{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,17]],"date-time":"2026-06-17T02:19:51Z","timestamp":1781662791672,"version":"3.54.5"},"reference-count":49,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2023,6,26]],"date-time":"2023-06-26T00:00:00Z","timestamp":1687737600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,6,26]],"date-time":"2023-06-26T00:00:00Z","timestamp":1687737600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Nat Comput Sci"],"DOI":"10.1038\/s43588-023-00467-6","type":"journal-article","created":{"date-parts":[[2023,6,26]],"date-time":"2023-06-26T16:02:10Z","timestamp":1687795330000},"page":"542-551","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":146,"title":["Theory of overparametrization in quantum neural networks"],"prefix":"10.1038","volume":"3","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8700-4308","authenticated-orcid":false,"given":"Mart\u00edn","family":"Larocca","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nathan","family":"Ju","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Diego","family":"Garc\u00eda-Mart\u00edn","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9879-8425","authenticated-orcid":false,"given":"Patrick J.","family":"Coles","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2757-3170","authenticated-orcid":false,"given":"Marco","family":"Cerezo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,6,26]]},"reference":[{"key":"467_CR1","unstructured":"Mohri, M., Rostamizadeh, A. & Talwalkar, A. Foundations of Machine Learning (MIT Press, 2018)."},{"key":"467_CR2","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1038\/s41573-019-0024-5","volume":"18","author":"J Vamathevan","year":"2019","unstructured":"Vamathevan, J. et al. Applications of machine learning in drug discovery and development. Nat. Rev. Drug Discov. 18, 463\u2013477 (2019).","journal-title":"Nat. Rev. Drug Discov."},{"key":"467_CR3","doi-asserted-by":"publisher","DOI":"10.1038\/s41524-019-0221-0","volume":"5","author":"J Schmidt","year":"2019","unstructured":"Schmidt, J., Marques, M. R., Botti, S. & Marques, M. A. Recent advances and applications of machine learning in solid-state materials science. npj Comput. Mater. 5, 83 (2019).","journal-title":"npj Comput. Mater."},{"key":"467_CR4","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1016\/S0893-6080(05)80010-3","volume":"5","author":"AL Blum","year":"1992","unstructured":"Blum, A. L. & Rivest, R. L. Training a 3-node neural network is NP-complete. Neural Netw. 5, 117\u2013127 (1992).","journal-title":"Neural Netw."},{"key":"467_CR5","doi-asserted-by":"publisher","unstructured":"Neyshabur, B., Li, Z., Bhojanapalli, S., LeCun, Y., & Srebro, N. Towards understanding the role of over-parametrization in generalization of neural networks. Preprint at arXiv https:\/\/doi.org\/10.48550\/arXiv.1805.12076 (2018).","DOI":"10.48550\/arXiv.1805.12076"},{"key":"467_CR6","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1145\/3446776","volume":"64","author":"C Zhang","year":"2021","unstructured":"Zhang, C., Bengio, S., Hardt, M., Recht, B. & Vinyals, O. Understanding deep learning (still) requires rethinking generalization. Commun. ACM 64, 107\u2013115 (2021).","journal-title":"Commun. ACM"},{"key":"467_CR7","unstructured":"Allen-Zhu, Z., Li, Y. & Song, Z. A convergence theory for deep learning via over-parameterization. In Proceedings of the 36th International Conference on Machine Learning 242\u2013252 (PMLR, 2019)."