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 $\tildeΩ\left(2^{n/4}\right)$ 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.
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 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 forr(f,g)=1 and forr(f,g)=−1. We show that this problem can be solved with one quantum query and success probability one, yet requires Ω~(2n/4) 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.
We study the forrelation problem: given a pair of n-bit Boolean functions f and g, estimate the correlation between f and the Fourier transform of g. This problem is known to provide the largest possible quantum speedup in terms of its query complexity and achieves the landmark oracle separation between the complexity class BQP and the Polynomial Hierarchy. Our first result is a classical algorithm for the forrelation problem which has runtime O(n2n/2). This is a nearly quadratic improvement over the best previously known algorithm. Secondly, we show that quantum query algorithm that makes t queries to an n-bit oracle can be simulated by classical query algorithm making only O(2n(1−1/2t)) queries. This fixes a gap in the literature arising from a recently discovered critical error in a previous proof; it matches recently established lower bounds (up to poly(n,t)) factors) and thus characterizes the maximal separation in query complexity between quantum and classical algorithms. Finally, we introduce a graph-based forrelation problem where n binary variables live at vertices of some fixed graph and the functions f,g are products of terms describing interactions between nearest-neighbor variables. We show that the graph-based forrelation problem can be solved on a classical computer in time O(n) for any bipartite graph, any planar graph, or, more generally, any graph which can be partitioned into two subgraphs of constant treewidth. The graph-based forrelation is simply related to the variational energy achieved by the Quantum Approximate Optimization Algorithm (QAOA) with two entangling layers and Ising-type cost functions. By exploiting the connection between QAOA and the graph-based forrelation we were able to simulate the recently proposed Recursive QAOA with two entangling layers and 225 qubits on a laptop computer.
The Forrelation problem, first introduced by Aaronson [Scott Aaronson, 2010] and Aaronson and Ambainis [Scott Aaronson and Andris Ambainis, 2015], is a well studied computational problem in the context of separating quantum and classical computational models. Variants of this problem were used to give tight separations between quantum and classical query complexity [Scott Aaronson and Andris Ambainis, 2015]; the first separation between poly-logarithmic quantum query complexity and bounded-depth circuits of super-polynomial size, a result that also implied an oracle separation of the classes BQP and PH [Ran Raz and Avishay Tal, 2019]; and improved separations between quantum and classical communication complexity [Uma Girish et al., 2021]. In all these separations, the lower bound for the classical model only holds when the advantage of the protocol (over a random guess) is more than ≈ 1/√N, that is, the success probability is larger than ≈ 1/2 + 1/√N. This is unavoidable as ≈ 1/√N is the correlation between two coordinates of an input that is sampled from the Forrelation distribution, and hence there are simple classical protocols that achieve advantage ≈ 1/√N, in all these models. To achieve separations when the classical protocol has smaller advantage, we study in this work the xor of k independent copies of (a variant of) the Forrelation function (where k≪ N). We prove a very general result that shows that any family of Boolean functions that is closed under restrictions, whose Fourier mass at level 2k is bounded by α^k (that is, the sum of the absolute values of all Fourier coefficients at level 2k is bounded by α^k), cannot compute the xor of k independent copies of the Forrelation function with advantage better than O((α^k)/(N^{k/2})). This is a strengthening of a result of [Eshan Chattopadhyay et al., 2019], that gave a similar statement for k = 1, using the technique of [Ran Raz and Avishay Tal, 2019]. We give several applications of our result. In particular, we obtain the following separations: Quantum versus Classical Communication Complexity. We give the first example of a partial Boolean function that can be computed by a simultaneous-message quantum protocol with communication complexity polylog(N) (where Alice and Bob also share polylog(N) EPR pairs), and such that, any classical randomized protocol of communication complexity at most õ(N^{1/4}), with any number of rounds, has quasipolynomially small advantage over a random guess. Previously, only separations where the classical protocol has polynomially small advantage were known between these models [Dmitry Gavinsky, 2016; Uma Girish et al., 2021]. Quantum Query Complexity versus Bounded Depth Circuits. We give the first example of a partial Boolean function that has a quantum query algorithm with query complexity polylog(N), and such that, any constant-depth circuit of quasipolynomial size has quasipolynomially small advantage over a random guess. Previously, only separations where the constant-depth circuit has polynomially small advantage were known [Ran Raz and Avishay Tal, 2019].
