[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82424-en":3,"doc-seo-82424-105":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},82424,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","Quantum Orchestras: a Concrete Semantics for Recursive Hybrid Programs","Quantum Orchestras develops a rigorous denotational semantics framework for hybrid quantum programming languages that model quantum effects by mutating an external quantum state and addressing qubits by reference. The work introduces the quantum orchestra monad with a concrete presentation via quantum instruments, enabling interpretation of divergent programs. It positions the monad as an extension of the classical state monad and the probabilistic powerdomain monad, while analyzing subtle issues that arise in a non-commutative quantum setting, especially with measurement and non-termination.","arXiv :2607 .09605v 1 [ cs .PL] 10 Jul 2026  \nQuantum Orchestras: a Concrete Semantics for Recursive Hybrid Programs  \nALEX RICE, The University of Edinburgh, United Kingdom DOMINIK LEICHTLE, The University of Edinburgh, United Kingdom KIM WORRALL, The University of Edinburgh, United Kingdom ROBERT I. BOOTH, University of Oxford, United Kingdom  \nMany production quantum programming languages represent hybrid quantum computations by extending a classical base language with a quantum effect, where qubits are addressed by reference, and quantum operations are understood to mutate some external quantum state. However, the semantics of this view of quantum computation remains underdeveloped, especially when the language allows mid-circuit measurements and non-termination.  \nIn this work, we provide a general method for building denotational semantics for such languages, by defining the quantum orchestra monad, which precisely captures this style of quantum effect. The monad has a concrete presentation, being based on the formalism of quantum instruments, a common tool in quantum information theory for capturing the action of a quantum process along with its classical outcomes. It acts on the category DCPO, and so enables the interpretation of divergent hybrid programs.  \nThe quantum orchestra monad serves as a natural extension of both the classical state monad and the probabilistic powerdomain monad. We investigate some of the subtleties present when trying to naïvely extend these definitions to the quantum non-commutative case.  \n1 Introduction  \nThe field of quantum computing is quickly progressing from small scale experiments in physics labs to increasingly capable industrial quantum computers. As these systems grow in scale, the software used to control and program them is becoming correspondingly more complex. This motivates the development of mathematical models that are both expressive and conceptually clear, supporting the design and analysis of modern quantum programming languages.  \nTraditionally, quantum programs have been expressed using the circuit model [12]: a static list of quantum gates is applied sequentially to an input quantum state, which is then measured atthe end of the computation. While the semantics of circuits are well understood and their static nature makes it possible for them to be heavily optimised, programming directly in terms of circuits provides little support for programs that adapt their quantum behaviour in response to intermediate measurement outcomes.  \nMore recently, there has been increased interest in hybrid quantum programs, which combine quantum operations with classical computation and control flow (see for example IBM introducing classical registers in OpenQASM 3 [10], Quantinuum’s HUGR [30], QIRO [22] which is used in the Pennylane toolkit [4], and QIR [37]). This hybrid functionality is essential for describing many aspects of quantum computing, including variational algorithms [8], measurement-based quantum computing [38], repeat-until-success circuits [1], error mitigation [48], qubit teleportation [13], iterative phase estimation [18], circuit knitting [36], and quantum error correction (e.g. [40]) .  \nAt a high level, many such languages follow the quantum random-access machine (QRAM) model of quantum computation [29]: a classical host program dynamically controls a quantum processor, issuing quantum commands and observing the outcomes of measurements. This view is particularly natural for languages that extend a classical base language with operations for  \nAuthors’ Contact Information: Alex Rice, The University of Edinburgh, United Kingdom, [alex.rice@ed.ac.uk](alex.rice@ed.ac.uk); Dominik Leichtle, The University of Edinburgh, United Kingdom, [dominik.leichtle@ed.ac.uk](dominik.leichtle@ed.ac.uk); Kim Worrall, The University of Edinburgh, United Kingdom, [kim.worrall@ed.ac.uk](kim.worrall@ed.ac.uk); Robert I. Booth, University of Oxford, United Kingdom, firstname. 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