},{"key":"467_CR8","doi-asserted-by":"publisher","unstructured":"Du, S. S., Zhai, X., Poczos, B., & Singh, A. Gradient descent provably optimizes over-parameterized neural networks. Preprint at arXiv https:\/\/doi.org\/10.48550\/arXiv.1810.02054 (2018).","DOI":"10.48550\/arXiv.1810.02054"},{"key":"467_CR9","doi-asserted-by":"publisher","first-page":"79","DOI":"10.22331\/q-2018-08-06-79","volume":"2","author":"J Preskill","year":"2018","unstructured":"Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).","journal-title":"Quantum"},{"key":"467_CR10","doi-asserted-by":"publisher","first-page":"625","DOI":"10.1038\/s42254-021-00348-9","volume":"3","author":"M Cerezo","year":"2021","unstructured":"Cerezo, M. et al. Variational quantum algorithms. Nat. Rev. Phys. 3, 625\u2013644 (2021).","journal-title":"Nat. Rev. Phys."},{"key":"467_CR11","doi-asserted-by":"publisher","first-page":"015004","DOI":"10.1103\/RevModPhys.94.015004","volume":"94","author":"K Bharti","year":"2022","unstructured":"Bharti, K. et al. Noisy intermediate-scale quantum algorithms. Rev. Mod. Phys. 94, 015004 (2022).","journal-title":"Rev. Mod. Phys."},{"key":"467_CR12","doi-asserted-by":"publisher","first-page":"2631","DOI":"10.1038\/s41467-021-22539-9","volume":"12","author":"H-Y Huang","year":"2021","unstructured":"Huang, H.-Y. et al. Power of data in quantum machine learning. Nat. Commun. 12, 2631 (2021).","journal-title":"Nat. Commun."},{"key":"467_CR13","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1038\/s43588-021-00084-1","volume":"1","author":"A Abbas","year":"2021","unstructured":"Abbas, A. et al. The power of quantum neural networks. Nat. Comput. Sci. 1, 403\u2013409 (2021).","journal-title":"Nat. Comput. Sci."},{"key":"467_CR14","doi-asserted-by":"publisher","first-page":"120502","DOI":"10.1103\/PhysRevLett.127.120502","volume":"127","author":"L Bittel","year":"2021","unstructured":"Bittel, L. & Kliesch, M. Training variational quantum algorithms is NP-hard. Phys. Rev. Lett. 127, 120502 (2021).","journal-title":"Phys. Rev. Lett."},{"key":"467_CR15","doi-asserted-by":"publisher","first-page":"043246","DOI":"10.1103\/PhysRevResearch.2.043246","volume":"2","author":"D Wierichs","year":"2020","unstructured":"Wierichs, D., Gogolin, C. & Kastoryano, M. Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer. Phys. Rev. Res. 2, 043246 (2020).","journal-title":"Phys. Rev. Res."},{"key":"467_CR16","doi-asserted-by":"publisher","unstructured":"Anschuetz, E. R. Critical points in quantum generative models. Preprint at arXiv https:\/\/doi.org\/10.48550\/arXiv.2109.06957 (2021).","DOI":"10.48550\/arXiv.2109.06957"},{"key":"467_CR17","doi-asserted-by":"publisher","first-page":"4812","DOI":"10.1038\/s41467-018-07090-4","volume":"9","author":"JR McClean","year":"2018","unstructured":"McClean, J. R., Boixo, S., Smelyanskiy, V. N., Babbush, R. & Neven, H. Barren plateaus in quantum neural network training landscapes. Nat. Commun. 9, 4812 (2018).","journal-title":"Nat. Commun."},{"key":"467_CR18","doi-asserted-by":"publisher","first-page":"1791","DOI":"10.1038\/s41467-021-21728-w","volume":"12","author":"M Cerezo","year":"2021","unstructured":"Cerezo, M., Sone, A., Volkoff, T., Cincio, L. & Coles, P. J. Cost function dependent barren plateaus in shallow parametrized quantum circuits. Nat. Commun. 12, 1791 (2021).","journal-title":"Nat. Commun."},{"key":"467_CR19","doi-asserted-by":"publisher","first-page":"6961","DOI":"10.1038\/s41467-021-27045-6","volume":"12","author":"S Wang","year":"2021","unstructured":"Wang, S. et al. Noise-induced barren plateaus in variational quantum algorithms. Nat. Commun. 12, 6961 (2021).","journal-title":"Nat. Commun."