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This is in particular the case whenever the combination of subsystems is described by a nontrivial blend of direct sums and tensor products of Hilbert spaces. We therefore propose an extension to the framework of quantum circuits, given by <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">routed linear maps<\/mml:mtext><\/mml:mrow><\/mml:math> and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">routed quantum circuits<\/mml:mtext><\/mml:mrow><\/mml:math>. We prove that this new framework allows for a consistent and intuitive diagrammatic representation in terms of circuit diagrams, applicable to both pure and mixed quantum theory, and exemplify its use in several situations, including the superposition of quantum channels and the causal decompositions of unitaries. We show that our framework encompasses the `extended circuit diagrams' of Lorenz and Barrett [arXiv:2001.07774 (2020)], which we derive as a special case, endowing them with a sound semantics.<\/jats:p>","DOI":"10.22331\/q-2021-07-13-503","type":"journal-article","created":{"date-parts":[[2021,7,13]],"date-time":"2021-07-13T16:41:12Z","timestamp":1626194472000},"page":"503","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":17,"title":["Routed quantum circuits"],"prefix":"10.22331","volume":"5","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9022-8655","authenticated-orcid":false,"given":"Augustin","family":"Vanrietvelde","sequence":"first","affiliation":[{"name":"Quantum Group, Department of Computer Science, University of Oxford"},{"name":"Department of Physics, Imperial College London"},{"name":"HKU-Oxford Joint Laboratory for Quantum Information and Computation"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4465-2863","authenticated-orcid":false,"given":"Hl\u00e9r","family":"Kristj\u00e1nsson","sequence":"additional","affiliation":[{"name":"Quantum Group, Department of Computer Science, University of Oxford"},{"name":"HKU-Oxford Joint Laboratory for Quantum Information and Computation"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2222-0579","authenticated-orcid":false,"given":"Jonathan","family":"Barrett","sequence":"additional","affiliation":[{"name":"Quantum Group, Department of Computer Science, University of Oxford"}]}],"member":"9598","published-online":{"date-parts":[[2021,7,13]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"D. 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