{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T17:41:24Z","timestamp":1779212484339,"version":"3.51.4"},"reference-count":92,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2024,7,19]],"date-time":"2024-07-19T00:00:00Z","timestamp":1721347200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,7,19]],"date-time":"2024-07-19T00:00:00Z","timestamp":1721347200000},"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":["K\u00fcnstl Intell"],"published-print":{"date-parts":[[2024,12]]},"DOI":"10.1007\/s13218-024-00856-7","type":"journal-article","created":{"date-parts":[[2024,7,19]],"date-time":"2024-07-19T13:02:02Z","timestamp":1721394122000},"page":"277-291","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Quantum Supervised Learning"],"prefix":"10.1007","volume":"38","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1348-250X","authenticated-orcid":false,"given":"Antonio","family":"Macaluso","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,7,19]]},"reference":[{"issue":"7553","key":"856_CR1","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1038\/nature14539","volume":"521","author":"Y LeCun","year":"2015","unstructured":"LeCun Y, Bengio Y, Hinton G (2015) Deep learning. Nature 521(7553):436\u2013444","journal-title":"Nature"},{"key":"856_CR2","unstructured":"Krizhevsky A, Sutskever I, Hinton GE (2012) Imagenet classification with deep convolutional neural networks. Adv Neural Inf Process Syst 25"},{"key":"856_CR3","unstructured":"Thompson NC, Greenewald K, Lee K, Manso GF (2020) The computational limits of deep learning. arXiv preprint arXiv:2007.05558"},{"key":"856_CR4","doi-asserted-by":"crossref","unstructured":"Chambers JM (2017) Linear models. In Statistical models in S, pp 95\u2013144. Routledge","DOI":"10.1201\/9780203738535-4"},{"issue":"3","key":"856_CR5","doi-asserted-by":"crossref","first-page":"1171","DOI":"10.1214\/009053607000000677","volume":"36","author":"T Hofmann","year":"2008","unstructured":"Hofmann T, Sch\u00f6lkopf B, Smola AJ (2008) Kernel methods in machine learning. Ann Stat 36(3):1171\u20131220","journal-title":"Ann Stat"},{"key":"856_CR6","doi-asserted-by":"crossref","unstructured":"Boser BE, Guyon IM, Vapnik VN (1992) A training algorithm for optimal margin classifiers. In Proceedings of the fifth annual workshop on Computational learning theory, pp 144\u2013152","DOI":"10.1145\/130385.130401"},{"key":"856_CR7","unstructured":"Nair V, Hinton GE (2010) Rectified linear units improve restricted Boltzmann machines. In Proceedings of the 27th international conference on machine learning (ICML-10), pp 807\u2013814"},{"key":"856_CR8","unstructured":"Glorot X, Bengio Y (2010) Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the thirteenth international conference on artificial intelligence and statistics, pp 249\u2013256. JMLR Workshop and Conference Proceedings"},{"key":"856_CR9","volume-title":"Quantum computation and quantum information","author":"MA Nielsen","year":"2010","unstructured":"Nielsen MA, Chuang IL (2010) Quantum computation and quantum information. Cambridge University Press"},{"issue":"2","key":"856_CR10","volume":"2","author":"P Reddy","year":"2021","unstructured":"Reddy P, Bhattacherjee AB (2021) A hybrid quantum regression model for the prediction of molecular atomization energies. Mach Learn 2(2):025019","journal-title":"Mach Learn"},{"key":"856_CR11","unstructured":"Wang C-CJ, Bennink RS (2023) Variational quantum regression algorithm with encoded data structure. arXiv preprint arXiv:2307.03334"},{"issue":"3","key":"856_CR12","volume":"101","author":"M Schuld","year":"2020","unstructured":"Schuld M, Bocharov A, Svore KM, Wiebe N (2020) Circuit-centric quantum classifiers. Phys Rev A 101(3):032308","journal-title":"Phys Rev A"},{"key":"856_CR13","doi-asserted-by":"crossref","unstructured":"Inajetovic MA, Orazi F, Macaluso A, Lodi S, Sartori