},{"key":"467_CR20","doi-asserted-by":"publisher","first-page":"1221","DOI":"10.1038\/s41567-021-01356-3","volume":"17","author":"D Stilck Fran\u00e7a","year":"2021","unstructured":"Stilck Fran\u00e7a, D. & Garcia-Patron, R. Limitations of optimization algorithms on noisy quantum devices. Nat. Phys. 17, 1221\u20131227 (2021).","journal-title":"Nat. Phys."},{"key":"467_CR21","doi-asserted-by":"publisher","first-page":"020319","DOI":"10.1103\/PRXQuantum.1.020319","volume":"1","author":"R Wiersema","year":"2020","unstructured":"Wiersema, R. et al. Exploring entanglement and optimization within the Hamiltonian variational ansatz. PRX Quantum 1, 020319 (2020).","journal-title":"PRX Quantum"},{"key":"467_CR22","doi-asserted-by":"publisher","unstructured":"Kiani, B. T., Lloyd, S. & Maity, R. Learning unitaries by gradient descent. Preprint at arXiv https:\/\/doi.org\/10.48550\/arXiv.2001.11897 (2020).","DOI":"10.48550\/arXiv.2001.11897"},{"key":"467_CR23","doi-asserted-by":"crossref","unstructured":"Funcke, L. et al. Best-approximation error for parametric quantum circuits. In 2021 IEEE International Conference on Web Services (ICWS) 693\u2013702 (IEEE, 2021).","DOI":"10.1109\/ICWS53863.2021.00096"},{"key":"467_CR24","doi-asserted-by":"publisher","first-page":"032401","DOI":"10.1103\/PhysRevA.104.032401","volume":"104","author":"J Lee","year":"2021","unstructured":"Lee, J., Magann, A. B., Rabitz, H. A. & Arenz, C. Progress toward favorable landscapes in quantum combinatorial optimization. Phys. Rev. A 104, 032401 (2021).","journal-title":"Phys. Rev. A"},{"key":"467_CR25","doi-asserted-by":"publisher","first-page":"113510","DOI":"10.1063\/1.3657939","volume":"52","author":"R Zeier","year":"2011","unstructured":"Zeier, R. & Schulte-Herbr\u00fcggen, T. Symmetry principles in quantum systems theory. J. Math. Phys. 52, 113510 (2011).","journal-title":"J. Math. Phys."},{"key":"467_CR26","doi-asserted-by":"publisher","first-page":"539","DOI":"10.22331\/q-2021-09-09-539","volume":"5","author":"JJ Meyer","year":"2021","unstructured":"Meyer, J. J. Fisher information in noisy intermediate-scale quantum applications. Quantum 5, 539 (2021).","journal-title":"Quantum"},{"key":"467_CR27","doi-asserted-by":"publisher","first-page":"134113","DOI":"10.1063\/1.4757133","volume":"137","author":"KW Moore","year":"2012","unstructured":"Moore, K. W. & Rabitz, H. Exploring constrained quantum control landscapes. J. Chem. Phys. 137, 134113 (2012).","journal-title":"J. Chem. Phys."},{"key":"467_CR28","doi-asserted-by":"publisher","first-page":"023410","DOI":"10.1103\/PhysRevA.101.023410","volume":"101","author":"M Larocca","year":"2020","unstructured":"Larocca, M., Calzetta, E. & Wisniacki, D. A. Exploiting landscape geometry to enhance quantum optimal control. Phys. Rev. A 101, 023410 (2020).","journal-title":"Phys. Rev. A"},{"key":"467_CR29","doi-asserted-by":"publisher","first-page":"013405","DOI":"10.1103\/PhysRevA.86.013405","volume":"86","author":"R-B Wu","year":"2012","unstructured":"Wu, R.-B., Long, R., Dominy, J., Ho, T.-S. & Rabitz, H. Singularities of quantum control landscapes. Phys. Rev. A 86, 013405 (2012).","journal-title":"Phys. Rev. A"},{"key":"467_CR30","doi-asserted-by":"publisher","first-page":"062343","DOI":"10.1103\/PhysRevA.92.062343","volume":"92","author":"N Rach","year":"2015","unstructured":"Rach, N., M\u00fcller, M. M., Calarco, T. & Montangero, S. Dressing the chopped-random-basis optimization: a bandwidth-limited access to the trap-free landscape. Phys. Rev. A 92, 062343 (2015).","journal-title":"Phys. Rev. A"},{"key":"467_CR31","doi-asserted-by":"publisher","first-page":"385305","DOI":"10.1088\/1751-8121\/aad657","volume":"51","author":"M Larocca","year":"2018","unstructured":"Larocca, M., Poggi, P. M. & Wisniacki, D. A. Quantum control landscape for a two-level system near the quantum speed limit. J. Phys. A Math. Theor. 