Parity $\notin$ QAC0 $\iff$ QAC0 is Fourier-Concentrated
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Quantum Physics arXiv:2604.02793 (quant-ph) [Submitted on 3 Apr 2026] Title:Parity $\notin$ QAC0 $\iff$ QAC0 is Fourier-Concentrated Authors:Lucas Gretta, Meghal Gupta, Malvika Raj Joshi View a PDF of the paper titled Parity $\notin$ QAC0 $\iff$ QAC0 is Fourier-Concentrated, by Lucas Gretta and 2 other authors View PDF Abstract:A major open problem in understanding shallow quantum circuits (QAC$^0$) is whether they can compute Parity. We show that this question is solely about the Fourier spectrum of QAC$^0$: any QAC$^0$ circuit with non-negligible high-level Fourier mass suffices to exactly compute PARITY in QAC$^0$. Thus, proving a quantum analog of the seminal LMN theorem for AC$^0$ is necessary to bound the quantum circuit complexity of PARITY. In the other direction, LMN does not fully capture the limitations of AC$^0$. For example, despite MAJORITY having $99\%$ of its weight on low-degree Fourier coefficients, no AC$^0$ circuit can non-trivially correlate with it. In contrast, we provide a QAC$^0$ circuit that achieves $(1-o(1))$ correlation with MAJORITY, establishing the first average-case decision separation between AC$^0$ and QAC$^0$. This suggests a uniquely quantum phenomenon: unlike in the classical setting, Fourier concentration may largely characterize the power of QAC$^0$. PARITY is also known to be equivalent in QAC$^0$ to inherently quantum tasks such as preparing GHZ states to high fidelity. We extend this equivalence to a broad class of state-synthesis tasks. We demonstrate that existing metrics such as trace distance, fidelity, and mutual information are insufficient to capture these states and introduce a new measure, felinity. We prove that preparing any state with non-negligible felinity, or derived states such as poly(n)-weight Dicke states, implies PARITY $\in$ QAC$^0$. Subjects: Quantum Physics (quant-ph); Computational Complexity (cs.CC) Cite as: arXiv:2604.02793 [quant-ph] (or arXiv:2604.02793v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2604.02793 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Malvika Raj Joshi [view email] [v1] Fri, 3 Apr 2026 06:59:01 UTC (45 KB) Full-text links: Access Paper: View a PDF of the paper titled Parity $\notin$ QAC0 $\iff$ QAC0 is Fourier-Concentrated, by Lucas Gretta and 2 other authorsView PDFTeX Source view license Current browse context: quant-ph new | recent | 2026-04 Change to browse by: cs cs.CC References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: 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?)