[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-125971-en":3,"doc-seo-125971-105":31,"detail-sidebar-cat-0-en-105":93},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},125971,687207024478,"Liam","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Geometric Quantum Machine Learning with Horizontal Quantum Gates","Geometric Quantum Machine Learning often enforces symmetry by restricting variational circuits to be equivariant, but strict equivariance can severely limit expressivity, especially for continuous symmetries. This work introduces a symmetry-informed alternative using homogeneous spaces and horizontal quantum gates, which transform only in directions orthogonal to symmetry-generated ones. The proposed gates are shown to be far more expressive than equivariant gates, enabling tasks like ground-state search for SU(2)-symmetric models with unknown spin sectors. For gates from symmetric spaces, efficient decompositions follow from the KAK theorem, and a special subclass achieves quadratic parameter reduction for generic problems.","arXiv :2406 .04418v1 [ quant-ph] 6 Jun 2024  \nGeometric Quantum Machine Learning with Horizontal Quantum Gates  \nRoeland Wiersema, 1, 2, 3 Alexander F. Kemper,4 Bojko N. Bakalov,5 and Nathan Killoran3  \n1 Vector Institute, MaRS Centre, Toronto, Ontario, M5G 1M1, Canada  \n2 Department of Physics and Astronomy, University of Waterloo, Ontario, N2L 3G1, Canada  \n3 Xanadu, Toronto, ON, M5G 2C8, Canada  \n4 Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA  \n5 Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695, USA (Dated: May 30, 2024)  \nIn the current framework of Geometric Quantum Machine Learning, the canonical method for constructing a variational ansatz that respects the symmetry of some group action is by forcing the circuit to be equivariant, i.e., to commute with the action of the group. This can, however, be an overzealous constraint that greatly limits the expressivity of the circuit, especially in the case of continuous symmetries. We propose an alternative paradigm for the symmetry-informed construction of variational quantum circuits, based on homogeneous spaces, relaxing the overly stringent requirement of equivariance. We achieve this by introducing horizontal quantum gates, which only transform the state with respect to the directions orthogonal to those of the symmetry. We show that horizontal quantum gates are much more expressive than equivariant gates, and thus can solve problems that equivariant circuits cannot. For instance, a circuit comprised of horizontal gates can find the ground state of an SU(2)-symmetric model where the ground state spin sector is unknown–a task where equivariant circuits fall short. Moreover, for a particular subclass of horizontal gates based on symmetric spaces, we can obtain efficient circuit decompositions for our gates through the KAK theorem. Finally, we highlight a particular class of horizontal quantum gates that behave similarly to general SU(4) gates, while achieving a quadratic reduction in the number of parameters for a generic problem.  \nI. INTRODUCTION  \nSymmetries play a critical role in many scientific endeavours. In physics, if we can identify the symmetries of a system, then we can incorporate them into the equations we use to describe the system. This can help us simplify and solve otherwise intractable problems, identify important physical quantities (quantum numbers, conserved currents, invariants), and deepen our understanding of the system under study. In machine learning, symmetries have also proven extremely powerful [1] . Understanding the underlying geometric regularities of real-world data, and adapting machine learning models to account for it, enables us to overcome the dreaded curse of dimensionality. The most powerful modern machine learning models, such as convolutional networks, graph neural networks, and transformers, all have deep symmetry underpinnings.  \nIt is evident that quantum computing and quantum machine learning (QML) may also benefit greatly when we can identify and leverage symmetries. In recent years, the first forays into a study of Geometric QML have begun [2–5] . Notably, a standard recipe has emerged for“geometrizing” a QML model (specifically, a quantum circuit) under a known symmetry group. This process, called twirling, converts every gate (or layer) in the circuit into a new version that now has the property of equivariance under the symmetry [4, 6 , 7] . Combined with a final measurement which is invariant under the same symmetry, the modified circuit is now guaranteed to respect the symmetry, i.e., it gives the same output for all inputs differing only by a symmetry transformation.  \nEquivariance—essentially, the property where a gate commutes with group transformations—provides a natural mathematical condition for enforcing symmetries. It also presents a clear-cut recipe, based on twirling, for how to incorporate those symmetries into a circuit. How","cbCaie8QMb8racEY","https://ap.wps.com/l/cbCaie8QMb8racEY","pdf",1734195,5,1,18,"English","en",105,"# Introduction\n## Symmetry and geometric structure\n## Limitations of equivariance and twirling\n## Horizontal quantum gates and new paradigm","[{\"question\":\"Why is strict equivariance in variational quantum circuits often too restrictive?\",\"answer\":\"Because enforcing equivariance can greatly limit circuit expressivity and may even remove legitimate symmetry-respecting evolutions from the variational search space, leading to underparameterization and poorer performance.\"},{\"question\":\"What are horizontal quantum gates in Geometric Quantum Machine Learning?\",\"answer\":\"Horizontal quantum gates are defined to transform only along directions orthogonal to those generated by the symmetry. They operate effectively on symmetry-equivalence classes rather than individual state vectors.\"},{\"question\":\"How do horizontal gates improve expressivity compared with equivariant gates?\",\"answer\":\"They are shown to be much more expressive, allowing solutions to problems where equivariant circuits fail, such as finding the ground state of an SU(2)-symmetric model when the ground state spin sector is unknown.\"}]","Geometric Quantum Machine Learning with Horizontal Quantum Gates | PDF",1785902313,45,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":88,"head_meta":90,"extra_data":92,"updated_unix":29},"geometric-quantum-machine-learning-with-horizontal-quantum-gates","",{"@graph":37,"@context":87},[38,55,70],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":54},"https://docshare.wps.com/document/geometric-quantum-machine-learning-with-horizontal-quantum-gates/125971/",4,{"url":53,"name":13,"@type":56,"author":57,"headline":13,"publisher":59,"fileFormat":62,"inLanguage":24,"description":14,"dateModified":63,"datePublished":64,"encodingFormat":62,"isAccessibleForFree":65,"interactionStatistic":66},"DigitalDocument",{"name":9,"@type":58},"Person",{"url":42,"name":60,"@type":61},"DocShare","Organization","application/pdf","2026-08-23","2026-08-05",true,{"@type":67,"interactionType":68,"userInteractionCount":20},"InteractionCounter",{"@type":69},"ViewAction",{"@type":71,"mainEntity":72},"FAQPage",[73,79,83],{"name":74,"@type":75,"acceptedAnswer":76},"Why is strict equivariance in variational quantum circuits often too restrictive?","Question",{"text":77,"@type":78},"Because enforcing equivariance can greatly limit circuit expressivity and may even remove legitimate symmetry-respecting evolutions from the variational search space, leading to underparameterization and poorer performance.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"What are horizontal quantum gates in Geometric Quantum Machine Learning?",{"text":82,"@type":78},"Horizontal quantum gates are defined to transform only along directions orthogonal to those generated by the symmetry. They operate effectively on symmetry-equivalence classes rather than individual state vectors.",{"name":84,"@type":75,"acceptedAnswer":85},"How do horizontal gates improve expressivity compared with equivariant gates?",{"text":86,"@type":78},"They are shown to be much more expressive, allowing solutions to problems where equivariant circuits fail, such as finding the ground state of an SU(2)-symmetric model when the ground state spin sector is unknown.","https://schema.org",{"og:url":53,"og:type":89,"og:title":13,"og:site_name":60,"og:description":14},"article",{"robots":91,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":94},[95,99,103,107,111,116,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":96,"show_sort_weight":97,"slug":98},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":100,"show_sort_weight":101,"slug":102},"Literature",80,"literature",{"id":54,"doc_module":4,"doc_module_name":47,"category_name":104,"show_sort_weight":105,"slug":106},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":108,"show_sort_weight":109,"slug":110},"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":20,"slug":139},19,"General","general"]