Parity $\notin$ QAC0 $\iff$ QAC0 is Fourier-Concentrated

Quantum Physics [Submitted on 3 Apr 2026] Parity \notin QAC0 \iff QAC0 is Fourier-Concentrated Lucas Gretta, Meghal Gupta, Malvika Raj Joshi 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 Submission history From: Malvika Raj Joshi [view email] [v1] Fri, 3 Apr 2026 06:59:01 UTC (45 KB) Access Paper: view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 Change to browse by: cs cs.CC References & Citations INSPIRE HEP NASA ADS 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 Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)