We achieve essentially the largest possible separation between quantum and classical query complexities. We do so using a property-testing problem called Forrelation, where one needs to decide whether one Boolean function is highly correlated with the Fourier transform of a second function. This problem can be solved using 1 quantum query, yet we show that any randomized algorithm needs $\Omega(\sqrt{N}/\log N)$ queries (improving an $\Omega(N^{1/4})$ lower bound of Aaronson). Conversely, we show that this 1 versus $\widetilde{\Omega}(\sqrt{N})$ separation is optimal: indeed, any $t$ -query quantum algorithm whatsoever can be simulated by an $O(N^{1-1/2t})$ -query randomized algorithm. Thus, resolving an open question of Buhrman et al. [ SIAM J. Comput., 37 (2008), pp. 1387--1400] from 2002, there is no partial Boolean function whose quantum query complexity is constant and whose randomized query complexity is linear. We conjecture that a natural generalization of Forrelation achieves the optimal $t$ versus $\Omega(N^{1-1/2t})$ separation for all $t$ . As a bonus, we show that this generalization is ${BQP}$ -complete. This yields what is arguably the simplest ${BQP}$ -complete problem yet known and gives a second sense in which Forrelation “captures the maximum power of quantum computation.”We achieve essentially the largest possible separation between quantum and classical query complexities. We do so using a property-testing problem called Forrelation, where one needs to decide whether one Boolean function is highly correlated with the Fourier transform of a second function. This problem can be solved using 1 quantum query, yet we show that any randomized algorithm needs $\Omega(\sqrt{N}/\log N)$ queries (improving an $\Omega(N^{1/4})$ lower bound of Aaronson). Conversely, we show that this 1 versus $\widetilde{\Omega}(\sqrt{N})$ separation is optimal: indeed, any $t$-query quantum algorithm whatsoever can be simulated by an $O(N^{1-1/2t})$-query randomized algorithm. Thus, resolving an open question of Buhrman et al. [ SIAM J. Comput., 37 (2008), pp. 1387--1400] from 2002, there is no partial Boolean function whose quantum query complexity is constant and whose randomized query complexity is linear. We conjecture that a natural generalization of Forrelation achieves the optimal $t$ versus $\Omega(N^{1-1/2t})$ separation for all $t$. As a bonus, we show that this generalization is ${BQP}$-complete. This yields what is arguably the simplest ${BQP}$-complete problem yet known and gives a second sense in which Forrelation “captures the maximum power of quantum computation.”

arXiv:1411.5729SIAM Journal on ComputingJan 1, 201853 citations

Towards Optimal Separations between Quantum and Randomized Query Complexities

Avishay Tal

The query model offers a concrete setting where quantum algorithms are provably superior to randomized algorithms. Beautiful results by Bernstein-Vazirani, Simon, Aaronson, and others presented partial Boolean functions that can be computed by quantum algorithms making much fewer queries compared to their randomized analogs. To date, separations of O(1) vs. √N between quantum and randomized query complexities remain the state-of-the-art (where N is the input length), leaving open the question of whether O(1) vs. N1/2+Ω(1)separations are possible? We answer this question in the affirmative. Our separating problem is a variant of the Aaronson-Ambainis k-fold Forrelation problem. We show that our variant: 1)Can be solved by a quantum algorithm making 2O(k)queries to the inputs. 2)Requires at least ~Ω(N2(k-1)/(3k-1)) queries for any randomized algorithm. For any constant , this gives a O(1) vs. N1/2-εseparation between the quantum and randomized query complexities of partial Boolean functions. Our proof is Fourier analytical and uses new bounds on the Fourier spectrum of classical decision trees, which could be of independent interest. Looking forward, we conjecture that the Fourier bounds could be further improved in a precise manner, and show that such conjectured bounds imply optimal O(1) vs. N1-εseparations between the quantum and randomized query complexities of partial Boolean functions.The query model offers a concrete setting where quantum algorithms are provably superior to randomized algorithms. Beautiful results by Bernstein-Vazirani, Simon, Aaronson, and others presented partial Boolean functions that can be computed by quantum algorithms making much fewer queries compared to their randomized analogs. To date, separations of O(1) vs. √N between quantum and randomized query complexities remain the state-of-the-art (where N is the input length), leaving open the question of whether O(1) vs. N1/2+Ω(1)separations are possible? We answer this question in the affirmative. Our separating problem is a variant of the Aaronson-Ambainis k-fold Forrelation problem. We show that our variant: 1)Can be solved by a quantum algorithm making 2O(k)queries to the inputs. 2)Requires at least ~Ω(N2(k-1)/(3k-1)) queries for any randomized algorithm. For any constant , this gives a O(1) vs. N1/2-εseparation between the quantum and randomized query complexities of partial Boolean functions. Our proof is Fourier analytical and uses new bounds on the Fourier spectrum of classical decision trees, which could be of independent interest. Looking forward, we conjecture that the Fourier bounds could be further improved in a precise manner, and show that such conjectured bounds imply optimal O(1) vs. N1-εseparations between the quantum and randomized query complexities of partial Boolean functions.

