{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T15:36:48Z","timestamp":1772120208240,"version":"3.50.1"},"reference-count":65,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,7,17]],"date-time":"2023-07-17T00:00:00Z","timestamp":1689552000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,7,17]],"date-time":"2023-07-17T00:00:00Z","timestamp":1689552000000},"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":["Quantum Mach. Intell."],"published-print":{"date-parts":[[2023,12]]},"DOI":"10.1007\/s42484-023-00120-5","type":"journal-article","created":{"date-parts":[[2023,7,17]],"date-time":"2023-07-17T11:01:57Z","timestamp":1689591717000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Estimating the degree of non-Markovianity using variational quantum circuits"],"prefix":"10.1007","volume":"5","author":[{"given":"Hossein T.","family":"Dinani","sequence":"first","affiliation":[]},{"given":"Diego","family":"Tancara","sequence":"additional","affiliation":[]},{"given":"Felipe F.","family":"Fanchini","sequence":"additional","affiliation":[]},{"given":"Ariel","family":"Norambuena","sequence":"additional","affiliation":[]},{"given":"Raul","family":"Coto","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,7,17]]},"reference":[{"key":"120_CR1","doi-asserted-by":"publisher","first-page":"13723","DOI":"10.1007\/s00521-021-06009-3","volume":"33","author":"G Acampora","year":"2021","unstructured":"Acampora G, Schiattarella R (2021) Deep neural networks for quantum circuit mapping. Neural Comput Appl 33:13723. https:\/\/doi.org\/10.1007\/s00521-021-06009-3","journal-title":"Neural Comput Appl"},{"key":"120_CR2","doi-asserted-by":"publisher","first-page":"808","DOI":"10.1038\/s41467-020-14454-2","volume":"11","author":"K Beer","year":"2020","unstructured":"Beer K, Bondarenko D, Farrelly T, Osborne TJ, Salzmann R, Scheiermann D, Wolf R (2020) Training deep quantum neural networks. Nat Comms 11:808. https:\/\/doi.org\/10.1038\/s41467-020-14454-2","journal-title":"Nat Comms"},{"issue":"16","key":"120_CR3","doi-asserted-by":"publisher","first-page":"160502","DOI":"10.1103\/PhysRevLett.99.160502","volume":"99","author":"B Bellomo","year":"2007","unstructured":"Bellomo B, Lo Franco R, Compagno G (2007) Non-Markovian effects on the dynamics of entanglement. Phys Rev Lett 99(16):160502. https:\/\/doi.org\/10.1103\/PhysRevLett.99.160502","journal-title":"Phys Rev Lett"},{"key":"120_CR4","doi-asserted-by":"publisher","DOI":"10.1088\/2058-9565\/ab4eb5","volume":"4","author":"M Benedetti","year":"2019","unstructured":"Benedetti M et al (2019) Parameterized quantum circuits as machine learning models. Quantum Sci Technol 4:043001. https:\/\/doi.org\/10.1088\/2058-9565\/ab4eb5","journal-title":"Quantum Sci Technol"},{"key":"120_CR5","unstructured":"Bergholm V et al (2020) PennyLane: Automatic differentiation of hybrid quantumclassical computations. Preprint at https:\/\/arxiv.org\/abs\/1811.04968v3"},{"key":"120_CR6","doi-asserted-by":"publisher","unstructured":"Berk GD, Garner BAJP, Yadin, Modi K, Pollock FA (2021) Resource theories of multi-time processes: a window into quantum non-Markovianity. Quantum 5:435. https:\/\/doi.org\/10.22331\/q-2021-04-20-435","DOI":"10.22331\/q-2021-04-20-435"},{"key":"120_CR7","doi-asserted-by":"crossref","unstructured":"Bharti K, Cervera-Lierta A, Kyaw TH, Haug T, Alperin-Lea, S, Anand A, Degroote M, Heimonen H, Kottmann JS, Menke T, Mok W-K, Sim S, Kwek L-C, Aspuru-Guzik A (2022) Noisy intermediate-scale quantum (NISQ) algorithms. Rev Mod Phys 94:015004. https:\/\/doi.org\/10.1103\/RevModPhys.94.015004","DOI":"10.1103\/RevModPhys.94.015004"},{"key":"120_CR8","doi-asserted-by":"publisher","unstructured":"Biamonte J, Wittek P, Pancotti N, Rebentrost P, Wiebe N, Lloyd S (2017) Quantum machine learning. Nature 549, 195\u2013202. https:\/\/doi.org\/10.1038\/nature23474","DOI":"10.1038\/nature23474"},{"key":"120_CR9","doi-asserted-by":"publisher","unstructured":"Blais A, Girvin SM, Oliver WD (2020) Quantum information processing and quantum