{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T14:05:15Z","timestamp":1781273115413,"version":"3.54.1"},"reference-count":69,"publisher":"Association for Computing Machinery (ACM)","issue":"2","license":[{"start":{"date-parts":[[2021,6,30]],"date-time":"2021-06-30T00:00:00Z","timestamp":1625011200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/501100003050","name":"Ministry of Economy, Trade and Industry","doi-asserted-by":"publisher","award":["17H06293"],"award-info":[{"award-number":["17H06293"]}],"id":[{"id":"10.13039\/501100003050","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Transactions on Quantum Computing"],"published-print":{"date-parts":[[2021,6,30]]},"abstract":"<jats:p>Applying quantum processors to model a high-dimensional function approximator is a typical method in quantum machine learning with potential advantage. It is conjectured that the unitarity of quantum circuits provides possible regularization to avoid overfitting. However, it is not clear how the regularization interplays with the expressibility under the limitation of current Noisy-Intermediate Scale Quantum devices. In this article, we perform simulations and theoretical analysis of the quantum circuit learning problem with hardware-efficient ansatz. Thorough numerical simulations show that the expressibility and generalization error scaling of the ansatz saturate when the circuit depth increases, implying the automatic regularization to avoid the overfitting issue in the quantum circuit learning scenario. This observation is supported by the theory on PAC learnability, which proves that VC dimension is upper bounded due to the locality and unitarity of the hardware-efficient ansatz. Our study provides supporting evidence for automatic regularization by unitarity to suppress overfitting and guidelines for possible performance improvement under hardware constraints.<\/jats:p>","DOI":"10.1145\/3466797","type":"journal-article","created":{"date-parts":[[2021,7,9]],"date-time":"2021-07-09T10:06:14Z","timestamp":1625825174000},"page":"1-24","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":39,"title":["On the Expressibility and Overfitting of Quantum Circuit Learning"],"prefix":"10.1145","volume":"2","author":[{"given":"Chih-Chieh","family":"Chen","sequence":"first","affiliation":[{"name":"Grid Inc., 107-0061 Tokyo, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Masaya","family":"Watabe","sequence":"additional","affiliation":[{"name":"Engineering Department, The University of Electro-Communications, 182-8585 Tokyo, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kodai","family":"Shiba","sequence":"additional","affiliation":[{"name":"Grid Inc., 107-0061 Tokyo, Japan and Engineering Department, The University of Electro-Communications, 182-8585 Tokyo, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Masaru","family":"Sogabe","sequence":"additional","affiliation":[{"name":"Grid Inc., 107-0061 Tokyo, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Katsuyoshi","family":"Sakamoto","sequence":"additional","affiliation":[{"name":"Engineering Department, The University of Electro-Communications, 182-8585 Tokyo, Japan and i-PERC, The University of Electro-Communications, 182-8585 Tokyo, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tomah","family":"Sogabe","sequence":"additional","affiliation":[{"name":"Engineering Department, The University of Electro-Communications, 182-8585 Tokyo, Japan and i-PERC, The University of Electro-Communications, 182-8585 Tokyo, Japan and Grid Inc., 107-0061 Tokyo, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2021,7,9]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.1985.0070"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02650179"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.58.5355"},{"key":"e_1_2_1_4_1","volume-title":"Chuang","author":"Nielsen Michael A.","year":"2011"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1109\/SFCS.1994.365700"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1038\/s41567-018-0124-x"},{"key":"e_1_2_1_7_1","volume-title":"Quantum