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For each sensing platform, pinpointing the optimal controls to enhance the sensor&apos;s precision remains a challenging task. While an analytical solution might be out of reach, machine learning offers a promising avenue for many systems of interest, especially given the capabilities of contemporary hardware. We have introduced a versatile procedure capable of optimizing a wide range of problems in quantum metrology, estimation, and hypothesis testing by combining model-aware reinforcement learning (RL) with Bayesian estimation based on particle filtering. To achieve this, we had to address the challenge of incorporating the many non-differentiable steps of the estimation in the training process, such as measurements and the resampling of the particle filter. Model-aware RL is a gradient-based method, where the derivatives of the sensor&apos;s precision are obtained through automatic differentiation (AD) in the simulation of the experiment. Our approach is suitable for optimizing both non-adaptive and adaptive strategies, using neural networks or other agents. We provide an implementation of this technique in the form of a Python library called qsensoropt, alongside several pre-made applications for relevant physical platforms, namely NV centers, photonic circuits, and optical cavities. This library will be released soon on PyPI. Leveraging our method, we&apos;ve achieved results for many examples that surpass the current state-of-the-art in experimental design. In addition to Bayesian estimation, leveraging model-aware RL, it is also possible to find optimal controls for the minimization of the Cram\u00e9r-Rao bound, based on Fisher information.<\/jats:p>","DOI":"10.22331\/q-2024-12-10-1555","type":"journal-article","created":{"date-parts":[[2024,12,10]],"date-time":"2024-12-10T16:16:38Z","timestamp":1733847398000},"page":"1555","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":4,"title":["Model-aware reinforcement learning for high-performance Bayesian experimental design in quantum metrology"],"prefix":"10.22331","volume":"8","author":[{"given":"Federico","family":"Belliardo","sequence":"first","affiliation":[{"name":"NEST, Scuola Normale Superiore, I-56126 Pisa, Italy"}]},{"given":"Fabio","family":"Zoratti","sequence":"additional","affiliation":[{"name":"Scuola Normale Superiore, I-56126 Pisa, Italy"}]},{"given":"Florian","family":"Marquardt","sequence":"additional","affiliation":[{"name":"Max Planck Institute for the Science of Light and Physics Department, University of Erlangen-Nuremberg, 91058 Erlangen, Germany"}]},{"given":"Vittorio","family":"Giovannetti","sequence":"additional","affiliation":[{"name":"NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa, Italy"}]}],"member":"9598","published-online":{"date-parts":[[2024,12,10]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Fulvio Flamini, Arne Hamann, Sofi\u00e8ne Jerbi, Lea M Trenkwalder, Hendrik Poulsen Nautrup, and Hans J Briegel. ``Photonic architecture for reinforcement learning''. 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(2015). url: http:\/\/arxiv.org\/abs\/1412.6980."},{"key":"67","unstructured":"Peter Karkus, David Hsu, and Wee Sun Lee. ``Particle Filter Networks with Application to Visual Localization''. In Proceedings of The 2nd Conference on Robot Learning. Pages 169\u2013178. PMLR (2018). url: https:\/\/proceedings.mlr.press\/v87\/karkus18a.html."},{"key":"68","unstructured":"Adam \u015acibior and Frank Wood. ``Differentiable Particle Filtering without Modifying the Forward Pass'' (2021)."},{"key":"69","doi-asserted-by":"publisher","unstructured":"Adam Gali. ``Ab initio theory of the nitrogen-vacancy center in diamond''. Nanophotonics 8, 1907\u20131943 (2019).","DOI":"10.1515\/nanoph-2019-0154"},{"key":"70","doi-asserted-by":"publisher","unstructured":"Marcus W. Doherty, Chunhui Rita Du, and Gregory D. Fuchs. ``Quantum science and technology based on color centers with accessible spin''. 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Journal of Statistical Physics 1, 231\u2013252 (1969).","DOI":"10.1007\/BF01007479"},{"key":"102","doi-asserted-by":"publisher","unstructured":"A. S. Holevo. ``Statistical problems in quantum physics''. In G. Maruyama and Yu. V. Prokhorov, editors, Proceedings of the Second Japan-USSR Symposium on Probability Theory. Pages 104\u2013119. Lecture Notes in MathematicsBerlin, Heidelberg (1973). Springer.","DOI":"10.1007\/BFb0061483"},{"key":"103","doi-asserted-by":"publisher","unstructured":"Masahito Hayashi. ``Asymptotic Theory of Quantum Statistical Inference''. WORLD SCIENTIFIC. (2005).","DOI":"10.1142\/5630"},{"key":"104","doi-asserted-by":"publisher","unstructured":"Antonio A. Gentile, Brian Flynn, Sebastian Knauer, Nathan Wiebe, Stefano Paesani, Christopher E. Granade, John G. Rarity, Raffaele Santagati, and Anthony Laing. ``Learning models of quantum systems from experiments''. 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