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We consider three control models: one where time evolution can be augmented with instantaneous quantum operations, one where the Hamiltonian itself can be augmented by adding constant terms, and one where the experimentalist has no control over the system's time evolution. With continuous quantum control, we provide an adaptive algorithm for learning a many-body Hamiltonian at the Heisenberg limit: <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>T<\/mml:mi><mml:mo>=<\/mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O<\/mml:mi><\/mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msup><mml:mi>&amp;#x03F5;<\/mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>&amp;#x2212;<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>, where <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>T<\/mml:mi><\/mml:math> is the total amount of time evolution across all experiments and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>&amp;#x03F5;<\/mml:mi><\/mml:math> is the target precision. This requires only preparation of product states, time-evolution, and measurement in a product basis. In the absence of quantum control, we prove that learning is standard quantum limited, <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>T<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi mathvariant=\"normal\">&amp;#x03A9;<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msup><mml:mi>&amp;#x03F5;<\/mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>&amp;#x2212;<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>, for large classes of many-body Hamiltonians, including any Hamiltonian that thermalizes via the eigenstate thermalization hypothesis. These results establish a quadratic advantage in experimental runtime for learning with quantum control.<\/jats:p>","DOI":"10.22331\/q-2024-11-26-1537","type":"journal-article","created":{"date-parts":[[2024,11,26]],"date-time":"2024-11-26T14:08:13Z","timestamp":1732630093000},"page":"1537","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":17,"title":["The advantage of quantum control in many-body Hamiltonian learning"],"prefix":"10.22331","volume":"8","author":[{"given":"Alicja","family":"Dutkiewicz","sequence":"first","affiliation":[{"name":"Google Quantum AI, Munich 80636, Germany"},{"name":"Instituut-Lorentz, Universiteit Leiden, 2300RA Leiden, The Netherlands"}]},{"given":"Thomas E.","family":"O&apos;Brien","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Munich 80636, Germany"},{"name":"Instituut-Lorentz, Universiteit Leiden, 2300RA Leiden, The Netherlands"}]},{"given":"Thomas","family":"Schuster","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Munich 80636, Germany"},{"name":"Department of Physics, University of California, Berkeley, California 94720 USA"}]}],"member":"9598","published-online":{"date-parts":[[2024,11,26]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Tillmann Baumgratz and Animesh Datta. 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