The Quantumly Fast and the Classically Forrious
Clément L. Canonne, Kenny Chen, Julián Mestre
TL;DR
The paper addresses the extremal Forrelation problem, where a quantum algorithm can solve with a single query but classical approaches face exponential query complexity. It develops a novel construction based on partial spread bent functions to create hard instances and uses nonuniform balls-into-bins and subspace counting to prove a near-tight classical lower bound of $Ω(2^{0.499n})$, nearly matching the conjectured $Ω(2^{n/2})$ up to an $o(1)$ exponent. The techniques extend to the Generalized Simon's Problem, yielding a tight lower bound of $Ω(k,\sqrt{kp^{n-k}})$ and thereby resolving related questions about quantum speedups in this framework. The results demonstrate a robust, collision-based indistinguishability approach that remains effective under adaptive querying and scales to higher-collision regimes, underscoring the depth of classical hardness for extremal Forrelation and its variants.
Abstract
We study the extremal Forrelation problem, where, provided with oracle access to Boolean functions $f$ and $g$ promised to satisfy either $\textrm{forr}(f,g)=1$ or $\textrm{forr}(f,g)=-1$, one must determine (with high probability) which of the two cases holds while performing as few oracle queries as possible. It is well known that this problem can be solved with \emph{one} quantum query; yet, Girish and Servedio (TQC 2025) recently showed this problem requires $\widetildeΩ(2^{n/4})$ classical queries, and conjectured that the optimal lower bound is $\widetildeΩ(2^{n/2})$. Through a completely different construction, we improve on their result and prove a lower bound of $Ω(2^{0.4999n})$, which matches the conjectured lower bound up to an arbitrarily small constant in the exponent.