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We study whether such causal structure can be understood in terms of compositional structure of the unitary. Given a quantum circuit with no path from input systemA<\/mml:mi><\/mml:math>to output systemB<\/mml:mi><\/mml:math>, systemA<\/mml:mi><\/mml:math>cannot influence systemB<\/mml:mi><\/mml:math>. Conversely, given a unitaryU<\/mml:mi><\/mml:math>with a no-influence relation from inputA<\/mml:mi><\/mml:math>to outputB<\/mml:mi><\/mml:math>, it follows from [B. Schumacher and M. D. Westmoreland, Quantum Information Processing 4 no. 1, (Feb, 2005)] that there exists a circuit decomposition ofU<\/mml:mi><\/mml:math>with no path fromA<\/mml:mi><\/mml:math>toB<\/mml:mi><\/mml:math>. However, as we argue, there are unitaries for which there does not exist a circuit decomposition that makes all causal constraints evidentsimultaneously<\/mml:mtext><\/mml:mrow><\/mml:math>. To address this, we introduce a new formalism of `extended circuit diagrams', which goes beyond what is expressible with quantum circuits, with the core new feature being the ability to represent direct sum structures in addition to sequential and tensor product composition. Acausally faithful<\/mml:mtext><\/mml:mrow><\/mml:math>extended circuit decomposition, representing a unitaryU<\/mml:mi><\/mml:math>, is then one for which there is a path from an inputA<\/mml:mi><\/mml:math>to an outputB<\/mml:mi><\/mml:math>if and only if there actually is influence fromA<\/mml:mi><\/mml:math>toB<\/mml:mi><\/mml:math>inU<\/mml:mi><\/mml:math>. We derive causally faithful extended circuit decompositions for a large class of unitaries, where in each case, the decomposition is implied by the unitary's respective causal structure. We hypothesize that every finite-dimensional unitary transformation has a causally faithful extended circuit decomposition.<\/jats:p>","DOI":"10.22331\/q-2021-07-28-511","type":"journal-article","created":{"date-parts":[[2021,7,28]],"date-time":"2021-07-28T10:01:55Z","timestamp":1627466515000},"page":"511","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":15,"title":["Causal and compositional structure of unitary transformations"],"prefix":"10.22331","volume":"5","author":[{"given":"Robin","family":"Lorenz","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK"},{"name":"Cambridge Quantum Computing Ltd, 17 Beaumont Street, Oxford OX1 2NA, UK"}]},{"given":"Jonathan","family":"Barrett","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK"}]}],"member":"9598","published-online":{"date-parts":[[2021,7,28]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"G. 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