C (2023) Enabling non-linear quantum operations through variational quantum splines. In: Miky\u0161ka J, de\u00a0Mulatier C, Paszynski M, Krzhizhanovskaya VV, Dongarra JJ and Sloot PMA (eds) Computational Science \u2013 ICCS 2023, pp 177\u2013192, Cham, 2023. Springer Nature Switzerland","DOI":"10.1007\/978-3-031-36030-5_14"},{"key":"856_CR14","doi-asserted-by":"crossref","unstructured":"Macaluso A, Clissa L, Lodi S, Sartori C (2020) Quantum splines for non-linear approximations. In Proceedings of the 17th ACM International Conference on Computing Frontiers, pp 249\u2013252","DOI":"10.1145\/3387902.3394032"},{"key":"856_CR15","volume-title":"Pattern recognition and machine learning (information science and statistics)","author":"CM Bishop","year":"2006","unstructured":"Bishop CM (2006) Pattern recognition and machine learning (information science and statistics). Springer-Verlag, Berlin"},{"key":"856_CR16","unstructured":"Vapnik V (1991) Principles of risk minimization for learning theory. Adv Neural Inf Process Syst 4"},{"key":"856_CR17","unstructured":"Russell SJ, Norvig P (2010) Artificial intelligence a modern approach. London"},{"key":"856_CR18","volume-title":"Applied linear statistical models. McGrwa-Hill international edition","author":"MH Kutner","year":"2005","unstructured":"Kutner MH (2005) Applied linear statistical models. McGrwa-Hill international edition. McGraw-Hill, Irwin"},{"key":"856_CR19","series-title":"Springer series in statistics","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-21606-5","volume-title":"The elements of statistical learning","author":"T Hastie","year":"2001","unstructured":"Hastie T, Tibshirani R, Friedman J (2001) The elements of statistical learning. Springer series in statistics. Springer New York Inc., New York"},{"key":"856_CR20","doi-asserted-by":"crossref","unstructured":"Hastie T, Tibshirani R, Friedman J, Hastie T, Tibshirani R, Friedman J (2009) Basis expansions and regularization. The elements of statistical learning: data mining, inference, and prediction, pp 139\u2013189","DOI":"10.1007\/978-0-387-84858-7_5"},{"key":"856_CR21","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-6333-3","volume-title":"A practical guide to splines","author":"C De Boor","year":"1978","unstructured":"De Boor C, De Boor C, Math\u00e9maticien E-U, De Boor C, De Boor C (1978) A practical guide to splines, vol 27. Springer-verlag, New York"},{"key":"856_CR22","volume-title":"Learning with kernels","author":"AJ Smola","year":"1998","unstructured":"Smola AJ, Sch\u00f6lkopf B (1998) Learning with kernels, vol 4. Citeseer"},{"issue":"13","key":"856_CR23","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevLett.113.130503","volume":"113","author":"P Rebentrost","year":"2014","unstructured":"Rebentrost P, Mohseni M, Lloyd S (2014) Quantum support vector machine for big data classification. Phys Rev Lett 113(13):130503","journal-title":"Phys Rev Lett"},{"key":"856_CR24","unstructured":"Ye J, Xiong T (2007) Svm versus least squares svm. In Artificial intelligence and statistics, pp 644\u2013651. PMLR"},{"key":"856_CR25","volume":"96","author":"G Wang","year":"2017","unstructured":"Wang G (2017) Quantum algorithm for linear regression. Phys Rev A 96:012335","journal-title":"Phys Rev A"},{"key":"856_CR26","doi-asserted-by":"crossref","unstructured":"Macaluso A, Clissa L, Lodi S, Sartori C (2020) A variational algorithm for quantum neural networks. In: Computational Science\u2013ICCS 2020: 20th International Conference, Amsterdam, The Netherlands, June 3\u20135, 2020, Proceedings, Part VI 20, pp 591\u2013604. Springer","DOI":"10.1007\/978-3-030-50433-5_45"},{"issue":"3","key":"856_CR27","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/s42484-019-00007-4","volume":"1","author":"R Mengoni","year":"2019","unstructured":"Mengoni R, Di Pierro A (2019) Kernel methods in quantum