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Starting from a time-evolved state, an ensemble of pure states supported on a small subsystem can be generated by performing projective measurements on the remainder of the system, leading to a <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\">projected ensemble<\/mml:mtext><\/mml:mrow><\/mml:math>. In chaotic quantum systems it was conjectured that such projected ensembles become indistinguishable from the uniform Haar-random ensemble and lead to a <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\">quantum state design<\/mml:mtext><\/mml:mrow><\/mml:math>. Exact results were recently presented by Ho and Choi [Phys. Rev. Lett. 128, 060601 (2022)] for the kicked Ising model at the self-dual point. We provide an alternative construction that can be extended to general chaotic dual-unitary circuits with solvable initial states and measurements, highlighting the role of the underlying dual-unitarity and further showing how dual-unitary circuit models exhibit both exact solvability and random matrix behaviour. Building on results from biunitary connections, we show how complex Hadamard matrices and unitary error bases both lead to solvable measurement schemes.<\/jats:p>","DOI":"10.22331\/q-2022-06-15-738","type":"journal-article","created":{"date-parts":[[2022,6,15]],"date-time":"2022-06-15T11:37:31Z","timestamp":1655293051000},"page":"738","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":64,"title":["Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics"],"prefix":"10.22331","volume":"6","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7150-8459","authenticated-orcid":false,"given":"Pieter W.","family":"Claeys","sequence":"first","affiliation":[{"name":"Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany"},{"name":"TCM Group, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0707-0488","authenticated-orcid":false,"given":"Austen","family":"Lamacraft","sequence":"additional","affiliation":[{"name":"TCM Group, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2022,6,15]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"L. 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