The Quantumly Fast and the Classically Forrious

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.

💡 Research Summary

The paper investigates the extremal Forrelation problem, where an algorithm receives oracle access to two Boolean functions f and g on n bits with the promise that their Forrelation value is either +1 or −1, and must decide which case holds. It is well‑known that a quantum algorithm can solve this task with a single query, achieving exact success. In contrast, the best known classical lower bound prior to this work, due to Girish and Servedio (TQC 2025), was Ω̃(2^{n/4}), and the authors conjectured the true bound should be Ω̃(2^{n/2}).

The authors dramatically improve the classical lower bound to Ω(2^{0.4999 n}), i.e., they show that any classical (randomized) algorithm needs at least 2^{0.4999 n} queries to distinguish the two cases with high probability. This matches the conjectured Ω(2^{n/2}) bound up to an arbitrarily small constant in the exponent, essentially closing the gap.

The technical advance rests on a new construction of hard instances based on a class of bent Boolean functions called partial spread functions, originally introduced by Dillon. Unlike the Maiorana‑McFarland bent functions used by Girish and Servedio, partial spread functions are built from a partition of F₂^{n} into 2^{n/2}+1 subspaces of dimension n/2, with a carefully chosen selection of subspaces to define the function’s sign pattern. This construction yields a much richer linear‑algebraic structure: each subspace is defined by only n/2 independent vectors, allowing the authors to argue about k‑collisions (multiple queries landing in the same subspace) for values of k as large as Θ(n).

The core of the lower‑bound proof is an indistinguishability argument. The authors show that, as long as a classical algorithm’s query transcript contains fewer than a certain number of such collisions, the distribution of answers on yes‑instances (where g = + 2^{n/2} · \hat f) and no‑instances (where g = − 2^{n/2} · \hat f) is statistically indistinguishable. To make this rigorous they develop a non‑uniform balls‑and‑bins analysis that accounts for the linear constraints imposed by the queries, and they combine it with subspace‑counting techniques based on Gaussian binomial coefficients to guarantee that a large number of bent functions remain consistent with the observed transcript. Consequently, any algorithm that tries to brute‑force the hidden function would still need an exponential number of additional queries.

Beyond the extremal Forrelation problem, the paper shows that the same ideas yield a tight classical lower bound for the Generalized Simon’s Problem. In that setting a function f over F_{p}^{n} contains a hidden subgroup of dimension k, and the goal is to identify the subgroup. The authors prove that any query algorithm requires Ω(k, p^{k}·p^{n−k}) queries, matching known upper bounds and resolving an open question from prior work.

In summary, the paper makes three major contributions: (1) it introduces a novel family of bent functions (partial spread functions) that enable the construction of extremal Forrelation instances with perfect correlation/anticorrelation; (2) it develops a refined collision‑based indistinguishability framework that pushes the classical query lower bound from 2^{n/4} to essentially 2^{n/2}; and (3) it applies the same methodology to obtain optimal lower bounds for a broader class of hidden‑subgroup problems. The work deepens our understanding of the maximal separation between quantum and classical query complexities and provides new tools that may be useful for proving tight bounds in other oracle‑based problems.