machine learning. Quant Mach Intell 1(3):65\u201371","journal-title":"Quant Mach Intell"},{"key":"856_CR28","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevLett.103.150502","volume":"103","author":"AW Harrow","year":"2009","unstructured":"Harrow AW, Hassidim A, Lloyd S (2009) Quantum algorithm for linear systems of equations. Phys Rev Lett 103:150502","journal-title":"Phys Rev Lett"},{"issue":"5","key":"856_CR29","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1162\/neco.1994.6.5.851","volume":"6","author":"V Vapnik","year":"1994","unstructured":"Vapnik V, Levin E, Le Cun Y (1994) Measuring the vc-dimension of a learning machine. Neural Comput 6(5):851\u2013876","journal-title":"Neural Comput"},{"key":"856_CR30","unstructured":"Berezniuk O, Figalli A, Ghigliazza R, Musaelian K (2020) A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872"},{"issue":"3","key":"856_CR31","doi-asserted-by":"crossref","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 (2021) Understanding deep learning (still) requires rethinking generalization. Commun ACM 64(3):107\u2013115","journal-title":"Commun ACM"},{"key":"856_CR32","unstructured":"Allen-Zhu Z, Li Y, Liang Y (2019) Learning and generalization in overparameterized neural networks, going beyond two layers. Adv Neural Inf Process Syst 32"},{"key":"856_CR33","unstructured":"Zhang C, Bengio S, Moritz H, Recht B, Oriol V (2017) Understanding deep learning requires rethinking generalization. In International Conference on Learning Representations"},{"key":"856_CR34","volume-title":"Advances in neural information processing systems","author":"B Neyshabur","year":"2017","unstructured":"Neyshabur B, Bhojanapalli S, Mcallester D, Srebro N (2017) Exploring generalization in deep learning. In: Guyon I, Von Luxburg U, Bengio S, Wallach H, Fergus R, Vishwanathan S, Garnett R (eds) Advances in neural information processing systems, vol 30. Curran Associates, Inc"},{"key":"856_CR35","first-page":"247","volume-title":"Fault-tolerant quantum machine learning","author":"M Schuld","year":"2021","unstructured":"Schuld M, Petruccione F (2021) Fault-tolerant quantum machine learning. Springer International Publishing, Cham, pp 247\u2013272"},{"key":"856_CR36","doi-asserted-by":"crossref","unstructured":"Gujju Y, Matsuo A, Raymond R (2023) Quantum machine learning on near-term quantum devices: current state of supervised and unsupervised techniques for real-world applications. arXiv preprint arXiv:2307.00908","DOI":"10.1103\/PhysRevApplied.21.067001"},{"key":"856_CR37","doi-asserted-by":"crossref","unstructured":"Wang Y, Liu J (2024) Quantum machine learning: from nisq to fault tolerance. arXiv preprint arXiv:2401.11351","DOI":"10.1088\/1361-6633\/ad7f69"},{"issue":"2","key":"856_CR38","doi-asserted-by":"crossref","first-page":"21308","DOI":"10.1007\/s11467-022-1249-z","volume":"18","author":"B Cheng","year":"2023","unstructured":"Cheng B, Deng X-H, Xiu G, He Yu, Guangchong H, Huang P, Li J, Lin B-C, Dawei L, Yao L et al (2023) Noisy intermediate-scale quantum computers. Front Phys 18(2):21308","journal-title":"Front Phys"},{"key":"856_CR39","doi-asserted-by":"crossref","first-page":"79","DOI":"10.22331\/q-2018-08-06-79","volume":"2","author":"J Preskill","year":"2018","unstructured":"Preskill J (2018) Quantum computing in the nisq era and beyond. Quantum 2:79","journal-title":"Quantum"},{"key":"856_CR40","doi-asserted-by":"crossref","unstructured":"Ciliberto C, Herbster M, Ialongo AD, Pontil M, Rocchetto A, Severini S, Wossnig L (2018) (2209) Quantum machine learning: a classical perspective. Proc R Soc A 474:20170551","DOI":"10.1098\/rspa.2017.0551"},{"issue":"7671","key":"856_CR41","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1038\/nature23474","volume":"549","author":"J Biamonte","year":"2017","unstructured":"Biamonte J, Wittek P, Pancotti N, Rebentrost P, Wiebe N, Lloyd