IQP circuits for 2-Forrelation

Abstract: 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, two $\mathsf{IQP}$ circuits with two quantum queries and efficient classical processing suffice. For the signed variant of 2-Forrelation, even a single $\mathsf{IQP}$ circuit and query suffices. This answers a recent open question of Girish (arXiv:2510.06385) on the power of commuting quantum computations. We use this to show that $(\mathsf{BPP}^{\mathsf{IQP}})^O \not\subseteq \mathsf{PH}^O$ relative to an oracle $O$, strengthening the result of Raz and Tal (STOC 2019). Our results show that $\mathsf{IQP}$ circuits can be used for classically hard decision problems, thus providing a new route for showing quantum advantage with $\mathsf{IQP}$ circuits, avoiding the verification difficulties associated with sampling tasks. We also prove Fourier growth bounds for $\mathsf{IQP}$ circuits in terms of the size of their accepting set. 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 paper's main contribution is the construction of IQP circuits that efficiently solve both signed and absolute value variants of the 2-Forrelation problem.
• It employs algebraic identities and Fourier growth bounds to achieve quantum-classical query separations with precise acceptance probabilities.
• The work has significant complexity-theoretic implications, strengthening oracle separations by demonstrating IQP's practical viability in restricted quantum models.

IQP Circuits for 2-Forrelation: Complexity, Construction, and Implications

Introduction

The paper "IQP circuits for 2-Forrelation" (2604.15248) investigates the quantum resources required to solve the 2-Forrelation problem, a canonical oracle task exhibiting maximal quantum-classical query separation and underlying BQP vs PH oracle separations. The authors demonstrate that 2-Forrelation can be solved using Instantaneous Quantum Polynomial-time (IQP) circuits, which only employ commuting gates—substantially weaker than generic BQP. Furthermore, the work provides explicit IQP constructions for both signed and absolute value variants, delivers new Fourier growth bounds for IQP circuits, and uses these results to strengthen complexity-theoretic oracle separations.

2-Forrelation and Quantum-Classical Separation

2-Forrelation consists of distinguishing whether a forrelation quantity $\Phi(f,g)$, computed between two Boolean functions $f,g : \{0,1\}^n \to \{-1,+1\}$, is large or small. Quantum circuits achieve this task with $O(1)$ queries, while randomized classical algorithms require $\Theta(2^{n/2})$ queries [AA15, SSW21]. This achieves maximal known quantum-vs-classical separation in query complexity. Moreover, Raz and Tal [RT22] used 2-Forrelation as a basis to construct an oracle separation between BQP and the Polynomial Hierarchy (PH).

A critical research direction is identifying the minimal quantum resources required for these exponential speedups—specifically, whether restricted quantum models suffice.

IQP Circuits and Computational Power

IQP circuits are characterized by gates diagonal in the $X$ basis (equivalently, circuits of the form $H^{\otimes m} D H^{\otimes m}$, with $D$ diagonal in $Z$), making all gates commute. IQP circuits are easier to implement and have simpler error correction protocols in practical settings, motivating their study for quantum advantage demonstrations [BJS11, BMS16].

Previous works established IQP sampling hardness under complexity assumptions, and that IQP sits strictly between DQC$_1$ ("one clean qubit") and $BQP$ [JM24]. The open problem was whether 2-Forrelation could be solved efficiently by IQP circuits, or whether Fourier growth obstructions would apply [Gir25].

Explicit IQP Computations for 2-Forrelation

The central result is the construction of IQP computations that solve both the signed and absolute value variants of 2-Forrelation:

• For signed 2-Forrelation, a single IQP circuit on $f,g : \{0,1\}^n \to \{-1,+1\}$0 qubits, with one quantum oracle query, achieves acceptance probability $f,g : \{0,1\}^n \to \{-1,+1\}$1 (odd $f,g : \{0,1\}^n \to \{-1,+1\}$2) or $f,g : \{0,1\}^n \to \{-1,+1\}$3 (even $f,g : \{0,1\}^n \to \{-1,+1\}$4).
• For absolute value (i.e., $f,g : \{0,1\}^n \to \{-1,+1\}$5), two IQP circuit calls suffice, yielding acceptance probability $f,g : \{0,1\}^n \to \{-1,+1\}$6 (odd $f,g : \{0,1\}^n \to \{-1,+1\}$7) or $f,g : \{0,1\}^n \to \{-1,+1\}$8 (even $f,g : \{0,1\}^n \to \{-1,+1\}$9).

