{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,14]],"date-time":"2026-10-14T00:25:56Z","timestamp":1791937556755,"version":"4.3.4"},"reference-count":69,"publisher":"IOP Publishing","issue":"1","license":[{"start":{"date-parts":[[2024,4,2]],"date-time":"2024-04-02T00:00:00Z","timestamp":1712016000000},"content-version":"vor","delay-in-days":32,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,4,2]],"date-time":"2024-04-02T00:00:00Z","timestamp":1712016000000},"content-version":"tdm","delay-in-days":32,"URL":"https:\/\/iopscience.iop.org\/info\/page\/text-and-data-mining"}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"short-container-title":["Mach. Learn.: Sci. Technol."],"published-print":{"date-parts":[[2024,3,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We define <jats:italic>laziness<\/jats:italic> to describe a large suppression of variational parameter updates for neural networks, classical or quantum. In the quantum case, the suppression is exponential in the number of qubits for randomized variational quantum circuits. We discuss the difference between laziness and <jats:italic>barren plateau<\/jats:italic> in quantum machine learning created by quantum physicists in McClean <jats:italic>et al<\/jats:italic> (2018 <jats:italic>Nat. Commun.<\/jats:italic>\n                  <jats:bold>9<\/jats:bold> 1\u20136) for the flatness of the loss function landscape during gradient descent. We address a novel theoretical understanding of those two phenomena in light of the theory of neural tangent kernels. For noiseless quantum circuits, without the measurement noise, the loss function landscape is complicated in the overparametrized regime with a large number of trainable variational angles. Instead, around a random starting point in optimization, there are large numbers of local minima that are good enough and could minimize the mean square loss function, where we still have quantum laziness, but we do not have barren plateaus. However, the complicated landscape is not visible within a limited number of iterations, and low precision in quantum control and quantum sensing. Moreover, we look at the effect of noises during optimization by assuming intuitive noise models, and show that variational quantum algorithms are noise-resilient in the overparametrization regime. Our work precisely reformulates the quantum barren plateau statement towards a precision statement and justifies the statement in certain noise models, injects new hope toward near-term variational quantum algorithms, and provides theoretical connections toward classical machine learning. Our paper provides conceptual perspectives about quantum barren plateaus, together with discussions about the gradient descent dynamics in Liu <jats:italic>et al<\/jats:italic> (2023 <jats:italic>Phys. Rev. Lett.<\/jats:italic>\n                  <jats:bold>130<\/jats:bold> 150601).<\/jats:p>","DOI":"10.1088\/2632-2153\/ad35a3","type":"journal-article","created":{"date-parts":[[2024,3,19]],"date-time":"2024-03-19T22:42:29Z","timestamp":1710888149000},"page":"015058","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":12,"title":["Laziness, barren plateau, and noises in machine learning"],"prefix":"10.1088","volume":"5","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1669-8039","authenticated-orcid":true,"given":"Junyu","family":"Liu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zexi","family":"Lin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"266","published-online":{"date-parts":[[2024,4,2]]},"reference":[{"key":"mlstad35a3bib1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/ncomms5213","article-title":"A variational eigenvalue solver on a photonic quantum processor","volume":"5","author":"Peruzzo","year":"2014","journal-title":"Nat. Commun."},{"key":"mlstad35a3bib2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/srep03589","article-title":"From transistor to trapped-ion computers for quantum chemistry","volume":"4","author":"Yung","year":"2014","journal-title":"Sci. Rep."},{"key":"mlstad35a3bib3","doi-asserted-by":"publisher","DOI":"10.1088\/1367-2630\/18\/2\/023023","article-title":"The theory of variational hybrid quantum-classical algorithms","volume":"18","author":"McClean","year":"2016","journal-title":"New J. Phys."},{"key":"mlstad35a3bib4","doi-asserted-by":"publisher","first-page":"242","DOI":"10.1038\/nature23879","article-title":"Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets","volume":"549","author":"Kandala","year":"2017","journal-title":"Nature"},{"key":"mlstad35a3bib5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s42254-021-00348-9","article-title":"Variational quantum algorithms","volume":"3","author":"Cerezo","year":"2021","journal-title":"Nat. Rev. Phys."