Intersection instance can be found efficiently by a quantum algorithm, and close the gap between existentially guaranteed satisfaction ratios and currently attainable quantum-algorithmic ratios.

Background

The paper develops quantum algorithms for OPI and also proves existential guarantees that are stronger in some parameter regimes than the guarantees currently achieved algorithmically. Thus, the exact efficiently attainable quantum satisfaction threshold remains unresolved. The open problem asks both for the optimal algorithmic ratio and, in particular, whether the gap between known existence results and quantum algorithms can be eliminated.

References

What is the largest fraction $s*_{\mathrm{alg}\le s*$ such that a solution to OPI with satisfaction at least $s*_{\mathrm{alg}$ can be found efficiently by a quantum algorithm? In particular, there are parameter regimes where we have shown the existence of solutions with satisfaction ratio beyond what is currently attainable quantum algorithmically; can this gap be closed?

— Improved Local Leakage Resilience of Shamir Secret Sharing and Worst-Case Optimal Polynomial Intersection  (2610.12357 - Sun et al., 8 Oct 2026) in Section ‘Open Problems’