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This framework allows us to enforce stabilizer commutation, <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>X<\/mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>\/<\/mml:mo><\/mml:mrow><mml:mi>Z<\/mml:mi><\/mml:math> balancing, finite rate, sparsity, and maximum-degree constraints simultaneously in a CSP that we can then solve numerically. Using a state-of-the-art CSP solver, we obtain convincing evidence for the existence of a satisfiability threshold. Furthermore, the extent of the satisfiable phase increases with the number of qubits. In that phase, finding sparse codes becomes an easy problem. Moreover, we observe that the sparse codes found in the satisfiable phase practically achieve the channel capacity for erasure noise. Our results show that intermediate-size finite-rate sparse quantum codes are easy to find, while also demonstrating a flexible methodology for generating good codes with custom properties. We therefore establish a complete and customizable pipeline for random quantum code discovery.<\/jats:p>","DOI":"10.22331\/q-2023-04-20-985","type":"journal-article","created":{"date-parts":[[2023,4,20]],"date-time":"2023-04-20T12:33:43Z","timestamp":1681994023000},"page":"985","update-policy":"http:\/\/dx.doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":["Finite-rate sparse quantum codes aplenty"],"prefix":"10.22331","volume":"7","author":[{"given":"Maxime","family":"Tremblay","sequence":"first","affiliation":[{"name":"D\u00e9partement de physique & Institut quantique, Universit\u00e9 de Sherbrooke, Sherbrooke, Qu\u00e9bec, Canada, J1K 2R1"}]},{"given":"Guillaume","family":"Duclos-Cianci","sequence":"additional","affiliation":[{"name":"D\u00e9partement de physique & Institut quantique, Universit\u00e9 de Sherbrooke, Sherbrooke, Qu\u00e9bec, Canada, J1K 2R1"}]},{"given":"Stefanos","family":"Kourtis","sequence":"additional","affiliation":[{"name":"D\u00e9partement de physique & Institut quantique, Universit\u00e9 de Sherbrooke, Sherbrooke, Qu\u00e9bec, Canada, J1K 2R1"}]}],"member":"9598","published-online":{"date-parts":[[2023,4,20]]},"reference":[{"key":"0","unstructured":"MiniZinc - Challenge, a. 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