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By allowing coherent evolution of quantum systems and entanglement across multiple probes, the precision of estimating a fully connected k<\/mml:mi><\/mml:math>-body interaction can scale up to (<\/mml:mo>n<\/mml:mi>k<\/mml:mi><\/mml:msup>t<\/mml:mi>)<\/mml:mo>&#x2212;<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math>, where n<\/mml:mi><\/mml:math> is the number of probes and t<\/mml:mi><\/mml:math> is the probing time. However, the optimal scaling may no longer be achievable under quantum noise, and it is important to apply quantum error correction in order to recover this limit. In this work, we study the performance of stabilizer quantum error correcting codes in estimating many-body Hamiltonians under noise. When estimating a fully connected Z<\/mml:mi>Z<\/mml:mi>Z<\/mml:mi><\/mml:math> interaction under single-qubit noise, we showcase three families of stabilizer codes \u2013 thin surface codes, quantum Reed\u2013Muller codes and Shor codes \u2013 that achieve the scalings of (<\/mml:mo>n<\/mml:mi>t<\/mml:mi>)<\/mml:mo>&#x2212;<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math>, (<\/mml:mo>n<\/mml:mi>2<\/mml:mn><\/mml:msup>t<\/mml:mi>)<\/mml:mo>&#x2212;<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math> and (<\/mml:mo>n<\/mml:mi>3<\/mml:mn><\/mml:msup>t<\/mml:mi>)<\/mml:mo>&#x2212;<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math>, respectively, all of which are optimal with t<\/mml:mi><\/mml:math>. We further discuss the relation between stabilizer structure and the scaling with n<\/mml:mi><\/mml:math>, and identify several no-go theorems. For instance, we find codes with constant-weight stabilizer generators can at most achieve the n<\/mml:mi>&#x2212;<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math> scaling, while the optimal n<\/mml:mi>&#x2212;<\/mml:mo>3<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math> scaling is achievable if and only if the code bears a repetition code substructure, like in Shor code.<\/jats:p>","DOI":"10.22331\/q-2025-06-05-1766","type":"journal-article","created":{"date-parts":[[2025,6,5]],"date-time":"2025-06-05T09:30:51Z","timestamp":1749115851000},"page":"1766","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Stabilizer codes for Heisenberg-limited many-body Hamiltonian estimation"],"prefix":"10.22331","volume":"9","author":[{"given":"Santanu Bosu","family":"Antu","sequence":"first","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada"},{"name":"Department of Applied Physics and Yale Quantum Institute, New Haven, Connecticut 06520, USA"}]},{"given":"Sisi","family":"Zhou","sequence":"additional","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada"},{"name":"Department of Physics and Astronomy, Department of Applied Mathematics, and Institute for Quantum Computing, University of Waterloo, Ontario N2L 3G1, Canada"}]}],"member":"9598","published-online":{"date-parts":[[2025,6,5]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Advances in quantum metrology. Nat. Photonics., 5(4):222, 2011.","DOI":"10.1038\/nphoton.2011.35"},{"key":"1","doi-asserted-by":"publisher","unstructured":"C. L. Degen, F. Reinhard, and P. Cappellaro. Quantum sensing. Rev. Mod. Phys., 89(3):035002, Jul 2017.","DOI":"10.1103\/RevModPhys.89.035002"},{"key":"2","doi-asserted-by":"publisher","unstructured":"Luca Pezz\u00e8, Augusto Smerzi, Markus K. Oberthaler, Roman Schmied, and Philipp Treutlein. Quantum metrology with nonclassical states of atomic ensembles. Rev. Mod. Phys., 90(3):035005, Sep 2018.","DOI":"10.1103\/RevModPhys.90.035005"},{"key":"3","doi-asserted-by":"publisher","unstructured":"Stefano Pirandola, Bhaskar Roy Bardhan, Tobias Gehring, Christian Weedbrook, and Seth Lloyd. Advances in photonic quantum sensing. Nat. Photonics., 12(12):724, 2018.","DOI":"10.1038\/s41566-018-0301-6"},{"key":"4","doi-asserted-by":"publisher","unstructured":"Lin Jiao, Wei Wu, Si-Yuan Bai, and Jun-Hong An. 