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When a global symmetry is present, the trial wavefunctions of the Bethe ansatz consist of plane wave superpositions. Previously, it has been shown that the Bethe ansatz can be recast as a deterministic quantum circuit. An analytical derivation of the quantum gates that form the circuit was lacking however. Here we present a comprehensive study of the transformation that brings the Bethe ansatz into a quantum circuit, which leads us to determine the analytical expression of the circuit gates. As a crucial step of the derivation, we present a simple set of diagrammatic rules that define a novel Matrix Product State network building Bethe wavefunctions. Remarkably, this provides a new perspective on the equivalence between the coordinate and algebraic versions of the Bethe ansatz.<\/jats:p>","DOI":"10.22331\/q-2024-05-23-1356","type":"journal-article","created":{"date-parts":[[2024,5,23]],"date-time":"2024-05-23T11:55:52Z","timestamp":1716465352000},"page":"1356","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":12,"title":["The Bethe Ansatz as a Quantum Circuit"],"prefix":"10.22331","volume":"8","author":[{"given":"Roberto","family":"Ruiz","sequence":"first","affiliation":[{"name":"Instituto de F\u00edsica Te\u00f3rica, UAM\/CSIC, Universidad Aut\u00f3noma de Madrid, Madrid, Spain"}]},{"given":"Alejandro","family":"Sopena","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica Te\u00f3rica, UAM\/CSIC, Universidad Aut\u00f3noma de Madrid, Madrid, Spain"}]},{"given":"Max Hunter","family":"Gordon","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica Te\u00f3rica, UAM\/CSIC, Universidad Aut\u00f3noma de Madrid, Madrid, Spain"},{"name":"Normal Computing Corporation, New York, New York, USA"}]},{"given":"Germ\u00e1n","family":"Sierra","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica Te\u00f3rica, UAM\/CSIC, Universidad Aut\u00f3noma de Madrid, Madrid, Spain"}]},{"given":"Esperanza","family":"L\u00f3pez","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica Te\u00f3rica, UAM\/CSIC, Universidad Aut\u00f3noma de Madrid, Madrid, Spain"}]}],"member":"9598","published-online":{"date-parts":[[2024,5,23]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"H. 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