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These processes can be described within the framework of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>h<\/mml:mi><mml:mi>i<\/mml:mi><mml:mi>g<\/mml:mi><mml:mi>h<\/mml:mi><mml:mi>e<\/mml:mi><mml:mi>r<\/mml:mi><mml:mo>&amp;#x2212;<\/mml:mo><mml:mi>o<\/mml:mi><mml:mi>r<\/mml:mi><mml:mi>d<\/mml:mi><mml:mi>e<\/mml:mi><mml:mi>r<\/mml:mi><\/mml:math> quantum theory which, starting from considering maps from quantum transformations to quantum transformations, recursively constructs a hierarchy of quantum maps of increasingly higher order. In this work, we develop a formalism for quantum computation with indefinite causal structures; namely, we characterize the computational structure of higher order quantum maps. Taking an axiomatic approach, the rules of this computation are identified as the most general compositions of higher order maps which are compatible with the mathematical structure of quantum theory. We provide a mathematical characterization of the admissible composition for arbitrary higher order quantum maps. We prove that these rules, which have a computational and information-theoretic nature, are determined by the more physical notion of the signalling relations between the quantum systems of the higher order quantum maps.<\/jats:p>","DOI":"10.22331\/q-2024-02-05-1241","type":"journal-article","created":{"date-parts":[[2024,2,5]],"date-time":"2024-02-05T14:42:53Z","timestamp":1707144173000},"page":"1241","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":5,"title":["No-signalling constrains quantum computation with indefinite causal structure"],"prefix":"10.22331","volume":"8","author":[{"given":"Luca","family":"Apadula","sequence":"first","affiliation":[{"name":"University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria"},{"name":"Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria"}]},{"given":"Alessandro","family":"Bisio","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica, Universit\u00e0 di Pavia, via Bassi 6, 27100 Pavia, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, Italy"}]},{"given":"Paolo","family":"Perinotti","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica, Universit\u00e0 di Pavia, via Bassi 6, 27100 Pavia, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, Italy"}]}],"member":"9598","published-online":{"date-parts":[[2024,2,5]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Carl W. 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