Hierarchical Group-Extended Quantum Fourier Transform for Robust and Interpretable Quantum Feature Extraction

Hierarchical Group-Extended Quantum Fourier Transform for Robust and Interpretable Quantum Feature Extraction | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Hierarchical Group-Extended Quantum Fourier Transform for Robust and Interpretable Quantum Feature Extraction Chenhao Huang, Zijun Guo, Hongyang Ma, Xingkui Fan, Guoliang Shentu, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9025597/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 4 You are reading this latest preprint version Abstract The Quantum Fourier Transform (QFT) is a fundamental technique for image feature extraction in quantum systems. However, its global spectral representation lacks local structural control, limiting its robustness on current noisy intermediate-scale quantum (NISQ) devices. In this paper, we propose a $Z_{2^6}$-group-extended Quantum Fourier Transform ($Z_{2^6}$-GQFT) that integrates cyclic group theory and hierarchical spectral decomposition to enhance feature interpretability and noise resilience. Based on the Cartan–Eilenberg (CE) condition, a two-level group extension using the normal subgroup 2$Z_{2^6}$ and quotient group $Z_{2}$ is constructed, introducing twiddle factors for cross-scale phase coupling. This structure enables multi-scale frequency-domain representation within a hardware-efficient circuit. The proposed $Z_{2^6}$-GQFT achieves a mean fidelity of 0.890 ± 0.020 on the Origin Wukong-72 superconducting quantum processor—representing a 12% improvement over the standard QFT—while maintaining stability across repeated runs. Integration into a quantum convolutional neural network (QCNN) framework with Ising and Heisenberg ansatzes further verifies its advantages in convergence speed, training stability, and generalization. These results demonstrate that group-extended spectral encoding provides an algebraically interpretable and hardware-adaptable foundation for quantum feature extraction and quantum machine learning on NISQ hardware. Group-Extended Quantum Fourier Transform Quantum Image Processing Quantum Neural Network (QCNN) Noise-Resilient Quantum Computing Multiscale Spectral Representation Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers invited by journal 30 Mar, 2026 Editor assigned by journal 05 Mar, 2026 Submission checks completed at journal 05 Mar, 2026 First submitted to journal 03 Mar, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9025597","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":614669084,"identity":"090e502b-86f0-452a-8a46-66394f1f2782","order_by":0,"name":"Chenhao Huang","email":"","orcid":"","institution":"Qingdao University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Chenhao","middleName":"","lastName":"Huang","suffix":""},{"id":614669085,"identity":"db5bd338-9658-4ab1-925b-808c290f4f05","order_by":1,"name":"Zijun Guo","email":"","orcid":"","institution":"Qingdao University of 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However, its global spectral representation lacks local structural control, limiting its robustness on current noisy intermediate-scale quantum (NISQ) devices. In this paper, we propose a $Z_{2^6}$-group-extended Quantum Fourier Transform ($Z_{2^6}$-GQFT) that integrates cyclic group theory and hierarchical spectral decomposition to enhance feature interpretability and noise resilience. Based on the Cartan–Eilenberg (CE) condition, a two-level group extension using the normal subgroup 2$Z_{2^6}$ and quotient group $Z_{2}$ is constructed, introducing twiddle factors for cross-scale phase coupling. This structure enables multi-scale frequency-domain representation within a hardware-efficient circuit. The proposed $Z_{2^6}$-GQFT achieves a mean fidelity of 0.890 ± 0.020 on the Origin Wukong-72 superconducting quantum processor—representing a 12\\% improvement over the standard QFT—while maintaining stability across repeated runs. Integration into a quantum convolutional neural network (QCNN) framework with Ising and Heisenberg ansatzes further verifies its advantages in convergence speed, training stability, and generalization. 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