optics with circuit quantum electrodynamics. Nat Phys 16: 247\u2013256 . https:\/\/doi.org\/10.1038\/s41567-020-0806-z","DOI":"10.1038\/s41567-020-0806-z"},{"key":"120_CR10","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780199213900.001.0001","volume-title":"The theory of open quantum systems","author":"HP Breuer","year":"2007","unstructured":"Breuer HP, Petruccione F (2007) The theory of open quantum systems. Oxford University Press, Oxford"},{"issue":"21","key":"120_CR11","doi-asserted-by":"publisher","first-page":"210401","DOI":"10.1103\/PhysRevLett.103.210401","volume":"103","author":"H-P Breuer","year":"2009","unstructured":"Breuer H-P, Laine EM, Piilo J (2009) Measure for the degree of non-Markovian behavior of quantum processes in open systems. Phys Rev Lett. 103(21):210401. https:\/\/doi.org\/10.1103\/PhysRevLett.103.210401","journal-title":"Phys Rev Lett."},{"issue":"2","key":"120_CR12","doi-asserted-by":"publisher","first-page":"021002","DOI":"10.1103\/RevModPhys.88.021002","volume":"88","author":"HP Breuer","year":"2016","unstructured":"Breuer HP, Laine EM, Piilo J, Vacchini B (2016) Colloquium: non-Markovian dynamics in open quantum systems. Rev Mod Phys 88(2):021002. https:\/\/doi.org\/10.1103\/RevModPhys.88.021002","journal-title":"Rev Mod Phys"},{"issue":"3","key":"120_CR13","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevX.8.031086","volume":"8","author":"M Bukov","year":"2018","unstructured":"Bukov M, Day GR, Sels D, Weinberg PAP, Mehta P (2018) Reinforcement learning in different phases of quantum control. Phys Rev X 8(3):031086. https:\/\/doi.org\/10.1103\/PhysRevX.8.031086","journal-title":"Phys Rev X"},{"key":"120_CR14","doi-asserted-by":"publisher","first-page":"5720","DOI":"10.1038\/srep05720","volume":"4","author":"B Bylicka","year":"2014","unstructured":"Bylicka B, Chruscinski D, Maniscalco S (2014) Non-Markovianity and reservoir memory of quantum channels: a quantum information theory perspective. Sci Rep 4:5720. https:\/\/doi.org\/10.1038\/srep05720","journal-title":"Sci Rep"},{"issue":"25","key":"120_CR15","doi-asserted-by":"publisher","first-page":"256903","DOI":"10.1103\/PhysRevLett.130.256903","volume":"130","author":"MC Cambria","year":"2023","unstructured":"Cambria MC, Norambuena A, Dinani HT, Thiering G, Gardill A, Kemeny I, Li Y, Lordi AV, Gali Maze JR, Kolkowitz S (2023) Temperature-dependent spin-lattice relaxation of the nitrogen-vacancy spin triplet in diamond. Phys. Rev. Lett. 130(25):256903. https:\/\/doi.org\/10.1103\/PhysRevLett.130.256903","journal-title":"Phys. Rev. Lett."},{"key":"120_CR16","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 (2021) Variational quantum algorithms. Nat Rev Phys 3:625. https:\/\/doi.org\/10.1038\/s42254-021-00348-9","journal-title":"Variational quantum algorithms. Nat Rev Phys"},{"issue":"23","key":"120_CR17","doi-asserted-by":"publisher","first-page":"233601","DOI":"10.1103\/PhysRevLett.109.233601","volume":"109","author":"AW Chin","year":"2012","unstructured":"Chin AW, Huelga SF, Plenio MB (2012) Quantum metrology in non-Markovian environments. Phys Rev Lett 109(23):233601. https:\/\/doi.org\/10.1103\/PhysRevLett.109.233601","journal-title":"Phys Rev Lett"},{"key":"120_CR18","doi-asserted-by":"publisher","first-page":"052128","DOI":"10.1103\/PhysRevA.83.052128","volume":"83","author":"D Chru\u015bci\u0144ski","year":"2011","unstructured":"Chru\u015bci\u0144ski D, Kossakowski A, Rivas A (2011) Measures of non-Markovianity: divisibility versus backflow of information. Phys Rev A 83:052128. https:\/\/doi.org\/10.1103\/PhysRevA.83.052128","journal-title":"Phys Rev A"},{"issue":"1","key":"120_CR19","doi-asserted-by":"publisher","first-page":"010304","DOI":"10.1103\/PhysRevA.70.010304","volume":"70","author":"S Daffer","year":"2004","unstructured":"Daffer S, W\u00f3dkiewicz K, Cresser JD, McIver JK (2004) Depolarizing channel as a completely positive map with memory. Phys Rev A 70(1):010304. https:\/\/doi.org\/10.1103\/PhysRevA.70.010304","journal-title":"Phys Rev