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L. de Matos Filho, and L Davidovich. General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology. Nat. Phys., 7(5):406, 2011.","DOI":"10.1038\/nphys1958"},{"key":"9","doi-asserted-by":"publisher","unstructured":"Rafa\u0142 Demkowicz-Dobrza\u0144ski, Jan Ko\u0142ody\u0144ski, and M\u0103d\u0103lin Gu\u0163\u0103. The elusive Heisenberg limit in quantum-enhanced metrology. Nat. Commun., 3:1063, 2012.","DOI":"10.1038\/ncomms2067"},{"key":"10","doi-asserted-by":"publisher","unstructured":"Rafa\u0142 Demkowicz-Dobrza\u0144ski, Jan Czajkowski, and Pavel Sekatski. Adaptive Quantum Metrology under General Markovian Noise. Phys. Rev. X, 7(4):041009, Oct 2017.","DOI":"10.1103\/PhysRevX.7.041009"},{"key":"11","doi-asserted-by":"publisher","unstructured":"Sisi Zhou, Mengzhen Zhang, John Preskill, and Liang Jiang. Achieving the Heisenberg limit in quantum metrology using quantum error correction. Nat. 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Probabilistic and statistical aspects of quantum theory, volume 1. Springer Science & Business Media, 2011.","DOI":"10.1007\/978-88-7642-378-9"},{"key":"57","doi-asserted-by":"publisher","unstructured":"Carl W Helstrom. Quantum detection and estimation theory. Journal of Statistical Physics, 1:231\u2013252, 1969.","DOI":"10.1007\/BF01007479"},{"key":"58","doi-asserted-by":"publisher","unstructured":"Samuel L Braunstein and Carlton M Caves. Statistical distance and the geometry of quantum states. Physical Review Letters, 72(22):3439, 1994.","DOI":"10.1103\/PhysRevLett.72.3439"},{"key":"59","doi-asserted-by":"publisher","unstructured":"Matteo GA Paris. Quantum estimation for quantum technology. Int. J. Quantum Inf., 7(supp01):125\u2013137, 2009.","DOI":"10.1142\/S0219749909004839"},{"key":"60","doi-asserted-by":"publisher","unstructured":"Matteo Paris and Jaroslav Rehacek. Quantum state estimation, volume 649. Springer Science & Business Media, 2004.","DOI":"10.1007\/b98673"},{"key":"61","doi-asserted-by":"publisher","unstructured":"Charles H Bennett, David P DiVincenzo, John A Smolin, and William K Wootters. Mixed-state entanglement and quantum error correction. Physical Review A, 54(5):3824, 1996.","DOI":"10.1103\/PhysRevA.54.3824"},{"key":"62","doi-asserted-by":"publisher","unstructured":"Emanuel Knill and Raymond Laflamme. Theory of quantum error-correcting codes. Physical Review A, 55(2):900, 1997.","DOI":"10.1103\/PhysRevA.55.900"},{"key":"63","doi-asserted-by":"publisher","unstructured":"Google Quantum AI. Suppressing quantum errors by scaling a surface code logical qubit. Nature, 614(7949):676\u2013681, 2023.","DOI":"10.1038\/s41586-022-05434-1"},{"key":"64","doi-asserted-by":"publisher","unstructured":"Ciaran Ryan-Anderson, Justin G Bohnet, Kenny Lee, Daniel Gresh, Aaron Hankin, J. P. Gaebler, David Francois, Alexander Chernoguzov, Dominic Lucchetti, Natalie C. Brown, et al. 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Physical Review Letters, 77(5):793, 1996.","DOI":"10.1103\/PhysRevLett.77.793"},{"key":"68","doi-asserted-by":"publisher","unstructured":"Jonas T Anderson, Guillaume Duclos-Cianci, and David Poulin. Fault-tolerant conversion between the steane and reed-muller quantum codes. Physical Review Letters, 113(8):080501, 2014.","DOI":"10.1103\/PhysRevLett.113.080501"},{"key":"69","doi-asserted-by":"publisher","unstructured":"Florence Jessie MacWilliams and Neil James Alexander Sloane. The theory of error-correcting codes, volume 16. Elsevier, 1977.","DOI":"10.1002\/9781118032749"},{"key":"70","doi-asserted-by":"publisher","unstructured":"Bei Zeng, Andrew Cross, and Isaac L Chuang. Transversality versus universality for additive quantum codes. IEEE Transactions on Information Theory, 57(9):6272\u20136284, 2011.","DOI":"10.1109\/TIT.2011.2161917"},{"key":"71","doi-asserted-by":"publisher","unstructured":"Hunter S Snevily. A generalization of the ray-chaudhuri-wilson theorem. 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