[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86413-en":3,"doc-seo-86413-105":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":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":13,"seo_description":14,"update_tm":28,"read_time":29},86413,4810365810221,"Aurora","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Affine Filtering Measurements and Their Applications to Quantum Decoding","Unambiguous state discrimination enables conclusive, error-free outcomes alongside inconclusive erasures. The work introduces affine filtering measurements, a structured USD variant for decoding classical linear codes over pure-state classical–quantum channels: conclusive results identify an affine subspace containing the transmitted codeword, while inconclusive results are treated as erasures. For group-covariant indexed pure-state codewords, optimal measurement design reduces to a semidefinite program that can be transformed into a linear program via character-based diagonalization, enabling a quantum decoding framework for local codes.","arXiv :2606 .07852v2 [ quant-ph] 12 Jul 2026  \nAffine Filtering Measurements and Their Applications to Quantum Decoding  \nAvijit Mandal 1,2,3 , Noah Shutty 1 , Henry D. Pfister2,3 , and Stephen P. Jordan 1  \n1 Google Quantum AI, Venice, CA 90291  \n2 Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708 3 Duke Quantum Center, Durham, NC 27708  \nUnambiguous state discrimination (USD) measurements are attractive because outcomes are either marked as conclusive (i.e., error free) or inconclusive (i.e., erased) . We study affine filtering measurements, a structured variant of USD for decoding classical linear codes over pure-state classical-quantum channels, where a conclusive outcome identifies an affine subspace containing the transmitted codeword and an inconclusive outcome is treated as an erasure. For a group-covariant indexing of pure-state codewords, we show that the optimal design of affine filtering measurements is a semidefinite program that can be reduced to a linear program via character-based diagonalization. We use the resulting measurement to build a quantum decoding framework for local codes, and we demonstrate (via simulations on regular LDPC codes from Gallager ensembles using single parity check local constraints) that affine filtering based decoding can outperform symbol-wise USD and symbol-wise pretty good measurement based decoding methods on i.i.d. pure-state channels. Inan independent and concurrent work, Buzet and Chailloux study similar finegrained USD measurements for symmetric families of states. Their focus is on the code-agnostic setting whereas our focus is on code-aware constructions and decoding.  \n1 Introduction  \nCoding theory for classical–quantum (CQ) channels traces back to the foundational work by Holevo [15] and by Schumacher and Westmoreland [31], who established that one can communicate reliably at any rate up to the channel capacity by performing joint (collective) measurements across long codeword blocks. For many practically important structured CQ channels such as pure-state signal constellations that arise in optical communications [18 , 7], the main hurdle in practice is the implementation of these large collective measurements. This challenge has spurred the development of structured receiver designs whose complexity grows slowly with blocklength while still offering provable performance.  \nRecent progress on quantum algorithms based on Regev’s reduction has renewed interest in designing efficient algorithms for the quantum decoding problem [3 , 4 , 6 , 35] . In the same spirit, a recently introduced method called decoded quantum interferometry (DQI) [17] uses reversible decoding together with quantum Fourier sampling to transform  \nAvijit Mandal: [avijit.mandal@duke.edu](avijit.mandal@duke.edu)  \ncertain structured optimization problems into decoding problems. In [32], the authors develop locally quantum decoders for LDPC codes and use them, through the Regev and DQI framework, to obtain improved approximate solutions for average-case D-regular max-kXORSAT problems, further motivating the study of efficient quantum decoding algorithms.  \nSeveral works have investigated the performance of classical codes, such as polar and Reed-Muller codes [33 , 34 , 21], on classical quantum channels in the context of achieving the Holevo bound. Although these works studied the capacity-achieving potential of such codes, they did not provide efficient constructions to design practical quantum decoding methods.  \nIn pursuit of designing low complexity quantum decoders, Belief propagation with quantum messages (BPQM) was introduced by Renes [28] as a quantum analogue of classical BP for decoding binary linear codes on pure-state channels (PSCs) . Subsequent works developed BPQM-based density-evolution (DE) for symmetric binary CQ channels and used them to study LDPC and polar constructions [1 , 24 , 25], with further extensions to turbo-style constructions [27] . Recent","cbCaikULn31LE4kM","https://ap.wps.com/l/cbCaikULn31LE4kM","pdf",966251,4,1,42,"English","en",105,"# Introduction\n## Unambiguous state discrimination and erasure-based decoding\n## Pretty good measurement approaches\n## Optimal USD and Gaussian-elimination decoding\n## Affine filtering measurements and design optimization","[{\"question\":\"What are affine filtering measurements in the context of quantum decoding?\",\"answer\":\"They are a code-aware structured variant of unambiguous state discrimination. A conclusive outcome specifies an affine subspace containing the transmitted codeword, while an inconclusive outcome is treated as an erasure.\"},{\"question\":\"How is the optimal design of affine filtering measurements computed?\",\"answer\":\"For group-covariant indexing of pure-state codewords, the optimal measurement design is formulated as a semidefinite program. It can then be reduced to a linear program using character-based diagonalization.\"},{\"question\":\"How does affine filtering based decoding compare to other methods?\",\"answer\":\"Simulations on regular LDPC codes using single parity check local constraints show affine filtering can outperform symbol-wise USD and symbol-wise pretty good measurement based decoding on i.i.d. pure-state channels.\"}]",1784211581,106,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"affine-filtering-measurements-and-their-applications-to-quantum-decoding","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"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":20},"https://docshare.wps.com/document/affine-filtering-measurements-and-their-applications-to-quantum-decoding/86413/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What are affine filtering measurements in the context of quantum decoding?","Question",{"text":75,"@type":76},"They are a code-aware structured variant of unambiguous state discrimination. A conclusive outcome specifies an affine subspace containing the transmitted codeword, while an inconclusive outcome is treated as an erasure.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the optimal design of affine filtering measurements computed?",{"text":80,"@type":76},"For group-covariant indexing of pure-state codewords, the optimal measurement design is formulated as a semidefinite program. It can then be reduced to a linear program using character-based diagonalization.",{"name":82,"@type":73,"acceptedAnswer":83},"How does affine filtering based decoding compare to other methods?",{"text":84,"@type":76},"Simulations on regular LDPC codes using single parity check local constraints show affine filtering can outperform symbol-wise USD and symbol-wise pretty good measurement based decoding on i.i.d. pure-state channels.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]