S (2017) Quantum machine learning. Nature 549(7671):195","journal-title":"Nature"},{"key":"856_CR42","doi-asserted-by":"crossref","unstructured":"Paler A, Devitt SJ (2015) An introduction into fault-tolerant quantum computing. In Proceedings of the 52nd Annual Design Automation Conference, pp 1\u20136","DOI":"10.1145\/2744769.2747911"},{"issue":"5","key":"856_CR43","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevLett.109.050505","volume":"109","author":"N Wiebe","year":"2012","unstructured":"Wiebe N, Braun D, Lloyd S (2012) Quantum algorithm for data fitting. Phys Rev Lett 109(5):050505","journal-title":"Phys Rev Lett"},{"key":"856_CR44","volume-title":"Algebraic complexity theory","author":"P B\u00fcrgisser","year":"2013","unstructured":"B\u00fcrgisser P, Clausen M, Shokrollahi MA (2013) Algebraic complexity theory, vol 315. Springer Science & Business Media"},{"key":"856_CR45","doi-asserted-by":"crossref","unstructured":"Coppersmith D, Winograd S (1987) Matrix multiplication via arithmetic progressions. In Proceedings of the nineteenth annual ACM symposium on Theory of computing, pp 1\u20136","DOI":"10.1145\/28395.28396"},{"issue":"4","key":"856_CR46","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1007\/BF02165411","volume":"13","author":"V Strassen","year":"1969","unstructured":"Strassen V (1969) Gaussian elimination is not optimal. Numer Math 13(4):354\u2013356","journal-title":"Numer Math"},{"key":"856_CR47","unstructured":"Shewchuk JR et\u00a0al (1994) An introduction to the conjugate gradient method without the agonizing pain"},{"issue":"4","key":"856_CR48","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1038\/nphys3272","volume":"11","author":"S Aaronson","year":"2015","unstructured":"Aaronson S (2015) Read the fine print. Nat Phys 11(4):291","journal-title":"Nat Phys"},{"issue":"3","key":"856_CR49","first-page":"28","volume":"128","author":"A Abdiansah","year":"2015","unstructured":"Abdiansah A, Wardoyo R (2015) Time complexity analysis of support vector machines (svm) in libsvm. Int J Comput Appl 128(3):28\u201334","journal-title":"Int J Comput Appl"},{"key":"856_CR50","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-6333-3","volume-title":"A practical guide to splines","author":"C de Boor","year":"1978","unstructured":"de Boor C (1978) A practical guide to splines. Springer Verlag, New York"},{"issue":"1","key":"856_CR51","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/0021-9045(72)90080-9","volume":"6","author":"C De Boor","year":"1972","unstructured":"De Boor C (1972) On calculating with b-splines. J Approx Theory 6(1):50\u201362","journal-title":"J Approx Theory"},{"key":"856_CR52","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevLett.87.167902","volume":"87","author":"H Buhrman","year":"2001","unstructured":"Buhrman H, Cleve R, Watrous J, de Wolf R (2001) Quantum fingerprinting. Phys Rev Lett 87:167902","journal-title":"Phys Rev Lett"},{"key":"856_CR53","doi-asserted-by":"crossref","unstructured":"Macaluso A, Orazi F, Klusch M, Lodi S, Sartori C (2022) A variational algorithm for quantum single layer perceptron. In International Conference on Machine Learning, Optimization, and Data Science, pp 341\u2013356. Springer","DOI":"10.1007\/978-3-031-25891-6_26"},{"issue":"1","key":"856_CR54","volume":"94","author":"B Kishor","year":"2022","unstructured":"Kishor B, Alba C-L, Ha KT, Tobias H, Sumner A-L, Abhinav A, Matthias D, Hermanni H, Kottmann Jakob S, Tim M et al (2022) Noisy intermediate-scale quantum algorithms. Rev Mod Phys 94(1):015004","journal-title":"Rev Mod Phys"},{"key":"856_CR55","volume":"92","author":"D Wecker","year":"2015","unstructured":"Wecker D, Hastings MB, Troyer M (2015) Progress towards practical quantum variational algorithms. Phys Rev A 92:042303","journal-title":"Phys Rev A"},{"issue":"3","key":"856_CR56","volume":"3","author":"N