{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,5]],"date-time":"2026-01-05T11:20:01Z","timestamp":1767612001787,"version":"3.41.0"},"reference-count":39,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T00:00:00Z","timestamp":1750118400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>We propose a novel quantum algorithm for solving linear autonomous ordinary differential equations (ODEs) using the Pad\u00e9 approximation. For linear autonomous ODEs, the discretized solution can be represented by a product of matrix exponentials. The proposed algorithm approximates the matrix exponential by the diagonal Pad\u00e9 approximation, which is then encoded into a large, block-sparse linear system and solved via quantum linear system algorithms (QLSA). The detailed quantum circuit is given based on quantum oracle access to the matrix, the inhomogeneous term, and the initial state. The complexity of the proposed algorithm is analyzed. Compared to the method based on Taylor approximation, which approximates the matrix exponential using a <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>k<\/mml:mi><\/mml:math>-th order Taylor series, the proposed algorithm improves the approximation order <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>k<\/mml:mi><\/mml:math> from two perspectives: 1) the explicit complexity dependency on <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>k<\/mml:mi><\/mml:math> is improved, and 2) a smaller <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>k<\/mml:mi><\/mml:math> suffices for the same precision. Numerical experiments demonstrate the advantages of the proposed algorithm comparing to other related algorithms.<\/jats:p>","DOI":"10.22331\/q-2025-06-17-1770","type":"journal-article","created":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T10:37:41Z","timestamp":1750156661000},"page":"1770","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":["A quantum algorithm for linear autonomous differential equations via Pad\u00e9 approximation"],"prefix":"10.22331","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-5553-0231","authenticated-orcid":false,"given":"Dekuan","family":"Dong","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Fudan University"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1852-3750","authenticated-orcid":false,"given":"Yingzhou","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Fudan University"},{"name":"Shanghai Key Laboratory for Contemporary Applied Mathematics"}]},{"given":"Jungong","family":"Xue","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Fudan University"}]}],"member":"9598","published-online":{"date-parts":[[2025,6,17]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Awad H. 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