The Quantumly Fast and the Classically Forrious

TL;DR

This paper establishes a near-optimal classical query lower bound for the extremal Forrelation problem, matching the conjectured exponential lower bound and highlighting the quantum advantage in query complexity.

Contribution

It provides a new proof technique that improves the classical query lower bound for the extremal Forrelation problem, nearly matching the conjectured exponential bound.

Findings

Quantum query complexity is constant (one query).

Classical query complexity has a lower bound of approximately 2^{0.4999n}.

The result nearly confirms the conjectured exponential separation.

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 $\widetilde\Omega(2^{n/4})$ classical queries, and conjectured that the optimal lower bound is $\widetilde\Omega(2^{n/2})$. Through a completely different construction, we improve on their result and prove a lower bound of $\Omega(2^{0.4999n})$, which matches the conjectured lower bound up to an arbitrarily small constant in the exponent.