[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82392-en":3,"doc-seo-82392-105":29,"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":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":13,"seo_description":14,"update_tm":27,"read_time":28},82392,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","Higher-Order Programs with Indefinite Causal Orders: a Linear Approach to Coherent Control of Quantum Processes","Higher-order quantum processes with indefinite causal orders (ICOs), such as the quantum switch, superpose the order of quantum operations and enable coherent control with potential computational advantages. Existing quantum programming languages fail to capture this power: they are limited to unitary behavior or treat coherent control nonlinearly, and they cannot combine ICOs with measurement while synchronizing outcomes across branches. The paper introduces a higher-order quantum functional language with linear type discipline, operational and denotational semantics, and a stronger typing framework ensuring physical well-defined programs.","arXiv :2607 .09534v 1 [ cs .LO] 10 Jul 2026  \nHigher-Order Programs with Indefinite Causal Orders: a Linear Approach to Coherent Control of Quantum Processes  \nKATHLEEN BARSSE, Université de Lorraine, CNRS, Inria, LORIA, France ROMAIN PÉCHOUX, Université de Lorraine, CNRS, Inria, LORIA, France SIMON PERDRIX, Université de Lorraine, CNRS, Inria, LORIA, France  \nProcesses with indefinite causal orders (ICOs), such as the quantum switch, are higher-order quantum processes that superpose the order in which quantum operations are performed. Such coherent control yields computational advantages but is not faithfully captured by existing quantum programming languages: either they are restricted to the unitary case, and thus cannot combine ICOs with measurement, or they treat coherent control nonlinearly. In both cases, they do not realize the full computational power ofICOs. We introduce a higher-order quantum functional language that supports general quantum computation, not merely the permutation of channels, and whose linear type system allows quantum control to be well-defined beyond the unitary case, on arbitrary quantum channels. We equip this language with a small-step operational semantics that synchronizes measurement outcomes across superposed branches, using device references and a memory function. We also give a denotational semantics by means of completely positive maps. With linearity asthe only constraint, some well-typed terms would denote unphysical maps. We therefore impose a typing discipline that goes beyond linearity, and interpret programs in the causal category Caus[CPM], under which every well-typed program is physically meaningful, a property that can be checked statically and efficiently. We prove soundness, and study the language’s expressive power: it can express every quantum channel at first order, and at second order a large subclass of the so-called quantum circuits with quantum control (QC-QCs), containing the quantum switch. Last but not least, we show that this language is well-designed enough to be extended to the nonlinear setting with recursion.  \n1 Introduction  \n1.1 Context and Motivations  \nCoherent control, or quantum control, is a concept that has attracted significant attention in the development of quantum programming languages. It refers to the ability to express not only superpositions of data, but also superpositions of programs, which are typically controlled by a quantum state. A fundamental example of coherent control is the quantum switch [Chiribella et al. 2013] . The quantum switch is the higher order quantum process whose input is pair of quantum operations, say C and D, and whose output is the operation that consists in applying D ◦ C and C ◦ D in a superposition, based on the state of a control qubit. The two programs in superposition are the two possible orderings ofthe input operations: C before D or D before C. This is called an indefinite causal order.  \nIndefinite causal orders [Oreshkov et al. 2012] are a subclass of coherent control that arises from higher-order quantum processes. They can be described as processes in which the ordering between a set of processes is in a superposition. Indefinite causal orders are of particular interest in quantum foundations [Hoffreumon and Oreshkov 2026; Vanrietvelde et al. 2025; Wechs et al. 2021], but have also been shown to grant an advantage in computational complexity in a variety of tasks [Abbott, Mhalla, et al. 2024; Araújo, Costa, et al. 2014; Araújo, Guérin, et al. 2017; Kristjánsson et al. 2024; Renner and Brukner 2022, 2021; Taddei et al. 2021] . A significant number of these examples use the quantum switch. Moreover, access to processes with indefinite causal orders can also provide advantages in some communication problems [Abbott, Wechs, et al. 2020; Ebler et al. 2018] or information processing tasks [Wechs et al. 2021] .  \nYet, existing quantum programming languages do not fully capture the expressive power of processes ","cbCaip4mPcndhqpq","https://ap.wps.com/l/cbCaip4mPcndhqpq","pdf",854955,1,43,"English","en",105,"# Introduction\n## Context and Motivations","[{\"question\":\"What problem do existing quantum programming languages have with indefinite causal orders?\",\"answer\":\"They either restrict coherent control to the unitary case and cannot combine ICOs with measurement, or they model coherent control in a nonlinear way. In both cases, they do not realize the full computational power of ICOs.\"},{\"question\":\"What is the key challenge in supporting measurements for coherent control?\",\"answer\":\"Measurements are probabilistic, and the observed outcome must be the same across all superposed branches. Synchronizing measurement outcomes across different causal orders is the core difficulty.\"},{\"question\":\"How does the proposed language ensure programs denote physically meaningful quantum processes?\",\"answer\":\"It uses a typing discipline beyond linearity, interpreting programs in the causal category Caus[CPM]. This makes every well-typed program physically meaningful and can be checked statically and efficiently.\"}]",1784180102,108,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"higher-order-programs-with-indefinite-causal-orders-a-linear-approach-to-coherent-control-of-quantum-processes","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/higher-order-programs-with-indefinite-causal-orders-a-linear-approach-to-coherent-control-of-quantum-processes/82392/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 do existing quantum programming languages have with indefinite causal orders?","Question",{"text":75,"@type":76},"They either restrict coherent control to the unitary case and cannot combine ICOs with measurement, or they model coherent control in a nonlinear way. In both cases, they do not realize the full computational power of ICOs.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the key challenge in supporting measurements for coherent control?",{"text":80,"@type":76},"Measurements are probabilistic, and the observed outcome must be the same across all superposed branches. Synchronizing measurement outcomes across different causal orders is the core difficulty.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed language ensure programs denote physically meaningful quantum processes?",{"text":84,"@type":76},"It uses a typing discipline beyond linearity, interpreting programs in the causal category Caus[CPM]. 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