supremacy and the complexity of random circuit sampling. arXiv","author":"Bouland Adam","year":"1803"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1038\/s41586-019-1666-5"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1038\/d41586-019-02936-3"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1002\/1521-3978(200009)48:9\/11<771::AID-PROP771>3.0.CO;2-E"},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.86.032324"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.22331\/q-2018-08-06-79"},{"key":"e_1_2_1_13_1","volume-title":"Hybrid programming for near-term quantum computing systems. arXiv","author":"McCaskey Alexander","year":"1805"},{"key":"e_1_2_1_14_1","volume-title":"Retrieved","author":"Farhi Edward","year":"2014"},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.3390\/a12020034"},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.1021\/acs.chemrev.8b00803"},{"key":"e_1_2_1_17_1","doi-asserted-by":"crossref","first-page":"20990","DOI":"10.1039\/D0CP03745A","article-title":"A probabilistic spin annihilation method for quantum chemical calculations on quantum computers","volume":"22","author":"Sugisaki Kenji","year":"2020","journal-title":"Phys Chem Chem Phys"},{"key":"e_1_2_1_18_1","doi-asserted-by":"publisher","DOI":"10.1038\/ncomms5213"},{"key":"e_1_2_1_19_1","volume-title":"Retrieved","author":"Romero Jonathan","year":"2018"},{"key":"e_1_2_1_20_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.122.230401"},{"key":"e_1_2_1_21_1","doi-asserted-by":"crossref","first-page":"3007","DOI":"10.1038\/s41467-019-10988-2","article-title":"An adaptive variational algorithm for exact molecular simulations on a quantum computer","volume":"10","author":"Grimsley Harper R.","year":"2019","journal-title":"Nat Commun"},{"key":"e_1_2_1_22_1","doi-asserted-by":"publisher","DOI":"10.5555\/3327546.3327696"},{"key":"e_1_2_1_23_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.122.040504"},{"key":"e_1_2_1_24_1","doi-asserted-by":"publisher","DOI":"10.1038\/s41586-019-0980-2"},{"key":"e_1_2_1_25_1","volume-title":"Proceedings of the 36th International Conference on Machine Learning (ICML\u201919)","author":"Li Tongyang","year":"2019"},{"key":"e_1_2_1_26_1","volume-title":"Retrieved","author":"Blank Carsten","year":"2019"},{"key":"e_1_2_1_27_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.99.032331"},{"key":"e_1_2_1_28_1","volume-title":"Carsten Blank, Alain Delgado, Soran Jahangiri, Keri McKiernan, Johannes Jakob Meyer, Zeyue Niu, Antal Sz\u00e1va, and Nathan Killoran.","author":"Bergholm Ville","year":"2018"},{"key":"e_1_2_1_29_1","volume-title":"Learning from data. AMLBook","author":"Abu-Mostafa Yaser S."},{"key":"e_1_2_1_30_1","doi-asserted-by":"publisher","DOI":"10.1038\/nature14539"},{"key":"e_1_2_1_31_1","doi-asserted-by":"publisher","DOI":"10.5555\/2627435.2670313"},{"key":"e_1_2_1_32_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.98.032309"},{"key":"e_1_2_1_33_1","doi-asserted-by":"publisher","DOI":"10.1038\/nature23879"},{"key":"e_1_2_1_34_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.scib.2018.06.007"},{"key":"e_1_2_1_35_1","volume-title":"A flexible high-performance simulator for verifying and benchmarking quantum circuits implemented on real hardware. npj Quantum Inf 5","author":"Villalonga Benjamin","year":"2019"},{"key":"e_1_2_1_36_1","doi-asserted-by":"crossref","first-page":"1900070","DOI":"10.1002\/qute.201900070","article-title":"Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms","volume":"2","author":"Sim Sukin","year":"2019","journal-title":"Adv Quantum Technol"},{"key":"e_1_2_1_37_1","volume-title":"Retrieved","author":"Nakaji Kouhei","year":"2020"},{"key":"e_1_2_1_38_1","volume-title":"Retrieved","author":"Hubregtsen