[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-124587-en":3,"doc-seo-124587-105":30,"detail-sidebar-cat-0-en-105":91},{"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":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},124587,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1784791008015729253",8,"Research & Report","Quaternion-based machine learning on topological quantum systems","Quaternion algebras are integrated into data analysis to classify two-dimensional Chern insulators using both unsupervised and supervised machine learning. Unsupervised classification applies principal component analysis to quaternion-transformed eigenstates to distinguish topological phases. Supervised learning adds a quaternion convolutional layer atop a conventional convolutional neural network, using quaternion-transformed configurations as inputs. The model classifies all distinct topological phases and generalizes to states with different distributions than training data, demonstrating advantages over conventional real-valued neural networks for topological phase classification.","arXiv :2209 . 14551v2 [ quant-ph] 5 May 2023  \nQuaternion-based machine learning on topological quantum systems  \nMin-Ruei Lin, 1, 􀀃 Wan-Ju Li, 1 and Shin-Ming Huang 1, 2, y  \n1 Department of Physics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan  \n2 Center of Crystal Research, National Sun Yat-sen University, Kaohsiung 80424, Taiwan (Dated: May 8, 2023)  \nTopological phase classi􀀌cations have been intensively studied via machine-learning techniques where di􀀋erent forms of the training data are proposed in order to maximize the information extracted from the systems of interests. Due to the complexity in quantum physics, advanced mathematical architecture should be considered in designing machines. In this work, we incorporate quaternion algebras into data analysis either in the frame of supervised and unsupervised learning to classify two-dimensional Chern insulators. For the unsupervised-learning aspect, we apply the principal component analysis (PCA) on the quaternion-transformed eigenstates to distinguish topological phases. For the supervised-learning aspect, we construct our machine by adding onequaternion convolutional layer on top of a conventional convolutional neural network. The machine takes quaternion-transformed con􀀌gurations as inputs and successfully classify all distinct topological phases, even for those states that have di􀀋erent distributuions from those states seen by the machine during the training process. Our work demonstrates the power of quaternion algebras on extracting crucial features from the targeted data and the advantages of quaternion-based neural networks than conventional ones in the tasks of topological phase classi􀀌cations.  \nI. INTRODUCTION  \nThe phase classi􀀌cation using machine-learning (ML) based techniques has been attracting intense attentions since the pioneering work in 2017 [1] . In addition to the classical phase detections [2, 3] where each phase is well de􀀌ned by the corresponding order parameters, detecting topological phase transitions [2] is interesting and challenging [4] due to the lack of local order parameters. Recently, the phase detections and classi􀀌cations have been performed via di􀀋erent ML techniques for classifying various topological invariants [4{36], including the Chern number [6, 10, 14{23], winding number [16, 18, 21, 24{ 26], Z2 index [4, 6, 10, 17, 26{36], to name a few. In addition to the applied ML architectures, the forms of the inputs for training the machine also play a crucial role in determining the resulting performance of the topological phase detections [4] .  \nFor the topological systems with the Chern numbers or the winding numbers as the topological invariants, various types of inputs are used to perform the phase classi􀀌cations. For instance, the quantum loop topography (QLT) is introduced to construct multi-dimensional images from raw Hamiltonians or wave functions as inputs [14, 17] . The Bloch Hamiltonians are arranged into an arrays to feed the neural networks [16, 24] . In addition, the real-space particle densities and local density of states [15] and the local projections of the density matrix [6] are also used as inputs. From cold-atom experiments, momentum-space density images were generated as inputs for classi􀀌cations [20] . The time-of-􀀍ight images [10, 19], spatial correlation function [10],  \n􀀃 [m082030021@student.nsysu.edu.tw](m082030021@student.nsysu.edu.tw)[ ](m082030021@student.nsysu.edu.tw)[y](y shinming@mail.nsysu.edu.tw)[ shinming@mail.nsysu.edu.tw](y shinming@mail.nsysu.edu.tw)  \ndensity{density correlation function [10] and the density pro􀀌les formed in quantum walks were also proposed as appropriate inputs [23] . Furthermore, the spin con􀀌gurations [18] and the Bloch Hamiltonians over the Brillouin zone (BZ) have also been treated as inputs for the neural networks [18, 21] . For these forms of inputs mentioned above, various ML techniques with distinct real-valued neural networks have been applied to discrimin","cbCaifXHtlGAaCAw","https://ap.wps.com/l/cbCaifXHtlGAaCAw","pdf",1469776,1,15,"English","en",105,"# Introduction\n## Machine-learning approaches for phase classification\n## Input representations for topological invariants\n## Motivation for quaternion-based neural networks\n# Quaternion-based supervised and unsupervised framework\n## Unsupervised PCA on quaternion-transformed eigenstates\n## Supervised learning with quaternion convolutional layer\n## Classification performance and generalization","[{\"question\":\"How does the paper perform unsupervised learning for topological phase classification?\",\"answer\":\"It transforms Chern-insulator eigenstates using quaternion algebra, then applies principal component analysis (PCA) to the quaternion-transformed representations to separate topological phases.\"},{\"question\":\"How is supervised learning implemented in this work?\",\"answer\":\"The method constructs a model by adding one quaternion convolutional layer on top of a conventional convolutional neural network and feeds quaternion-transformed configurations as inputs to classify the phases.\"},{\"question\":\"Does the model generalize beyond training data distributions?\",\"answer\":\"Yes. The approach successfully classifies distinct topological phases even for states whose distributions differ from those used during training.\"}]","Quaternion-based machine learning on topological quantum systems | PDF",1785893187,38,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"quaternion-based-machine-learning-on-topological-quantum-systems","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/quaternion-based-machine-learning-on-topological-quantum-systems/124587/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-05",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"How does the paper perform unsupervised learning for topological phase classification?","Question",{"text":75,"@type":76},"It transforms Chern-insulator eigenstates using quaternion algebra, then applies principal component analysis (PCA) to the quaternion-transformed representations to separate topological phases.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is supervised learning implemented in this work?",{"text":80,"@type":76},"The method constructs a model by adding one quaternion convolutional layer on top of a conventional convolutional neural network and feeds quaternion-transformed configurations as inputs to classify the phases.",{"name":82,"@type":73,"acceptedAnswer":83},"Does the model generalize beyond training data distributions?",{"text":84,"@type":76},"Yes. 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