Moll","year":"2018","unstructured":"Moll N, Barkoutsos P, Bishop LS, Chow JM, Cross A, Egger DJ, Filipp S, Fuhrer A, Gambetta JM, Ganzhorn M et al (2018) Quantum optimization using variational algorithms on near-term quantum devices. Quant Sci Technol 3(3):030503","journal-title":"Quant Sci Technol"},{"issue":"4","key":"856_CR57","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1049\/qtc2.12032","volume":"2","author":"M Weigold","year":"2021","unstructured":"Weigold M, Barzen J, Leymann F, Salm M (2021) Encoding patterns for quantum algorithms. IET Quant Commun 2(4):141\u2013152","journal-title":"IET Quant Commun"},{"key":"856_CR58","doi-asserted-by":"crossref","unstructured":"Gil-Fuster E, Eisert J, Dunjko V (2023) On the expressivity of embedding quantum kernels. arXiv preprint arXiv:2309.14419","DOI":"10.1088\/2632-2153\/ad2f51"},{"key":"856_CR59","doi-asserted-by":"crossref","unstructured":"Schuld M (2021) Supervised quantum machine learning models are kernel methods. arXiv preprint arXiv:2101.11020","DOI":"10.1007\/978-3-030-83098-4_6"},{"key":"856_CR60","doi-asserted-by":"crossref","first-page":"226","DOI":"10.22331\/q-2020-02-06-226","volume":"4","author":"A P\u00e9rez-Salinas","year":"2020","unstructured":"P\u00e9rez-Salinas A, Cervera-Lierta A, Gil-Fuster E, Latorre JI (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226","journal-title":"Quantum"},{"issue":"1","key":"856_CR61","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1038\/s41467-023-36159-y","volume":"14","author":"S Jerbi","year":"2023","unstructured":"Jerbi S, Fiderer LJ, Nautrup HP, K\u00fcbler JM, Briegel HJ, Dunjko V (2023) Quantum machine learning beyond kernel methods. Nat Commun 14(1):517","journal-title":"Nat Commun"},{"issue":"7747","key":"856_CR62","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1038\/s41586-019-0980-2","volume":"567","author":"V Havl\u00ed\u010dek","year":"2019","unstructured":"Havl\u00ed\u010dek V, C\u00f3rcoles AD, Temme K, Harrow AW, Kandala A, Chow JM, Gambetta JM (2019) Supervised learning with quantum-enhanced feature spaces. Nature 567(7747):209\u2013212","journal-title":"Nature"},{"issue":"12","key":"856_CR63","doi-asserted-by":"crossref","first-page":"1273","DOI":"10.1038\/s41567-019-0648-8","volume":"15","author":"I Cong","year":"2019","unstructured":"Cong I, Choi S, Lukin MD (2019) Quantum convolutional neural networks. Nat Phys 15(12):1273\u20131278","journal-title":"Nat Phys"},{"issue":"1","key":"856_CR64","doi-asserted-by":"crossref","first-page":"4919","DOI":"10.1038\/s41467-022-32550-3","volume":"13","author":"MC Caro","year":"2022","unstructured":"Caro MC, Huang H-Y, Cerezo M, Sharma K, Sornborger A, Cincio L, Coles PJ (2022) Generalization in quantum machine learning from few training data. Nat Commun 13(1):4919","journal-title":"Nat Commun"},{"issue":"3","key":"856_CR65","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1007\/s11128-023-03901-w","volume":"22","author":"A Macaluso","year":"2023","unstructured":"Macaluso A, Klusch M, Lodi S, Sartori C (2023) Maqa: a quantum framework for supervised learning. Quantum Inf Process 22(3):159","journal-title":"Quantum Inf Process"},{"issue":"5","key":"856_CR66","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/0893-6080(89)90020-8","volume":"2","author":"K Hornik","year":"1989","unstructured":"Hornik K, Stinchcombe M, White H (1989) Multilayer feedforward networks are universal approximators. Neural Netw 2(5):359\u2013366","journal-title":"Neural Netw"},{"key":"856_CR67","doi-asserted-by":"crossref","unstructured":"Macaluso A, Clissa L, Lodi S, Sartori C (2024) An efficient quantum algorithm for ensemble classification using bagging. IET Quant Commun 1\u201316","DOI":"10.1049\/qtc2.12087"},{"key":"856_CR68","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1613\/jair.731","volume":"12","author":"J Baxter","year":"2000","unstructured":"Baxter