[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82781-en":3,"doc-seo-82781-105":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},82781,137441390410,"Hazel","https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984",8,"Research & Report","The Delayed Stabilizer ZX-Calculus","The Delayed Stabilizer ZX-calculus is introduced as a finite graphical language for infinite, translation-invariant stabilizer quantum error-correcting processes. It extends the odd-prime-dimensional stabilizer ZX-calculus with a single new delay generator that propagates information across successive time steps. Two semantics are provided: one via equivalence classes of quantum-channel sequences yielding a unique infinite stabilizer group, and another using delay as a formal variable with generating-function tableaux. A complete axiomatization proves soundness, universality, and completeness.","arXiv :2607 .04015v1 [ quant-ph] 4 Jul 2026  \nThe Delayed Stabilizer ZX-Calculus  \nCole Comfort 1 Giovanni de Felice2  \n1Universit Paris-Saclay, CNRS, ENS Paris-Saclay, Inria, CentraleSuplec,  \nLaboratoire Mthodes Formelles, 91190 Gif-sur-Yvette, France  \n2Relational Intelligence Ltd.  \nMany stabilizer quantum error-correcting codes are built from a finite pattern repeated across space or time, such as lattice codes, translation-invariant graph states, and quantum convolutional codes.  \nOrdinary stabilizer ZX-diagrams capture only finite truncations of such systems, obscuring the repeated structure that defines them. We introduce the delayed stabilizer ZX-calculus, a finite graphical language for these infinite, translation-invariant processes. It extends the odd-prime-dimensional stabilizer ZX-calculus with a single new generator, the delay, which feeds data from one time step to the next.  \nWe equip the calculus with two semantics. In the first semantics, we interpret the behaviour of a delayed ZX-diagram as an equivalence class of sequences of quantum channels; where two sequences are identified if they have the same information content. We show that the behaviour of a delayed ZX-diagram uniquely determines an infinite stabilizer group. In the second semantics, we interpret the delay as a formal variable, encoding the translation-invariant families of Pauli operators as generating functions. This allows us to represent a delayed ZX-diagram in terms of a tableau of generating functions, from which the infinite stabilizer group can be recovered.  \nFinally, we give a complete axiomatization of the delayed stabilizer ZX-calculus, featuring generalised Euler decomposition and colour change rules. Using generalised forms of local complementation and pivoting, we reduce every diagram to a unique normal form. This establishes soundness, universality, and completeness for the generating tableau semantics.  \n1 Introduction  \nStabilizer quantum mechanics is a cornerstone of quantum information theory and underlies much of quantum error correction and fault-tolerant quantum computation, describing quantum processes compactly through Pauli operators rather than exponentially large matrices [17, 30] . The ZX-calculus, and more specifically the stabilizer ZX-calculus, allows such processes to be reasoned about using graph rewriting [12,13,36] . The stabilizer ZX-calculus is sound, universal, and complete for stabilizer quantum mechanics in all prime dimensions [1,6,33], meaning that reasoning about stabilizer quantum mechanics can be done exclusively using this graphical notation. This has been particularly useful for quantum error correction and measurement-based quantum computation, where diagrammatic rewrites correspond to transformations of graph states and stabilizer codes [25] .  \nThe stabilizer ZX-calculus becomes inadequate, however, for processes defined by structure repeated across space or time. The quantum error-correction literature is full of examples: topological codes [16, 26], in which a fixed local stabilizer pattern repeats across space, and convolutional and serial turbo codes [22, 31, 32, 38, 39], where the pattern repeats across time. Further examples include quantum channels with memory [27], resource states for measurement-based quantum computing [29], and quantum cellular automata [21] .  \nSuch processes can be truncated to finite circuits, which themselves can be represented by ordinary ZX-diagrams. However, working with these truncations quickly becomes intractable, obscuring the re-  \n2 The Delayed Stabilizer ZX-Calculus  \ncursive procedure that generates the circuit. For example, consider the ZX-calculus representation of the surface code from [24]:  \n(1)  \nProperties of this diagram, representing a single truncation of the lattice, may not extend to larger truncations, forcing unwieldy use of ellipses.  \nThis leaves a gap. We want a graphical language in which these families of codes are represented and manip","cbCaifwnr1umMTI8","https://ap.wps.com/l/cbCaifwnr1umMTI8","pdf",4254371,2,1,53,"English","en",105,"# Introduction\n# Delayed Stabilizer ZX-Calculus","[{\"question\":\"What problem does the delayed stabilizer ZX-calculus address?\",\"answer\":\"It addresses the limitation of ordinary stabilizer ZX-diagrams, which only capture finite truncations and obscure the repeated translation-invariant structure of infinite quantum stabilizer processes.\"},{\"question\":\"How is the delay implemented in the ZX-calculus?\",\"answer\":\"The calculus is extended with a single new generator, the delay, which passes data from one time step to the next, allowing finite patterns to represent infinite iterated processes.\"},{\"question\":\"How are delayed ZX-diagrams interpreted in the paper?\",\"answer\":\"They are given two semantics: as equivalence classes of sequences of quantum channels that determine an infinite stabilizer group, and as generating-function families where the delay encodes translation-invariant Pauli operators.\"}]",1784182890,134,{"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},"the-delayed-stabilizer-zx-calculus","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/the-delayed-stabilizer-zx-calculus/82781/",4,{"url":51,"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","2026-07-24","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 problem does the delayed stabilizer ZX-calculus address?","Question",{"text":75,"@type":76},"It addresses the limitation of ordinary stabilizer ZX-diagrams, which only capture finite truncations and obscure the repeated translation-invariant structure of infinite quantum stabilizer processes.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the delay implemented in the ZX-calculus?",{"text":80,"@type":76},"The calculus is extended with a single new generator, the delay, which passes data from one time step to the next, allowing finite patterns to represent infinite iterated processes.",{"name":82,"@type":73,"acceptedAnswer":83},"How are delayed ZX-diagrams interpreted in the paper?",{"text":84,"@type":76},"They are given two semantics: as equivalence classes of sequences of quantum channels that determine an infinite stabilizer group, and as generating-function families where the delay encodes translation-invariant Pauli 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