51, 385305 (2018).","journal-title":"J. Phys. A Math. Theor."},{"key":"467_CR32","doi-asserted-by":"publisher","first-page":"824","DOI":"10.22331\/q-2022-09-29-824","volume":"6","author":"M Larocca","year":"2022","unstructured":"Larocca, M. et al. Diagnosing barren plateaus with tools from quantum optimal control. Quantum 6, 824 (2022).","journal-title":"Quantum"},{"key":"467_CR33","doi-asserted-by":"publisher","first-page":"040309","DOI":"10.1103\/PRXQuantum.2.040309","volume":"2","author":"T Haug","year":"2021","unstructured":"Haug, T., Bharti, K. & Kim, M. S. Capacity and quantum geometry of parametrized quantum circuits. PRX Quantum 2, 040309 (2021).","journal-title":"PRX Quantum"},{"key":"467_CR34","doi-asserted-by":"publisher","first-page":"033108","DOI":"10.1103\/PhysRevA.102.033108","volume":"102","author":"M Larocca","year":"2020","unstructured":"Larocca, M., Calzetta, E. & Wisniacki, D. Fourier compression: a customization method for quantum control protocols. Phys. Rev. A 102, 033108 (2020).","journal-title":"Phys. Rev. A"},{"key":"467_CR35","doi-asserted-by":"publisher","first-page":"015018","DOI":"10.1088\/2058-9565\/ac39f5","volume":"7","author":"S Efthymiou","year":"2021","unstructured":"Efthymiou, S. et al. Qibo: a framework for quantum simulation with hardware acceleration. Quantum Sci. Technol. 7, 015018 (2021).","journal-title":"Quantum Sci. Technol."},{"key":"467_CR36","doi-asserted-by":"publisher","first-page":"4213","DOI":"10.1038\/ncomms5213","volume":"5","author":"A Peruzzo","year":"2014","unstructured":"Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).","journal-title":"Nat. Commun."},{"key":"467_CR37","doi-asserted-by":"publisher","unstructured":"Garc\u00eda-Mart\u00edn, D., Larocca, M. & Cerezo, M. Effects of noise on the overparametrization of quantum neural networks. Preprint at arXiv https:\/\/doi.org\/10.48550\/arXiv.2302.05059 (2023).","DOI":"10.48550\/arXiv.2302.05059"},{"key":"467_CR38","doi-asserted-by":"publisher","first-page":"871","DOI":"10.1016\/0893-6080(95)00119-0","volume":"9","author":"K Fukumizu","year":"1996","unstructured":"Fukumizu, K. A regularity condition of the information matrix of a multilayer perceptron network. Neural Netw. 9, 871\u2013879 (1996).","journal-title":"Neural Netw."},{"key":"467_CR39","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1080\/00029890.1985.11971675","volume":"92","author":"N Chan","year":"1985","unstructured":"Chan, N. & Kwong, M. K. Hermitian matrix inequalities and a conjecture. Am. Math. Monthly 92, 533\u2013541 (1985).","journal-title":"Am. Math. Monthly"},{"key":"467_CR40","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1140\/epjd\/e2015-60464-1","volume":"69","author":"SJ Glaser","year":"2015","unstructured":"Glaser, S. J. et al. Training Schr\u00f6dinger\u2019s cat: quantum optimal control. Eur. Phys. J. D 69, 279 (2015).","journal-title":"Eur. Phys. J. D"},{"key":"467_CR41","doi-asserted-by":"publisher","first-page":"024701","DOI":"10.1116\/5.0006785","volume":"2","author":"P Rembold","year":"2020","unstructured":"Rembold, P. et al. Introduction to quantum optimal control for quantum sensing with nitrogen-vacancy centers in diamond. AVS Quantum Sci. 2, 024701 (2020).","journal-title":"AVS Quantum Sci."