J (2000) A model of inductive bias learning. J Artific Intell Res 12:149\u2013198","journal-title":"J Artific Intell Res"},{"key":"856_CR69","volume-title":"Machine learning of inductive bias","author":"PE Utgoff","year":"2012","unstructured":"Utgoff PE (2012) Machine learning of inductive bias, vol 15. Springer Science & Business Media"},{"key":"856_CR70","first-page":"12661","volume":"34","author":"J K\u00fcbler","year":"2021","unstructured":"K\u00fcbler J, Buchholz S, Sch\u00f6lkopf B (2021) The inductive bias of quantum kernels. Adv Neural Inf Process Syst 34:12661\u201312673","journal-title":"Adv Neural Inf Process Syst"},{"key":"856_CR71","unstructured":"Schuld M, Bocharov A, Svore K, Wiebe N (2018) Circuit-centric quantum classifiers. arXiv preprint arXiv:1804.00633"},{"key":"856_CR72","doi-asserted-by":"crossref","unstructured":"Jerbi S, Gyurik C, Marshall SC, Molteni R, Dunjko V (2023) Shadows of quantum machine learning. arXiv preprint arXiv:2306.00061","DOI":"10.1038\/s41467-024-49877-8"},{"issue":"10","key":"856_CR73","volume":"131","author":"J Schreiber Franz","year":"2023","unstructured":"Schreiber Franz J, Jens E, Jakob MJ (2023) Classical surrogates for quantum learning models. Phys Rev Lett 131(10):100803","journal-title":"Phys Rev Lett"},{"issue":"1","key":"856_CR74","doi-asserted-by":"crossref","first-page":"4812","DOI":"10.1038\/s41467-018-07090-4","volume":"9","author":"JR McClean","year":"2018","unstructured":"McClean JR, Boixo S, Smelyanskiy VN, Babbush R, Neven H (2018) Barren plateaus in quantum neural network training landscapes. Nat Commun 9(1):4812","journal-title":"Nat Commun"},{"key":"856_CR75","doi-asserted-by":"crossref","unstructured":"Kulshrestha A, Safro I (2022) Beinit: avoiding barren plateaus in variational quantum algorithms. In 2022 IEEE International Conference on Quantum Computing and Engineering (QCE), pp 197\u2013203. IEEE","DOI":"10.1109\/QCE53715.2022.00039"},{"key":"856_CR76","unstructured":"Cerezo M, Larocca M, Garc\u00eda-Mart\u00edn D, Diaz NL, Braccia P, Fontana E, Rudolph MS, Bermejo P, Ijaz A, Thanasilp S et\u00a0al (2023) Does provable absence of barren plateaus imply classical simulability? Or, why we need to rethink variational quantum computing. arXiv preprint arXiv:2312.09121"},{"issue":"6088","key":"856_CR77","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1038\/323533a0","volume":"323","author":"DE Rumelhart","year":"1986","unstructured":"Rumelhart DE, Hinton GE, Williams RJ (1986) Learning representations by back-propagating errors. Nature 323(6088):533\u2013536","journal-title":"Nature"},{"key":"856_CR78","unstructured":"LeCun Y, Boser B, Denker J, Henderson D, Howard R, Hubbard W, Jackel L (1989) Handwritten digit recognition with a back-propagation network. Adv Neural Inf Process Syst 2"},{"key":"856_CR79","unstructured":"Crooks GE (2019) Gradients of parameterized quantum gates using the parameter-shift rule and gate decomposition. arXiv preprint arXiv:1905.13311"},{"key":"856_CR80","unstructured":"Abbas A, King R, Huang H-Y, Huggins WJ, Movassagh R, Gilboa D, McClean J (2024) On quantum backpropagation, information reuse, and cheating measurement collapse. Adv Neural Inf Process Syst 36"},{"key":"856_CR81","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511809682","volume-title":"Kernel methods for pattern analysis","author":"J Shawe-Taylor","year":"2004","unstructured":"Shawe-Taylor J, Cristianini N (2004) Kernel methods for pattern analysis. Cambridge University Press"},{"key":"856_CR82","first-page":"14820","volume":"33","author":"B Ghorbani","year":"2020","unstructured":"Ghorbani B, Mei S, Misiakiewicz T, Montanari A (2020) When do neural networks outperform kernel methods? Adv Neural Inf Process Syst 33:14820\u201314830","journal-title":"Adv Neural Inf Process