Algorithmizing the entropic proof techniques of the Polynomial Freiman–Ruzsa theorem

Establish whether the entropic methods underlying the recent combinatorial proof of the Polynomial Freiman–Ruzsa theorem can be transformed into efficient algorithms that directly yield an algorithmic version without relying on quantum detours.

Background

The paper notes that a direct attempt to algorithmize each step of the recent proof of the PFR conjecture faces a major obstacle: the proof heavily uses entropic techniques that are inherently non-algorithmic. The authors circumvent this by leveraging quantum algorithms and then dequantizing them to obtain classical procedures, but they emphasize that it remains unclear whether the original entropic machinery itself can be made algorithmic.

Resolving this question would clarify whether one can derive efficient classical algorithms directly from the combinatorial proof framework, potentially simplifying the algorithmic pathway and removing the need for quantum techniques or dequantization.