Exponential quantum-classical query separations for cyclic Forrelation

Abstract

We study the randomized classical query complexity of cyclic Forrelation, a cyclic analogue of $k$-Forrelation [AA15, STOC'15; BS21, STOC'21]. Given $k$ Boolean functions $f_1,\ldots,f_k:\mathbb{F}_2^n\to\{-1,1\}$, the goal is to estimate $T_k=N^{-1}\operatorname{Tr}(D_{f_1}H^{\otimes n}\cdots D_{f_k}H^{\otimes n}),$ where $D_{f_i}=\operatorname{diag}(f_i(x):x\in\mathbb{F}_2^n)$, $H$ is the Hadamard gate, and $N=2^n$. This normalized-trace problem can be solved with $k$ quantum queries in the DQC1 model. Its classical query complexity was left open in [JLS+26, FOCS'26], which conjectured a lower bound of $\widetildeΩ(N^{1-2/k})$. We establish a nearly complete characterization of the nonadaptive classical query complexity of cyclic Forrelation. For even $k\ge 4$, we prove a tight bound of $Θ_k(N^{1-2/k})$ for constant additive error. For odd $k\ge 5$, we obtain a lower bound of $Ω_k(N^{1-2/(k-1)})$ and an upper bound of $O_k(N^{1-2/k})$. Surprisingly, $k=3$ admits a polylogarithmic-query nonadaptive classical algorithm, disproving the conjecture in this case. These results suggest an unexpected parity-dependent structure: the nonadaptive complexity may be governed by the independence number of the $k$-cycle, leading to the exponent $1-1/\lfloor k/2\rfloor$, rather than the previously conjectured $1-2/k$, which was motivated by a Fourier-growth heuristic. Finally, allowing adaptivity, we prove a lower bound of $Ω(N^{1/4})$ for every $k\ge4$. Although formally close to $k$-Forrelation, the cyclic structure leads to rather different classical behavior and requires different techniques. Our lower bounds are based on hidden-shift collision constructions, and our upper bounds follow from randomized sampling together with cycle-specific conditional-moment estimates. Our results provide a systematic understanding of its query advantages over classical computation.