Syst"},{"issue":"3","key":"856_CR83","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevA.103.032430","volume":"103","author":"M Schuld","year":"2021","unstructured":"Schuld M, Sweke R, Meyer JJ (2021) Effect of data encoding on the expressive power of variational quantum-machine-learning models. Phys Rev A 103(3):032430","journal-title":"Phys Rev A"},{"issue":"6","key":"856_CR84","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1038\/s43588-021-00084-1","volume":"1","author":"A Abbas","year":"2021","unstructured":"Abbas A, Sutter D, Zoufal C, Lucchi A, Figalli A, Woerner S (2021) The power of quantum neural networks. Nat Comput Sci 1(6):403\u2013409","journal-title":"Nat Comput Sci"},{"key":"856_CR85","unstructured":"Kapoor A, Wiebe N, Svore K (2016) Quantum perceptron models. Adv Neural Inf Process Syst 29"},{"issue":"01","key":"856_CR86","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1142\/S0129183102002948","volume":"13","author":"M Andrecut","year":"2002","unstructured":"Andrecut M, Ali MK (2002) A quantum neural network model. Int J Mod Phys C 13(01):75\u201388","journal-title":"Int J Mod Phys C"},{"issue":"1","key":"856_CR87","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevA.101.010301","volume":"101","author":"M Lubasch","year":"2020","unstructured":"Lubasch M, Joo J, Moinier P, Kiffner M, Jaksch D (2020) Variational quantum algorithms for nonlinear problems. Phys Rev A 101(1):010301","journal-title":"Phys Rev A"},{"issue":"4","key":"856_CR88","volume":"11","author":"A Pesah","year":"2021","unstructured":"Pesah A, Cerezo M, Wang S, Volkoff T, Sornborger AT, Coles PJ (2021) Absence of barren plateaus in quantum convolutional neural networks. Phys Rev X 11(4):041011","journal-title":"Phys Rev X"},{"issue":"9","key":"856_CR89","doi-asserted-by":"crossref","first-page":"1013","DOI":"10.1038\/s41567-021-01287-z","volume":"17","author":"Y Liu","year":"2021","unstructured":"Liu Y, Arunachalam S, Temme K (2021) A rigorous and robust quantum speed-up in supervised machine learning. Nat Phys 17(9):1013\u20131017","journal-title":"Nat Phys"},{"issue":"19","key":"856_CR90","volume":"126","author":"H-Y Huang","year":"2021","unstructured":"Huang H-Y, Kueng R, Preskill J (2021) Information-theoretic bounds on quantum advantage in machine learning. Phys Rev Lett 126(19):190505","journal-title":"Phys Rev Lett"},{"key":"856_CR91","doi-asserted-by":"crossref","unstructured":"Huang H-Y, Broughton M, Mohseni M, Babbush R, Boixo S, Neven H, McClean JR (2021) Power of data in quantum machine learning. Nat Commun 12(1)","DOI":"10.1038\/s41467-021-22539-9"},{"key":"856_CR92","doi-asserted-by":"crossref","unstructured":"Feynman RP et\u00a0al (2018) Simulating physics with computers. Int J Theor Phys 21(6\/7)","DOI":"10.1007\/BF02650179"}],"container-title":["KI - K\u00fcnstliche Intelligenz"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s13218-024-00856-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s13218-024-00856-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s13218-024-00856-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,2,24]],"date-time":"2025-02-24T12:55:32Z","timestamp":1740401732000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s13218-024-00856-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,19]]},"references-count":92,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2024,12]]}},"alternative-id":["856"],"URL":"https:\/\/doi.org\/10.1007\/s13218-024-00856-7","relation":{},"ISSN":["0933-1875","1610-1987"],"issn-type":[{"value":"0933-1875","type":"print"},{"value":"1610-1987","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,7,19]]},"assertion":[{"value":"1 March 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 July 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 July 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no conflicts of interest to disclose.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}