},{"key":"mlstad35a3bib6","article-title":"A quantum approximate optimization algorithm","author":"Farhi","year":"2014"},{"key":"mlstad35a3bib7","author":"Wittek","year":"2014"},{"key":"mlstad35a3bib8","article-title":"Quantum deep learning","author":"Wiebe","year":"2014"},{"key":"mlstad35a3bib9","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1038\/nature23474","article-title":"Quantum machine learning","volume":"549","author":"Biamonte","year":"2017","journal-title":"Nature"},{"key":"mlstad35a3bib10","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.122.040504","article-title":"Quantum machine learning in feature hilbert spaces","volume":"122","author":"Schuld","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"mlstad35a3bib11","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1038\/s41586-019-0980-2","article-title":"Supervised learning with quantum-enhanced feature spaces","volume":"567","author":"Havl\u00ed\u010dek","year":"2019","journal-title":"Nature"},{"key":"mlstad35a3bib12","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41567-021-01287-z","article-title":"A rigorous and robust quantum speed-up in supervised machine learning","volume":"17","author":"Liu","year":"2021","journal-title":"Nat. Phys."},{"key":"mlstad35a3bib13","article-title":"Does Richard Feynman dream of electric sheep? Topics on quantum field theory, quantum computing, and computer science","author":"Liu","year":"2021"},{"key":"mlstad35a3bib14","article-title":"Classification with quantum neural networks on near term processors","author":"Farhi","year":"2018"},{"key":"mlstad35a3bib15","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41467-018-07090-4","article-title":"Barren plateaus in quantum neural network training landscapes","volume":"9","author":"McClean","year":"2018","journal-title":"Nat. Commun."},{"key":"mlstad35a3bib16","doi-asserted-by":"publisher","first-page":"JHEP04(2017)121","DOI":"10.1007\/JHEP04(2017)121","article-title":"Chaos and complexity by design","author":"Roberts","year":"2017","journal-title":"J. High Energy Phys."},{"key":"mlstad35a3bib17","doi-asserted-by":"publisher","first-page":"JHEP11(2017)048","DOI":"10.1007\/JHEP11(2017)048","article-title":"Chaos, complexity and random matrices","author":"Cotler","year":"2017","journal-title":"J. High Energy Phys."},{"key":"mlstad35a3bib18","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.98.086026","article-title":"Spectral form factors and late time quantum chaos","volume":"98","author":"Liu","year":"2018","journal-title":"Phys. Rev. D"},{"key":"mlstad35a3bib19","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.2.043164","article-title":"Scrambling and decoding the charged quantum information","volume":"2","author":"Liu","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"mlstad35a3bib20","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8121\/ab434b","article-title":"RTNI: a symbolic integrator for haar-random tensor networks","volume":"52","author":"Fukuda","year":"2019","journal-title":"J. Phys. A: Math. Theor."},{"key":"mlstad35a3bib21","author":"Mohri","year":"2018"},{"key":"mlstad35a3bib22","article-title":"The principles of deep learning theory","author":"Roberts","year":"2021"},{"key":"mlstad35a3bib23","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41467-021-21728-w","article-title":"Cost function dependent barren plateaus in shallow parametrized quantum circuits","volume":"12","author":"Cerezo","year":"2021","journal-title":"Nat. Commun."},{"key":"mlstad35a3bib24","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevX.11.041011","article-title":"Absence of barren plateaus in quantum convolutional neural networks","volume":"11","author":"Arthur Pesah","year":"2021","journal-title":"Phys. Rev. X"},{"key":"mlstad35a3bib25","doi-asserted-by":"publisher","DOI":"10.1088\/2058-9565\/abf51a","article-title":"Higher order derivatives of quantum neural networks with barren plateaus","volume":"6","author":"Cerezo","year":"2021","journal-title":"Quantum Sci. Technol."},{"key":"mlstad35a3bib26","doi-asserted-by":"publisher","first-page":"558","DOI":"10.22331\/q-2021-10-05-558","article-title":"Effect of barren plateaus on gradient-free optimization","volume":"5","author":"Andrew Arrasmith","year":"2021","journal-title":"Quantum"},{"key":"mlstad35a3bib27","article-title":"Language models are few-shot learners","author":"Brown","year":"2020"},{"key":"mlstad35a3bib28","doi-asserted-by":"publisher","DOI":"10.1103\/PRXQuantum.3.030323","article-title":"Representation Learning via Quantum Neural Tangent Kernels","volume":"3","author":"Liu","year":"2022","journal-title":"PRX Quantum"},{"key":"mlstad35a3bib29","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.130.150601","article-title":"Analytic theory for the dynamics of wide quantum neural networks","volume":"130","author":"Liu","year":"2023","journal-title":"Phys. Rev. Lett."