The construction leverages an algebraic identity involving the quadratic form $O(1)$0, allowing phase extraction corresponding to $O(1)$1 within IQP circuits. By carefully crafting the diagonal operator $O(1)$2 and the acceptance set $O(1)$3, and through classical pre- and post-processing, the circuits realize the desired computational bias.

The impossibility of realizing the forrelation exactly for all $O(1)$4 using only IQP circuits is circumvented by splitting the computation into odd/even Hamming weight subproblems, then combining the results via classical randomization.

Oracle Separations and Complexity Consequences

By solving 2-Forrelation via IQP circuits, the paper extends and strengthens Raz-Tal's oracle separation: $O(1)$5. Raz-Tal's construction depends on a distribution $O(1)$6 making $O(1)$7 invertible polynomial in $O(1)$8, while for uniform draws it is negligible. The IQP-based procedure achieves distinguishing advantage $O(1)$9 with two queries, while PH algorithms require $\Theta(2^{n/2})$0 queries. Amplification via $\Theta(2^{n/2})$1 repetitions yields constant advantage in $\Theta(2^{n/2})$2, with all queries non-adaptive.

IQP circuits thus enable classically hard decision problems, without relying on sampling hardness, and with verifiable outcomes—a notable practical implication.

Fourier Growth Bounds and Accepting Set Characterization

The acceptance probability of any $\Theta(2^{n/2})$3-query IQP algorithm is a degree-$\Theta(2^{n/2})$4 polynomial in the oracle bits, with all higher-degree terms (degree $\Theta(2^{n/2})$5) vanishing. The level-$\Theta(2^{n/2})$6 Fourier growth $\Theta(2^{n/2})$7 quantifies the "weight" of degree-$\Theta(2^{n/2})$8 Fourier coefficients. The paper proves:

• For single-query IQP circuits, $\Theta(2^{n/2})$9, where $X$0 is the size of the accepting set. Therefore, $X$1 for any restriction.
• To achieve constant advantage for 2-Forrelation, any IQP circuit must have an accepting set of size $X$2; the constructed circuits attain $X$3, which is tight.

This sharply contrasts with BQP circuits, where the accepting set can have minimal dimension (e.g., accepting only $X$4), highlighting structural constraints within IQP.

Furthermore, the degree restriction implies IQP circuits cannot solve 3-Forrelation; this aligns with Girish's recent results proving 3-Forrelation is not efficiently computable in $X$5, and IQP is weaker than $X$6.

Practical and Theoretical Implications

IQP circuits, deployable on near-term hardware, can now be used to realize decision problems with provable quantum advantage, bypassing classical verification bottlenecks inherent in sampling tasks [AC17, Kah23]. By picking efficiently implementable oracles, experiments can be designed where the quantum outcome is verifiable classically, yet due to complexity-theoretic underpinnings, quantum circuits are required for efficiency.

Beyond practical implications, these results further resolve complexity-theoretic landscapes: IQP computations occupy a position outside PH even in oracle settings, but below the full power of BQP. The approach lays the foundation for probing the limitations and capabilities of commuting circuits for various oracle tasks.

Possible future directions include: exploring quantum advantage for higher-order Forrelation tasks, optimizing IQP architectures for practical deployment, and generalizing the algebraic construction techniques to other non-complete BQP problems.

Conclusion

This work unambiguously establishes that IQP circuits suffice for solving 2-Forrelation with constant quantum-classical separation, both for signed and unsigned variants. The construction leverages quadratic form identities, and Fourier analytic techniques to characterize the acceptance polynomial structure. The practical and theoretical implications include strengthened complexity separations, and new avenues for realizing and verifying quantum advantage with restricted quantum models. The explicit Fourier bounds and accepting set analyses contribute to understanding structural constraints in instantaneous computation, and delimit the reach of IQP paradigms within quantum complexity theory.

References:

(2604.15248) Quentin Buzet, André Chailloux, "IQP circuits for 2-Forrelation" [AA15], [SSW21], [RT22], [BJS11], [BMS16], [JM24], [Gir25], [AC17], [Kah23]