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We systematically expand on that paradigm, and use combinatorial tiling theory to study and construct new examples of fault-tolerant cluster states derived from crystal structures. Included among these is a robust self-dual cluster state requiring only degree-<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mn>3<\/mml:mn><\/mml:math>connectivity. We benchmark several of these cluster states in the presence of circuit-level noise, and find a variety of promising candidates whose performance depends on the specifics of the noise model. By eschewing the distinction between data and ancilla, this malleable framework lays a foundation for the development of creative and competitive fault-tolerance schemes beyond conventional error-correcting codes.<\/jats:p>","DOI":"10.22331\/q-2020-07-13-295","type":"journal-article","created":{"date-parts":[[2020,7,13]],"date-time":"2020-07-13T18:48:56Z","timestamp":1594666136000},"page":"295","source":"Crossref","is-referenced-by-count":22,"title":["Generating Fault-Tolerant Cluster States from Crystal Structures"],"prefix":"10.22331","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6640-1072","authenticated-orcid":false,"given":"Michael","family":"Newman","sequence":"first","affiliation":[{"name":"Departments of Electrical and Computer Engineering, Chemistry, and Physics, Duke University, Durham, NC, 27708, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2491-8126","authenticated-orcid":false,"given":"Leonardo Andreta","family":"de Castro","sequence":"additional","affiliation":[{"name":"Departments of Electrical and Computer Engineering, Chemistry, and Physics, Duke University, Durham, NC, 27708, USA"},{"name":"Q-CTRL Pty Ltd, Sydney, NSW, Australia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7716-1425","authenticated-orcid":false,"given":"Kenneth R.","family":"Brown","sequence":"additional","affiliation":[{"name":"Departments of Electrical and Computer Engineering, Chemistry, and Physics, Duke University, Durham, NC, 27708, USA"}]}],"member":"9598","published-online":{"date-parts":[[2020,7,13]]},"reference":[{"key":"0","doi-asserted-by":"crossref","unstructured":"Panos Aliferis, Daniel Gottesman, and John Preskill. 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