[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-290472-105":59,"doc-detail-290472-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","provably-trainable-rotationally-equivariant-quantum-machine-learning","Provably Trainable Rotationally Equivariant Quantum Machine Learning","","Quantum machine learning can outperform classical methods, yet training is often hindered by barren plateaus where gradients vanish exponentially as qubit count grows. This work introduces a family of rotationally equivariant quantum machine learning models based on the quantum Fourier transform. Using Lie-algebraic results, the paper proves that a subset of the proposed models avoids barren plateaus. Empirically, the models are benchmarked on simulated scanning tunnelling microscope images, showing dramatic gains over generic fully expressible counterparts.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/provably-trainable-rotationally-equivariant-quantum-machine-learning/290472/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/provably-trainable-rotationally-equivariant-quantum-machine-learning/290472.png","ImageObject",300,407,{"name":92,"@type":93},"Violet","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-09-17",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",13,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What problem does the paper address in quantum machine learning training?","Question",{"text":113,"@type":114},"It addresses barren plateaus, large training-landscape regions where cost-function gradients vanish exponentially with the number of qubits, making models effectively untrainable.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How do the proposed models differ from generic QML models?",{"text":118,"@type":114},"They are rotationally equivariant and are built using a quantum Fourier transform strategy that constrains the search space to a symmetry-respecting subset of Hilbert space.",{"name":120,"@type":111,"acceptedAnswer":121},"What proof and numerical evidence support the model’s trainability and performance?",{"text":122,"@type":114},"The paper uses Lie-algebraic techniques to prove (for a subset of models) freedom from barren plateaus and numerically benchmarks the approach on simulated STM images, where it significantly outperforms generic fully expressible counterparts.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},290472,1789641767,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":39,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":130,"read_time":46,"preview_page_count":4},4398048950312,"https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908","Provably Trainable Rotationally Equivariant Quantum Machine Learning  \nMaxwell T. West, 1, ∗ Jamie Heredge, 1 Martin Sevior, 1 and Muhammad Usman 1, 2,†  \n1 School of Physics, The University of Melbourne, Parkville, 3010, VIC, Australia  \n2 Data61, CSIRO, Clayton, 3168, VIC, Australia  \n14 Jan 2024  \nExploiting the power of quantum computation to realise superior machine learning algorithms has been a major research focus of recent years, but the prospects of quantum machine learning (QML) remain dampened by considerable technical challenges. A particularly significant issue is that generic QML models suffer from so-called barren plateaus in their training landscapes – large regions where cost function gradients vanish exponentially in the number of qubits employed, rendering large models effectively untrainable. A leading strategy for combating this effect is to build problem-specific models which take into account the symmetries of their data in order to focus on a smaller, relevant subset of Hilbert space. In this work, we introduce a family of rotationally equivariant QML models built upon the quantum Fourier transform, and leverage recent insights from the Lie-algebraic study of QML models to prove that (a subset of) our models do not exhibit barren plateaus. In addition to our analytical results we numerically test our rotationally equivariant modelson a dataset of simulated scanning tunnelling microscope images of phosphorus impurities in silicon, where rotational symmetry naturally arises, and find that they dramatically outperform their generic counterpartsin practice.  \narXiv :23 1 1 .05873v3  \nAmong the most important discoveries in quantum machine learning (QML) has been the existence of barren plateaus in the training landscapes of variational quantum models [12– 15], reminiscent of the vanishing gradients which plagued early neural networks [16] . Barren plateaus have come to be understood to be linked to the expressibility of the ansatz being optimised [14, 17–19], with generic, highly expressible models rendered effectively untrainable as the number of qubits increases. This suggests that the architectures of variational QML models should be tweaked on a per-problem basis, with domain specific knowledge employed to construct ansatze with inductive biases compatible with the optimal solution. A natural approach along these lines is to build models that explicitly respect the symmetries of their data, so-called geometric quantum machine learning (GQML) [20– 30] . For example, GQML has been used to classify images with spatial symmetries [28, 29], and more general data which satisfies permutation symmetries [23, 24 , 31] . By constraining the search space of the optimisation procedure, such symmetry-informed models have in a few instances been proved to be free of barren plateaus [23, 32] . Despite a general theory of the trainability of QML models starting to emerge [18, 19], however, the number of explicit examples of provably trainable models that have been developed and benchmarked on real data to test their performance in practice remains limited.  \n∗ [westm2@student.unimelb.edu.au](westm2@student.unimelb.edu.au)[ ](westm2@student.unimelb.edu.au)† [musman@unimelb.edu.au](musman@unimelb.edu.au)  \nIn this work, we introduce a family of rotationally equivariant QML models for classifying two dimensional data with labels that are invariant to rotations by 2π/2k for k ∈ N. In particular, these models are applicable to image data, the consideration of the symmetries of which in the GQML context has previously been limited to reflections [28] and 90° rotations [29] . Our proposed architectures allow for the fraction of resources allocated to the processing of the radial and angular degrees of freedom to be controlled, enabling for an interpolation between a model with no explicit notion of rotational symmetry (k = 0) to models which respect continuous rotations (k → ∞ ) . This is facilitated by choosing an encoding ","cbCaikR0RvG6uiIb","https://ap.wps.com/l/cbCaikR0RvG6uiIb","pdf",15829269,"English","# Introduction\n## Barren plateaus and symmetry-informed models\n## Rotationally equivariant model construction\n# Trainability analysis\n## Lie-algebraic proof for avoiding barren plateaus\n# Experimental benchmarking\n## STM image classification with rotational symmetry","[{\"question\":\"What problem does the paper address in quantum machine learning training?\",\"answer\":\"It addresses barren plateaus, large training-landscape regions where cost-function gradients vanish exponentially with the number of qubits, making models effectively untrainable.\"},{\"question\":\"How do the proposed models differ from generic QML models?\",\"answer\":\"They are rotationally equivariant and are built using a quantum Fourier transform strategy that constrains the search space to a symmetry-respecting subset of Hilbert space.\"},{\"question\":\"What proof and numerical evidence support the model’s trainability and performance?\",\"answer\":\"The paper uses Lie-algebraic techniques to prove (for a subset of models) freedom from barren plateaus and numerically benchmarks the approach on simulated STM images, where it significantly outperforms generic fully expressible counterparts.\"}]","Provably Trainable Rotationally Equivariant Quantum Machine Learning | PDF"]