Thomas","year":"2020"},{"key":"e_1_2_1_39_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.103.032430"},{"key":"e_1_2_1_40_1","doi-asserted-by":"publisher","DOI":"10.1145\/76359.76371"},{"key":"e_1_2_1_41_1","doi-asserted-by":"publisher","DOI":"10.1137\/1116025"},{"key":"e_1_2_1_42_1","doi-asserted-by":"publisher","DOI":"10.1145\/1968.1972"},{"key":"e_1_2_1_43_1","doi-asserted-by":"publisher","DOI":"10.1038\/323533a0"},{"key":"e_1_2_1_44_1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-018-07090-4"},{"key":"e_1_2_1_45_1","doi-asserted-by":"publisher","DOI":"10.5555\/1953048.2078195"},{"key":"e_1_2_1_46_1","doi-asserted-by":"publisher","DOI":"10.1145\/3313276.3316310"},{"key":"e_1_2_1_47_1","unstructured":"Gadi Aleksandrowicz Thomas Alexander Panagiotis Barkoutsos Luciano Bello Yael Ben-Haim David Bucher Francisco Jose Cabrera-Hern\u00e1ndez Jorge Carballo-Franquis Adrian Chen Chun-Fu Chen Jerry M. Chow Antonio D. C\u00f3rcoles-Gonzales Abigail J. Cross Andrew Cross Juan Cruz-Benito Chris Culver Salvador De La Puente Gonz\u00e1lez Enrique De La Torre Delton Ding Eugene Dumitrescu Ivan Duran Pieter Eendebak Mark Everitt Ismael Faro Sertage Albert Frisch Andreas Fuhrer Jay Gambetta Borja Godoy Gago Juan Gomez-Mosquera Donny Greenberg Ikko Hamamura Vojtech Havlicek Joe Hellmers \u0141ukasz Herok Hiroshi Horii Shaohan Hu Takashi Imamichi Toshinari Itoko Ali Javadi-Abhari Naoki Kanazawa Anton Karazeev Kevin Krsulich Peng Liu Yang Luh Yunho Maeng Manoel Marques Francisco Jose Mart\u00edn-Fern\u00e1ndez Douglas T. McClure David McKay Srujan Meesala Antonio Mezzacapo Nikolaj Moll Diego Moreda Rodr\u00edguez Giacomo Nannicini Paul Nation Pauline Ollitrault Lee James O'Riordan Hanhee Paik Jes\u00fas P\u00e9rez Anna Phan Marco Pistoia Viktor Prutyanov Max Reuter Julia Rice Abd\u00f3n Rodr\u00edguez Davila Raymond Harry Putra Rudy Mingi Ryu Ninad Sathaye Chris Schnabel Eddie Schoute Kanav Setia Yunong Shi Adenilton Silva Yukio Siraichi Seyon Sivarajah John A. Smolin Mathias Soeken Hitomi Takahashi Ivano Tavernelli Charles Taylor Pete Taylour Kenso Trabing Matthew Treinish Wes Turner Desiree Vogt-Lee Christophe Vuillot Jonathan A. Wildstrom Jessica Wilson Erick Winston Christopher Wood Stephen Wood Stefan W\u00f6rner Ismail Yunus Akhalwaya and Christa Zoufal. 2019. Qiskit: An open-source framework for quantum computing (Version 0.7.2). Zenodo. https:\/\/doi.org\/10.5281\/zenodo.2562111  Gadi Aleksandrowicz Thomas Alexander Panagiotis Barkoutsos Luciano Bello Yael Ben-Haim David Bucher Francisco Jose Cabrera-Hern\u00e1ndez Jorge Carballo-Franquis Adrian Chen Chun-Fu Chen Jerry M. Chow Antonio D. C\u00f3rcoles-Gonzales Abigail J. Cross Andrew Cross Juan Cruz-Benito Chris Culver Salvador De La Puente Gonz\u00e1lez Enrique De La Torre Delton Ding Eugene Dumitrescu Ivan Duran Pieter Eendebak Mark Everitt Ismael Faro Sertage Albert Frisch Andreas Fuhrer Jay Gambetta Borja Godoy Gago Juan Gomez-Mosquera Donny Greenberg Ikko Hamamura Vojtech Havlicek Joe Hellmers \u0141ukasz Herok Hiroshi Horii Shaohan Hu Takashi Imamichi Toshinari Itoko Ali Javadi-Abhari Naoki Kanazawa Anton Karazeev Kevin Krsulich Peng Liu Yang Luh Yunho Maeng Manoel Marques Francisco Jose Mart\u00edn-Fern\u00e1ndez Douglas T. McClure David McKay Srujan Meesala Antonio Mezzacapo Nikolaj Moll Diego Moreda Rodr\u00edguez Giacomo Nannicini Paul Nation Pauline Ollitrault Lee James O'Riordan Hanhee Paik Jes\u00fas P\u00e9rez Anna Phan Marco Pistoia Viktor Prutyanov Max Reuter Julia Rice Abd\u00f3n Rodr\u00edguez Davila Raymond Harry Putra Rudy Mingi Ryu Ninad Sathaye Chris Schnabel Eddie Schoute Kanav Setia Yunong Shi Adenilton Silva Yukio Siraichi Seyon Sivarajah John A. Smolin Mathias Soeken Hitomi Takahashi Ivano Tavernelli Charles Taylor Pete Taylour Kenso Trabing Matthew Treinish Wes Turner Desiree Vogt-Lee Christophe Vuillot Jonathan A. Wildstrom Jessica Wilson Erick Winston Christopher Wood Stephen Wood Stefan W\u00f6rner Ismail Yunus Akhalwaya and Christa Zoufal. 