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However, the optimization quickly becomes intractable when multiple control operations are applied sequentially. In this work, we propose efficient tensor network algorithms for optimizing strategies of quantum metrology enhanced by a long sequence of control operations. Our approach covers a general and practical scenario where the experimenter applies N<\/mml:mi>&#x2212;<\/mml:mo>1<\/mml:mn><\/mml:math> interleaved control operations between N<\/mml:mi><\/mml:math> queries of the channel to estimate and uses no or bounded ancilla. Tailored to different experimental capabilities, these control operations can be generic quantum channels or variational unitary gates. Numerical experiments show that our algorithm has a good performance in optimizing the metrological strategy for as many as N<\/mml:mi>=<\/mml:mo>100<\/mml:mn><\/mml:math> queries. In particular, our algorithm identifies a strategy that can outperform the state-of-the-art strategy when N<\/mml:mi><\/mml:math> is finite but large.<\/jats:p>","DOI":"10.22331\/q-2024-12-18-1571","type":"journal-article","created":{"date-parts":[[2024,12,18]],"date-time":"2024-12-18T14:15:58Z","timestamp":1734531358000},"page":"1571","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":4,"title":["Efficient tensor networks for control-enhanced quantum metrology"],"prefix":"10.22331","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7352-8840","authenticated-orcid":false,"given":"Qiushi","family":"Liu","sequence":"first","affiliation":[{"name":"QICI Quantum Information and Computation Initiative, Department of Computer Science, School of Computing and Data Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0531-8929","authenticated-orcid":false,"given":"Yuxiang","family":"Yang","sequence":"additional","affiliation":[{"name":"QICI Quantum Information and Computation Initiative, Department of Computer Science, School of Computing and Data Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China"}]}],"member":"9598","published-online":{"date-parts":[[2024,12,18]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. ``Quantum-enhanced measurements: Beating the standard quantum limit''. 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