IQP circuits for 2-Forrelation

[2604.15248] IQP circuits for 2-Forrelation Skip to main content Search Submit Donate Log in Quantum Physics arXiv:2604.15248 (quant-ph) [Submitted on 16 Apr 2026 ( v1 ), last revised 2 Oct 2026 (this version, v3)] Title: IQP circuits for 2-Forrelation Authors: Quentin Buzet , André Chailloux View PDF 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, 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. Comments: 30 pages, revised and strengthened Fourier growth section Subjects: Quantum Physics (quant-ph) ; Computational Complexity (cs.CC) Cite as: arXiv:2604.15248 [quant-ph] (or arXiv:2604.15248v3 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2604.15248 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Quentin Buzet [ view email ] [v1] Thu, 16 Apr 2026 17:24:25 UTC (23 KB) [v2] Mon, 24 Aug 2026 22:05:09 UTC (26 KB) [v3] Fri, 2 Oct 2026 17:54:17 UTC (28 KB) Full-text links: Access Paper: View PDF TeX Source view license Current browse context: quant-ph < prev | next > new | recent | 2026-04 Change to browse by: cs cs.CC References & Citations INSPIRE HEP SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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