2019. Qiskit: An open-source framework for quantum computing (Version 0.7.2). Zenodo. https:\/\/doi.org\/10.5281\/zenodo.2562111"},{"key":"e_1_2_1_48_1","doi-asserted-by":"publisher","DOI":"10.1109\/PGEC.1965.264137"},{"key":"e_1_2_1_49_1","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176995384"},{"key":"e_1_2_1_50_1","first-page":"69","article-title":"VC dimension of neural networks","volume":"168","author":"Sontag Eduardo D.","year":"1998","journal-title":"NATO ASI Series F Computer and Systems Sciences"},{"key":"e_1_2_1_51_1","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.aop.2010.09.012","article-title":"The density-matrix renormalization group in the age of matrix product states","volume":"326","author":"Schollwoeck Ulrich","year":"2011","journal-title":"Annals of Physics"},{"key":"e_1_2_1_52_1","volume-title":"Retrieved","author":"Goto Takahiro","year":"2020"},{"key":"e_1_2_1_53_1","volume-title":"Pattern Recognition and Machine Learning (Information Science and Statistics)","author":"Bishop Christopher M."},{"key":"e_1_2_1_54_1","volume-title":"A variational eigenvalue solver on a photonic quantum processor. Nat Commun 5, 4213","author":"Peruzzo Alberto","year":"2014"},{"key":"e_1_2_1_55_1","doi-asserted-by":"crossref","unstructured":"Jarrod R. McClean Jonathan Romero Ryan Babbush and Al\u00e1n Aspuru-Guzik. 2016. The theory of variational hybrid quantum-classical algorithms. New J Phys. 18 023023. https:\/\/doi.org\/10.1088\/1367-2630\/18\/2\/023023  Jarrod R. McClean Jonathan Romero Ryan Babbush and Al\u00e1n Aspuru-Guzik. 2016. The theory of variational hybrid quantum-classical algorithms. New J Phys. 18 023023. https:\/\/doi.org\/10.1088\/1367-2630\/18\/2\/023023","DOI":"10.1088\/1367-2630\/18\/2\/023023"},{"key":"e_1_2_1_56_1","volume-title":"Retrieved","author":"Farhi Edward","year":"2018"},{"key":"e_1_2_1_57_1","doi-asserted-by":"publisher","DOI":"10.5555\/2621980"},{"key":"e_1_2_1_58_1","volume-title":"Foundations of Machine Learning","author":"Mohri Mehryar"},{"key":"e_1_2_1_59_1","volume-title":"Proc R Soc A.4633089\u20133114","author":"Aaronson Scott","year":"2007"},{"key":"e_1_2_1_60_1","doi-asserted-by":"publisher","DOI":"10.5555\/3291125.3309633"},{"key":"e_1_2_1_61_1","doi-asserted-by":"publisher","DOI":"10.1145\/3106700.3106710"},{"key":"e_1_2_1_62_1","volume-title":"Retrieved","author":"Huang Hsin-Yuan","year":"2021"},{"key":"e_1_2_1_63_1","volume-title":"Retrieved","author":"Shiba Kodai","year":"2021"},{"key":"e_1_2_1_64_1","volume-title":"Mark Wiebe, Pearu Peterson, Pierre Gerard-Marchant, Kevin Sheppard, Tyler Reddy, Warren Weckesser, Hameer Abbasi, Christoph Gohlke, and Travis E. Oliphant.","author":"Harris Charles R.","year":"2020"},{"key":"e_1_2_1_65_1","doi-asserted-by":"crossref","volume-title":"Convergence of Stochastic Processes","author":"Pollard David","DOI":"10.1007\/978-1-4612-5254-2"},{"key":"e_1_2_1_66_1","volume-title":"Latorre","author":"Perez-Salinas Adrian","year":"2020"},{"key":"e_1_2_1_67_1","volume-title":"Retrieved","author":"Abbas Amira","year":"2020"},{"key":"e_1_2_1_68_1","volume-title":"Retrieved","author":"Watabe Masaya","year":"2019"},{"key":"e_1_2_1_69_1","volume-title":"Retrieved","author":"Gyurik Casper","year":"2021"}],"container-title":["ACM Transactions on Quantum Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3466797","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3466797","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T21:28:09Z","timestamp":1750195689000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3466797"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,30]]},"references-count":69,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,6,30]]}},"alternative-id":["10.1145\/3466797"],"URL":"https:\/\/doi.org\/10.1145\/3466797","relation":{},"ISSN":["2643-6809","2643-6817"],"issn-type":[{"value":"2643-6809","type":"print"},{"value":"2643-6817","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,30]]},"assertion":[{"value":"2020-10-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2021-05-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2021-07-09","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}