Forrelation is Extremally Hard

TL;DR

This paper introduces a new algebraic approach to the Forrelation problem, demonstrating its exponential classical hardness while highlighting the efficiency of quantum algorithms in solving extremal instances.

Contribution

It provides a linear algebraic perspective on Forrelation, connects it to bent functions, and proves classical hardness under cryptographic assumptions for restricted cases.

Findings

Quantum algorithm solves extremal Forrelation with one query.

Classical algorithms require exponential queries for extremal Forrelation.

Classical hardness established under cryptographic assumptions for small circuit inputs.

Abstract

The Forrelation problem is a central problem that demonstrates an exponential separation between quantum and classical capabilities. In this problem, given query access to $n$-bit Boolean functions $f$ and $g$, the goal is to estimate the Forrelation function $\mathrm{forr}(f,g)$, which measures the correlation between $g$ and the Fourier transform of $f$. In this work we provide a new linear algebraic perspective on the Forrelation problem, as opposed to prior analytic approaches. We establish a connection between the Forrelation problem and bent Boolean functions and through this connection, analyze an extremal version of the Forrelation problem where the goal is to distinguish between extremal instances of Forrelation, namely $(f,g)$ with $\mathrm{forr}(f,g)=1$ and $\mathrm{forr}(f,g)=-1$. We show that this problem can be solved with one quantum query and success probability one, yet requires $\tilde{\Omega}\left(2^{n/4}\right)$ classical randomized queries, even for algorithms with a one-third failure probability, highlighting the remarkable power of one exact quantum query. We also study a restricted variant of this problem where the inputs $f,g$ are computable by small classical circuits and show classical hardness under cryptographic assumptions.