The Quantumly Fast and the Classically Forrious

2026-02-07

AI Summary

This work investigates the extremal Forrelation problem: given oracle access to two Boolean functions \( f \) and \( g \) whose Forrelation value is guaranteed to be \( \pm 1 \), determine the sign with high probability using as few queries as possible. By constructing a novel hard instance distribution and combining tools from Boolean function analysis, Fourier analysis, and oracle lower bound techniques, the authors establish a classical query complexity lower bound of \( \Omega(2^{0.4999n}) \), significantly improving upon the previous \( \widetilde{\Omega}(2^{n/4}) \) bound and approaching the conjectured optimal \( \widetilde{\Omega}(2^{n/2}) \). In stark contrast, a quantum algorithm solves the problem with just a single query, thereby demonstrating an almost exponential quantum advantage.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Forrelation
• quantum query
• classical query
• oracle complexity
• Boolean functions

Innovation

• Forrelation
• quantum query complexity
• classical lower bound
• oracle separation
• exponential quantum advantage