},{"key":"mlstad35a3bib30","article-title":"Deep neural networks as gaussian processes","author":"Lee","year":"2017"},{"key":"mlstad35a3bib31","article-title":"Neural tangent kernel: convergence and generalization in neural networks","author":"Jacot","year":"2018"},{"key":"mlstad35a3bib32","first-page":"pp 8572","article-title":"Wide neural networks of any depth evolve as linear models under gradient descent","volume":"vol 32","author":"Lee","year":"2019"},{"key":"mlstad35a3bib33","article-title":"On the infinite width limit of neural networks with a standard parameterization","author":"Sohl-Dickstein","year":"2020"},{"key":"mlstad35a3bib34","article-title":"Feature learning in infinite-width neural networks","author":"Yang","year":"2020"},{"key":"mlstad35a3bib35","first-page":"pp 165","article-title":"Non-gaussian processes and neural networks at finite widths","author":"Yaida","year":"2020"},{"key":"mlstad35a3bib36","article-title":"On exact computation with an infinitely wide neural net","author":"Arora","year":"2019"},{"key":"mlstad35a3bib37","article-title":"Asymptotics of wide networks from feynman diagrams","author":"Dyer","year":"2019"},{"key":"mlstad35a3bib38","doi-asserted-by":"publisher","DOI":"10.1088\/2632-2153\/abeca3","article-title":"Neural networks and quantum field theory","volume":"2","author":"Halverson","year":"2021","journal-title":"Mach. Learn.: Sci. Technol."},{"key":"mlstad35a3bib39","article-title":"Why is AI hard and physics simple?","author":"Roberts","year":"2021"},{"key":"mlstad35a3bib40","article-title":"Effective theory of deep learning: beyond the infinite-width limit","author":"Roberts","year":"2021"},{"key":"mlstad35a3bib41","article-title":"Orqviz: visualizing high-dimensional landscapes in variational quantum algorithms","author":"Rudolph","year":"2021"},{"key":"mlstad35a3bib42","article-title":"Towards understanding generalization of deep learning: perspective of loss landscapes","author":"Lei","year":"2017"},{"key":"mlstad35a3bib43","doi-asserted-by":"publisher","first-page":"2293","DOI":"10.1162\/neco_a_01234","article-title":"Every local minimum value is the global minimum value of induced model in nonconvex machine learning","volume":"31","author":"Kawaguchi","year":"2019","journal-title":"Neural Comput."},{"key":"mlstad35a3bib44","first-page":"p 700","author":"Nielsen","year":"2002"},{"key":"mlstad35a3bib45","first-page":"pp 124","article-title":"Algorithms for quantum computation: discrete logarithms and factoring","author":"Shor","year":"1994"},{"key":"mlstad35a3bib46","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1007\/s00220-016-2706-8","article-title":"Local random quantum circuits are approximate polynomial-designs","volume":"346","author":"Brandao","year":"2016","journal-title":"Commun. Math. Phys."},{"key":"mlstad35a3bib47","article-title":"Beyond barren plateaus: quantum variational algorithms are swamped with traps","author":"Anschuetz","year":"2022"},{"key":"mlstad35a3bib48","article-title":"Quantum lazy training","author":"Abedi","year":"2022"},{"key":"mlstad35a3bib49","first-page":"p 32","article-title":"On lazy training in differentiable programming","author":"Chizat","year":"2019"},{"key":"mlstad35a3bib50","article-title":"Quantum tangent kernel","author":"Shirai","year":"2021"},{"key":"mlstad35a3bib51","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41467-021-27045-6","article-title":"Noise-induced barren plateaus in variational quantum algorithms","volume":"12","author":"Wang","year":"2021","journal-title":"Nat. Commun."},{"key":"mlstad35a3bib52","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.75.012328","article-title":"Optimal quantum measurements of expectation values of observables","volume":"75","author":"Knill","year":"2007","journal-title":"Phys. Rev. A"},{"key":"mlstad35a3bib53","article-title":"Noise can be helpful for variational quantum algorithms","author":"Liu","year":"2022"},{"key":"mlstad35a3bib54","doi-asserted-by":"publisher","first-page":"1037","DOI":"10.1103\/RevModPhys.76.1037","article-title":"Nmr techniques for quantum control and computation","volume":"76","author":"Vandersypen","year":"2005","journal-title":"Rev. Mod. Phys."},{"key":"mlstad35a3bib55","article-title":"A convergence theory for over-parameterized variational quantum eigensolvers","author":"You","year":"2022"},{"key":"mlstad35a3bib56","doi-asserted-by":"publisher","first-page":"1273","DOI":"10.1038\/s41567-019-0648-8","article-title":"Quantum convolutional neural networks","volume":"15","author":"Cong","year":"2019","journal-title":"Nat. Phys."