[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-123539-en":3,"doc-seo-123539-105":30,"detail-sidebar-cat-0-en-105":95},{"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},123539,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","Quantum kernel machine learning with continuous variables - Paper","Popular qubit-based quantum kernel machine learning lacks a comparative framework for continuous-variable (CV) quantum computing platforms. This paper models CV quantum kernels as closed-form functions, deriving a general solution showing each kernel equals a Gaussian times an algebraic function of feature-map parameters. For multi-mode kernels, it quantifies quantum-classical separation using a hierarchical “stellar rank” of the feature map. It proves kernels of infinite stellar rank, such as GKP-state encodings, can be approximated by finite-stellar-rank kernels. Simulations with a displaced Fock state encoding link accuracy to stellar rank and examine bandwidth tuning under underfitting and noisy overfitting.","arXiv :2401 .05647v6 [ quant-ph] 13 Dec 2024  \nQuantum kernel machine learning with continuous variables  \nLaura J. Henderson 1,2 , Rishi Goel 1 , and Sally Shrapnel 1,2  \n1 School of Mathematics and Physics, The University of Queensland, QLD 4072, Australia  \n2 ARC Centre for Engineered Quantum Systems, The University of Queensland, QLD, 4072, Australia.  \nThe popular qubit framework has dominated recent work on quantum kernel machine learning, with results characterising expressivity, learnability and generalisation. As yet, there is no comparative framework to understand these concepts for continuous variable (CV) quantum computing platforms. In this paper we represent CV quantum kernels as closed form functions and use this representation to provide several important theoretical insights. We derive a general closed form solution for all CV quantum kernels and show every such kernel can be expressed as the product of a Gaussian and an algebraic function of the parameters of the feature map. Furthermore, in the multi-mode case, we present quantification of a quantum-classical separation for all quantum kernels via a hierarchical notion of the “stellar rank” of the quantum kernel feature map. We then prove kernels defined by feature maps of infinite stellar rank, such as GKP-state encodings, can be approximated arbitrarily well by kernels defined by feature maps of finite stellar rank. Finally, we simulate learning with a single-mode displaced Fock state encoding and show that (i) accuracy on our specific task (an annular data set) increases with stellar rank,(ii) for underfit models, accuracy can be improved by increasing a bandwidth hyperparameter, and (iii) for noisy data that is overfit, decreasing the bandwidth will improve generalisation but does so at the cost of effective stellar rank.  \nContents  \n1 Introduction 2  \n2 Preliminaries 4  \n2.1 Notation ...................................... 4  \n2.2 Introduction to classical kernel machine learning ............... 5  \n2.3 Background on quantum kernel machine learning ............... 6  \n3 CV quantum kernels 7  \n3.1 Representing CV quantum states as holomorphic functions ......... 7  \n3.2 CV quantum feature maps ............................ 9  \n4 General CV kernels 9  \n5 Displaced Fock state kernel 12  \nAccepted in  2024-10-22, click title to verify. Published under CC-BY 4 .0. 1  \n5.1 Closed form & analytic properties ....................... 12  \n5.2 Bandwidth tuning ................................ 14  \n5.3 Learning experiments .............................. 15  \n6 Qudit kernels 19  \n7 Conclusions & future work 19  \nA Proof that inner products are positive semi-definite 24  \nB Segal-Bargmann space is a RKHS of the Gaussian kernel 24  \nC An integration required to calculate the closed form of CV kernels 25  \nC.1 Explicit calculation ................................ 26  \nC.2 Proof the confluent hypergeometric function 􀀀1 F1 (b + n; b ; ζ)􀀁 is a product of an exponential and a polynomial ....................... 27  \nD Explicit calculation of the general m-mode CV kernel 28  \nE Approximating CV kernels of infinite stellar rank 37  \nF Properties of the displaced Fock state kernel 40  \nF.1 Derivation of Eq. (42) .............................. 40  \nF.2 Explicit examples of the displaced Fock state kernel ............. 41  \nF.3 Showing the displaced Fock state kernel is translation invariant ....... 42  \nF.4 Showing the displaced Fock state kernel is rotation invariant ......... 43  \nF.5 Showing the displaced Fock state kernel is a radial kernel .......... 44  \nF.6 The Fourier transform of the displaced Fock state kernel ........... 44  \nF.7 Showing the displaced Fock state kernel integrates to π ........... 45  \nG Calculation of the qudit kernel 47  \nG.1 Calculating the value of nd,j ........................... 47  \nG.2 Calculation of the qudit kernel from the general multi-mode kernel ..... 47  \n1 Introduction  \nThe quantum machine learning (QML) community has recently begun t","cbCaifQQZrlSpx68","https://ap.wps.com/l/cbCaifQQZrlSpx68","pdf",2466908,1,49,"English","en",105,"# Introduction\n# Preliminaries\n## Notation\n## Classical kernel machine learning\n## Quantum kernel machine learning\n# CV quantum kernels\n## Representing CV quantum states as holomorphic functions\n## CV quantum feature maps\n# General CV kernels\n# Displaced Fock state kernel\n## Closed form & analytic properties\n## Bandwidth tuning\n## Learning experiments\n# Qudit kernels\n# Conclusions & future work\n# Appendix: Proofs and calculations","[{\"question\":\"How are continuous-variable (CV) quantum kernels characterized in this paper?\",\"answer\":\"CV quantum kernels are represented as closed-form functions, with a general solution that expresses each kernel as a Gaussian multiplied by an algebraic function of feature-map parameters.\"},{\"question\":\"What does “stellar rank” measure for quantum kernels?\",\"answer\":\"In the multi-mode setting, stellar rank hierarchically quantifies the quantum-classical separation via properties of the quantum kernel feature map.\"},{\"question\":\"How does the paper handle kernels with infinite stellar rank, such as GKP encodings?\",\"answer\":\"It proves that kernels defined by infinite stellar rank feature maps can be approximated arbitrarily well by kernels defined by finite stellar rank feature maps.\"},{\"question\":\"What role does bandwidth tuning play in the learning experiments?\",\"answer\":\"For underfit models, increasing a bandwidth hyperparameter improves accuracy; for noisy overfit cases, decreasing bandwidth improves generalisation while reducing effective stellar rank.\"}]","Quantum kernel machine learning with continuous variables - Paper | PDF",1785817199,123,{"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":90,"head_meta":92,"extra_data":94,"updated_unix":28},"quantum-kernel-machine-learning-with-continuous-variables-paper","",{"@graph":36,"@context":89},[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/quantum-kernel-machine-learning-with-continuous-variables-paper/123539/",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","2026-08-04",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81,85],{"name":72,"@type":73,"acceptedAnswer":74},"How are continuous-variable (CV) quantum kernels characterized in this paper?","Question",{"text":75,"@type":76},"CV quantum kernels are represented as closed-form functions, with a general solution that expresses each kernel as a Gaussian multiplied by an algebraic function of feature-map parameters.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What does “stellar rank” measure for quantum kernels?",{"text":80,"@type":76},"In the multi-mode setting, stellar rank hierarchically quantifies the quantum-classical separation via properties of the quantum kernel feature map.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper handle kernels with infinite stellar rank, such as GKP encodings?",{"text":84,"@type":76},"It proves that kernels defined by infinite stellar rank feature maps can be approximated arbitrarily well by kernels defined by finite stellar rank feature maps.",{"name":86,"@type":73,"acceptedAnswer":87},"What role does bandwidth tuning play in the learning experiments?",{"text":88,"@type":76},"For underfit models, increasing a bandwidth hyperparameter improves accuracy; 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