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 f1,…,fk:F2n→{−1,1}, the goal is to estimate Tk=N−1Tr(Df1H⊗n⋯DfkH⊗n), where Dfi=diag(fi(x):x∈F2n), H is the Hadamard gate, and N=2n. 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 Ω(N1−2/k).
We establish a nearly complete characterization of the nonadaptive classical query complexity of cyclic Forrelation. For even k≥4, we prove a tight bound of Θk(N1−2/k) for constant additive error. For odd k≥5, we obtain a lower bound of Ωk(N1−2/(k−1)) and an upper bound of Ok(N1−2/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/⌊k/2⌋, 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(N1/4) for every k≥4.
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.
We study the extremal Forrelation problem, where, provided with oracle access to Boolean functions f and g promised to satisfy either forr(f,g)=1 or 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 one quantum query; yet, Girish and Servedio (ITCS 2026) recently showed this problem requires Ω(2n/4) classical queries, and conjectured the optimal bound to be Θ(2n/2). By generalizing their construction, we build on their result and prove a non-adaptive lower bound of Ω(2(1/2−o(1))n), which matches the conjectured lower bound up to a vanishing constant in the exponent.
The 2-Forrelation problem provides an optimal separation between classical and quantum query complexity and is also the problem used for separating BQP and PH relative to an oracle. A natural question is therefore to ask what are the minimal quantum resources needed to solve this problem. We show that 2-Forrelation can be solved using Instantaneous Quantum Polynomial-time (IQP) circuits, a restricted model of quantum computation in which all gates commute. Concretely, signed 2-Forrelation can be solved by a classical random choice between two one-query IQP circuits, while the absolute-value variant uses two independent executions of this randomized procedure. This answers a recent open question of Girish (STOC 2026) on the power of commuting quantum computations. For the Raz-Tal distribution, this randomization is unnecessary. We use this to show that there is an oracle O such that IQPO⊆PHO, strengthening the result of Raz and Tal (STOC 2019). It also yields an oracle separation between IQP and DQC1. We prove Fourier growth bounds for multi-query IQP circuits, including bounds in terms of the size of their accepting set. Our results suggest a possible route toward decision-based quantum advantage within the restricted IQP model. The key ingredient is an algebraic identity of the quadratic function Q(x)=∑i<jxixj that allows extracting inner-product phases within an IQP circuit.
The Forrelation problem, introduced by Aaronson [A10] and Aaronson and Ambainis [AA15], is a well studied problem in the context of separating quantum and classical models. Variants of this problem were used to give exponential separations between quantum and classical query complexity [A10, AA15]; quantum query complexity and bounded-depth circuits [RT19]; and quantum and classical communication complexity [GRT19]. In all these separations, the lower bound for the classical model only holds when the advantage of the protocol (over a random guess) is more than ≈1/N, that is, the success probability is larger than ≈1/2+1/N. To achieve separations when the classical protocol has smaller advantage, we study in this work the XOR of k independent copies of the Forrelation function (where k≪N). We prove a very general result that shows that any family of Boolean functions that is closed under restrictions, whose Fourier mass at level 2k is bounded by αk, cannot compute the XOR of k independent copies of the Forrelation function with advantage better than O(Nk/2αk). This is a strengthening of a result of [CHLT19], that gave a similar result for k=1, using the technique of [RT19]. As an application of our result, we give the first example of a partial Boolean function that can be computed by a simultaneous-message quantum protocol of cost $\mbox{polylog}(N)$ (when players share $\mbox{polylog}(N)$ EPR pairs), however, any classical interactive randomized protocol of cost at most o~(N1/4), has quasipolynomially small advantage over a random guess. We also give the first example of a partial Boolean function that has a quantum query algorithm of cost $\mbox{polylog}(N)$, and such that, any constant-depth circuit of quasipolynomial size has quasipolynomially small advantage over a random guess.