arXiv:1912.12561Annual Symposium on Foundations of Computer ScienceNov 1, 202025 citations

Classical algorithms for Forrelation

Sergey B. Bravyi, David Gosset, Daniel Grier +1 more

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 $O(n2^{n/2})$ . 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 $O(2^{n(1-1/2t)})$ 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.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 $O(n2^{n/2})$. 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 $O(2^{n(1-1/2t)})$ 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.

arXiv:2102.06963Feb 13, 20211 citation

Exponential quantum-classical query separations for cyclic Forrelation

Yuxin Zhang, Changpeng Shao

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.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.

quant-pharXiv:2610.41694Nov 2, 2026

Lower Bounds for XOR of Forrelations.

Uma Girish, Ran Raz, Wei Zhan

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 o(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].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 o(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].

Electronic colloquium on computational complexityJan 1, 20201 citation

IQP circuits for 2-Forrelation

Quentin Buzet, André Chailloux

The $2$ -Forrelation problem provides an optimal separation between classical and quantum query complexity and is also the problem used for separating $\mathsf{BQP}$ and $\mathsf{PH}$ 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 ( $\mathsf{IQP}$ ) 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 $\mathsf{IQP}$ 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 $\mathsf{IQP}^O \not\subseteq \mathsf{PH}^O$ , strengthening the result of Raz and Tal (STOC 2019). It also yields an oracle separation between $\mathsf{IQP}$ and $\mathsf{DQC}_1$ . We prove Fourier growth bounds for multi-query $\mathsf{IQP}$ 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 $\mathsf{IQP}$ model. The key ingredient is an algebraic identity of the quadratic function $Q(x) = \sum_{i < j} x_ix_j$ that allows extracting inner-product phases within an $\mathsf{IQP}$ circuit.The $2$-Forrelation problem provides an optimal separation between classical and quantum query complexity and is also the problem used for separating $\mathsf{BQP}$ and $\mathsf{PH}$ 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 ($\mathsf{IQP}$) 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 $\mathsf{IQP}$ 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 $\mathsf{IQP}^O \not\subseteq \mathsf{PH}^O$, strengthening the result of Raz and Tal (STOC 2019). It also yields an oracle separation between $\mathsf{IQP}$ and $\mathsf{DQC}_1$. We prove Fourier growth bounds for multi-query $\mathsf{IQP}$ 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 $\mathsf{IQP}$ model. The key ingredient is an algebraic identity of the quadratic function $Q(x) = \sum_{i < j} x_ix_j$ that allows extracting inner-product phases within an $\mathsf{IQP}$ circuit.

quant-pharXiv:2604.15248Apr 17, 20260 citations

\Omega(\sqrt{N}/\log N)

queries (improving an

\Omega(N^{1/4})

lower bound of Aaronson). Conversely, we show that this 1 versus

\widetilde{\Omega}(\sqrt{N})

separation is optimal: indeed, any

t

-query quantum algorithm whatsoever can be simulated by an

O(N^{1-1/2t})

-query randomized algorithm. Thus, resolving an open question of Buhrman et al. [ SIAM J. Comput., 37 (2008), pp. 1387--1400] from 2002, there is no partial Boolean function whose quantum query complexity is constant and whose randomized query complexity is linear. We conjecture that a natural generalization of Forrelation achieves the optimal

t

versus

\Omega(N^{1-1/2t})

separation for all

t

. As a bonus, we show that this generalization is

{BQP}

-complete. This yields what is arguably the simplest

{BQP}

-complete problem yet known and gives a second sense in which Forrelation “captures the maximum power of quantum computation.”

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(n2^{n/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(2^{n(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.

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.

The

2

-Forrelation problem provides an optimal separation between classical and quantum query complexity and is also the problem used for separating

\mathsf{BQP}

and

\mathsf{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 (

\mathsf{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

\mathsf{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

\mathsf{IQP}^O \not\subseteq \mathsf{PH}^O

, strengthening the result of Raz and Tal (STOC 2019). It also yields an oracle separation between

\mathsf{IQP}

and

\mathsf{DQC}_1

. We prove Fourier growth bounds for multi-query

\mathsf{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

\mathsf{IQP}

model. The key ingredient is an algebraic identity of the quadratic function

Q(x) = \sum_{i < j} x_ix_j

that allows extracting inner-product phases within an

\mathsf{IQP}

circuit.