},{"key":"467_CR42","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1038\/s41586-021-03582-4","volume":"595","author":"S Ebadi","year":"2021","unstructured":"Ebadi, S. et al. Quantum phases of matter on a 256-atom programmable quantum simulator. Nature 595, 227\u2013232 (2021).","journal-title":"Nature"},{"key":"467_CR43","doi-asserted-by":"publisher","first-page":"250502","DOI":"10.1103\/PhysRevLett.129.250502","volume":"129","author":"AB Magann","year":"2022","unstructured":"Magann, A. B., Rudinger, K. M., Grace, M. D. & Sarovar, M. Feedback-based quantum optimization. Phys. Rev. Lett. 129, 250502 (2022).","journal-title":"Phys. Rev. Lett."},{"key":"467_CR44","doi-asserted-by":"publisher","first-page":"010101","DOI":"10.1103\/PRXQuantum.2.010101","volume":"2","author":"AB Magann","year":"2021","unstructured":"Magann, A. B. et al. From pulses to circuits and back again: a quantum optimal control perspective on variational quantum algorithms. PRX Quantum 2, 010101 (2021).","journal-title":"PRX Quantum"},{"key":"467_CR45","doi-asserted-by":"publisher","first-page":"104109","DOI":"10.1063\/1.2981796","volume":"130","author":"M Hsieh","year":"2009","unstructured":"Hsieh, M., Wu, R. & Rabitz, H. Topology of the quantum control landscape for observables. J. Chem. Phys. 130, 104109 (2009).","journal-title":"J. Chem. Phys."},{"key":"467_CR46","doi-asserted-by":"publisher","first-page":"013422","DOI":"10.1103\/PhysRevA.79.013422","volume":"79","author":"T-S Ho","year":"2009","unstructured":"Ho, T.-S., Dominy, J. & Rabitz, H. Landscape of unitary transformations in controlled quantum dynamics. Phys. Rev. A 79, 013422 (2009).","journal-title":"Phys. Rev. A"},{"key":"467_CR47","doi-asserted-by":"publisher","first-page":"013404","DOI":"10.1103\/PhysRevA.90.013404","volume":"90","author":"G Riviello","year":"2014","unstructured":"Riviello, G. et al. Searching for quantum optimal control fields in the presence of singular critical points. Phys. Rev. A 90, 013404 (2014).","journal-title":"Phys. Rev. A"},{"key":"467_CR48","doi-asserted-by":"publisher","unstructured":"Schatzki, L., Arrasmith, A., Coles, P. J. & Cerezo, M. Entangled datasets for quantum machine learning. Preprint at arXiv https:\/\/doi.org\/10.48550\/arXiv.2109.03400 (2021).","DOI":"10.48550\/arXiv.2109.03400"},{"key":"467_CR49","doi-asserted-by":"publisher","unstructured":"Garc\u00eda-Mart\u00edn, D. DiegoGM91\/theory-of-overparametrization: v0.0.1. Zenodo https:\/\/doi.org\/10.5281\/zenodo.7916659 (2023).","DOI":"10.5281\/zenodo.7916659"}],"container-title":["Nature Computational Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.nature.com\/articles\/s43588-023-00467-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s43588-023-00467-6","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s43588-023-00467-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,26]],"date-time":"2023-06-26T16:10:51Z","timestamp":1687795851000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.nature.com\/articles\/s43588-023-00467-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,26]]},"references-count":49,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["467"],"URL":"https:\/\/doi.org\/10.1038\/s43588-023-00467-6","relation":{},"ISSN":["2662-8457"],"issn-type":[{"value":"2662-8457","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,26]]},"assertion":[{"value":"11 April 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 May 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 June 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}