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.

翻译：2-Forrelation问题提供了经典查询复杂度与量子查询复杂度之间的最优分离，同时也是用于相对化预言机分离$\mathsf{BQP}$与$\mathsf{PH}$的问题。因此，一个自然的问题是求解该问题所需的最小量子资源。我们证明，2-Forrelation问题可通过瞬时量子多项式时间（$\mathsf{IQP}$）电路求解——这是一种所有门都可交换的受限量子计算模型。具体而言，两个具有两次量子查询和高效经典处理的$\mathsf{IQP}$电路就足以求解该问题。对于2-Forrelation的有符号变体，甚至单个$\mathsf{IQP}$电路和单次查询即可。这回答了Girish（arXiv:2510.06385）最近关于交换量子计算能力的开放问题。我们利用该结果证明，相对化于预言机$O$时$(\mathsf{BPP}^{\mathsf{IQP}})^O \not\subseteq \mathsf{PH}^O$，从而强化了Raz和Tal（STOC 2019）的结论。我们的结果表明，$\mathsf{IQP}$电路可用于经典困难判定问题，因此为展示$\mathsf{IQP}$电路的量子优势提供了新途径，避免了与采样任务相关的验证难题。我们还基于$\mathsf{IQP}$电路可接受集的大小证明了其傅里叶增长界。关键要素是二次函数$Q(x) = \sum_{i < j} x_ix_j$的代数恒等式，该恒等式允许在$\mathsf{IQP}$电路内提取内积相位。