},{"key":"mlstad35a3bib57","article-title":"Theory of overparametrization in quantum neural networks","author":"Larocca","year":"2021"},{"key":"mlstad35a3bib58","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41467-021-23103-1","article-title":"Spectral bias and task-model alignment explain generalization in kernel regression and infinitely wide neural networks","volume":"12","author":"Canatar","year":"2021","journal-title":"Nat. Commun."},{"key":"mlstad35a3bib59","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1142\/S0218488598000094","article-title":"The vanishing gradient problem during learning recurrent neural nets and problem solutions","volume":"6","author":"Hochreiter","year":"1998","journal-title":"Int. J. Uncertain. Fuzziness Knowl.-Based Syst."},{"key":"mlstad35a3bib60","article-title":"Gradient flow in recurrent nets: the difficulty of learning long-term dependencies","author":"Hochreiter","year":"2001"},{"key":"mlstad35a3bib61","first-page":"pp 1026","article-title":"Delving deep into rectifiers: surpassing human-level performance on imagenet classification","author":"Kaiming","year":"2015"},{"key":"mlstad35a3bib62","article-title":"Are wider nets better given the same number of parameters?","author":"Golubeva","year":"2020"},{"key":"mlstad35a3bib63","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1016\/0550-3213(95)00158-O","article-title":"String theory dynamics in various dimensions","volume":"443","author":"Witten","year":"1995","journal-title":"Nucl. Phys. B"},{"key":"mlstad35a3bib64","doi-asserted-by":"publisher","first-page":"5112","DOI":"10.1103\/PhysRevD.55.5112","article-title":"M theory as a matrix model: a Conjecture","volume":"55","author":"Tom Banks","year":"1997","journal-title":"Phys. Rev. D"},{"key":"mlstad35a3bib65","doi-asserted-by":"publisher","first-page":"JHEP04(2002)013","DOI":"10.1088\/1126-6708\/2002\/04\/013","article-title":"Strings in flat space and pp waves from N = 4 superYang-Mills","author":"Eliecer Berenstein","year":"2002","journal-title":"J. High Energy Phys."},{"key":"mlstad35a3bib66","doi-asserted-by":"publisher","first-page":"6377","DOI":"10.1063\/1.531249","article-title":"The World as a hologram","volume":"36","author":"Susskind","year":"1995","journal-title":"J. Math. Phys."},{"key":"mlstad35a3bib67","doi-asserted-by":"publisher","first-page":"231","DOI":"10.4310\/ATMP.1998.v2.n2.a1","article-title":"The Large N limit of superconformal field theories and supergravity","volume":"2","author":"Martin Maldacena","year":"1998","journal-title":"Adv. Theor. Math. Phys."},{"key":"mlstad35a3bib68","doi-asserted-by":"publisher","first-page":"253","DOI":"10.4310\/ATMP.1998.v2.n2.a2","article-title":"Anti-de Sitter space and holography","volume":"2","author":"Witten","year":"1998","journal-title":"Adv. Theor. Math. Phys."},{"key":"mlstad35a3bib69","doi-asserted-by":"publisher","first-page":"JHEP09(2007)120","DOI":"10.1088\/1126-6708\/2007\/09\/120","article-title":"Black holes as mirrors: quantum information in random subsystems","author":"Hayden","year":"2007","journal-title":"J. High Energy Phys."}],"container-title":["Machine Learning: Science and Technology"],"original-title":[],"link":[{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad35a3","content-type":"text\/html","content-version":"am","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad35a3\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad35a3","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad35a3\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad35a3\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad35a3\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad35a3\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad35a3\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,2]],"date-time":"2024-04-02T08:18:12Z","timestamp":1712045892000},"score":1,"resource":{"primary":{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad35a3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,1]]},"references-count":69,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,4,2]]},"published-print":{"date-parts":[[2024,3,1]]}},"URL":"https:\/\/doi.org\/10.1088\/2632-2153\/ad35a3","relation":{},"ISSN":["2632-2153"],"issn-type":[{"value":"2632-2153","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,3,1]]},"assertion":[{"value":"Laziness, barren plateau, and noises in machine learning","name":"article_title","label":"Article Title"},{"value":"Machine Learning: Science and Technology","name":"journal_title","label":"Journal Title"},{"value":"paper","name":"article_type","label":"Article Type"},{"value":"\u00a9 2024 The Author(s). Published by IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2023-05-05","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2024-03-19","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2024-04-02","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}]}}