A"},{"key":"120_CR20","doi-asserted-by":"publisher","first-page":"382","DOI":"10.1038\/srep00382","volume":"2","author":"G de Lange","year":"2012","unstructured":"de Lange G, van der Sar T, Blok M, Wang Z-H, Dobrovitski V, Hanson R (2012) Controlling the quantum dynamics of a mesoscopic spin bath in diamond ambient conditions. Sci Rep 2:382. https:\/\/doi.org\/10.1038\/srep00382","journal-title":"Sci Rep"},{"issue":"1","key":"120_CR21","doi-asserted-by":"publisher","first-page":"015001","DOI":"10.1103\/RevModPhys.89.015001","volume":"89","author":"I De Vega","year":"2017","unstructured":"De Vega I, Alonso D (2017) Dynamics of non-Markovian open quantum systems. Rev Mod Phys 89(1):015001. https:\/\/doi.org\/10.1103\/RevModPhys.89.015001","journal-title":"Rev Mod Phys"},{"key":"120_CR22","first-page":"2121","volume":"12","author":"J Duchi","year":"2011","unstructured":"Duchi J, Hazan E, Singer Y (2011) Adaptive subgradient methods for online learning and stochastic optimization. Journal of Machine Learning Research 12:2121","journal-title":"Journal of Machine Learning Research"},{"key":"120_CR23","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6633\/aab406","volume":"81","author":"V Dunjko","year":"2018","unstructured":"Dunjko V, Briegel H (2018) Machine learning and artificial intelligence in the quantum domain: a review of recent progress. Rep Prog Phys 81:074001. https:\/\/doi.org\/10.1088\/1361-6633\/aab406","journal-title":"Rep Prog Phys"},{"issue":"2","key":"120_CR24","doi-asserted-by":"publisher","first-page":"022425","DOI":"10.1103\/PhysRevA.103.022425","volume":"103","author":"F Fanchini","year":"2021","unstructured":"Fanchini F, Karpat G, Rossatto DZ, Norambuena A, Coto R (2021) Estimating the degree of non-Markovianity using machine learning. Phys Rev A 103(2):022425. https:\/\/doi.org\/10.1103\/PhysRevA.103.022425","journal-title":"Phys Rev A"},{"issue":"3","key":"120_CR25","doi-asserted-by":"publisher","first-page":"031084","DOI":"10.1103\/PhysRevX.8.031084","volume":"8","author":"T Fosel","year":"2018","unstructured":"Fosel T, Tighineanu P, Weiss TFM (2018) Reinforcement learning with neural networks for quantum feedback. Phys Rev X 8(3):031084. https:\/\/doi.org\/10.1103\/PhysRevX.8.031084","journal-title":"Phys Rev X"},{"key":"120_CR26","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41534-019-0235-y","volume":"6","author":"G Garc\u00eda-P\u00e9rez","year":"2020","unstructured":"Garc\u00eda-P\u00e9rez G, Rossi MAC, Maniscalco S (2020) IBM Q experience as a versatile experimental testbed for simulating open quantum systems. npj Quantum Information 6:1. https:\/\/doi.org\/10.1038\/s41534-019-0235-y","journal-title":"npj Quantum Information"},{"issue":"2","key":"120_CR27","doi-asserted-by":"publisher","first-page":"022432","DOI":"10.1103\/PhysRevA.104.022432","volume":"104","author":"K Goswami","year":"2021","unstructured":"Goswami K, Giarmatzi C, Monterola C, Shrapnel S, Romero J, Costa F (2021) Experimental characterisation of a non-Markovian quantum process. Phys Rev A 104(2):022432. https:\/\/doi.org\/10.1103\/PhysRevA.104.022432","journal-title":"Phys Rev A"},{"issue":"6","key":"120_CR28","doi-asserted-by":"publisher","first-page":"062414","DOI":"10.1103\/PhysRevA.102.062414","volume":"102","author":"C Guo","year":"2020","unstructured":"Guo C, Modi K, Poletti D (2020) Tensor-network-based machine learning of non-Markovian quantum processes. Phys Rev A 102(6):062414. https:\/\/doi.org\/10.1103\/PhysRevA.102.062414","journal-title":"Phys Rev A"},{"issue":"5","key":"120_CR29","doi-asserted-by":"publisher","first-page":"052103","DOI":"10.1103\/PhysRevA.81.052103","volume":"81","author":"P Haikka","year":"2010","unstructured":"Haikka P, Maniscalco S (2010) Non-Markovian dynamics of a damped driven two-state system. Phys Rev A 81(5):052103. https:\/\/doi.org\/10.1103\/PhysRevA.81.052103","journal-title":"Phys Rev A"},{"issue":"1","key":"120_CR30","doi-asserted-by":"publisher","first-page":"013182","DOI":"10.1103\/PhysRevResearch.3.013182","volume":"3","author":"K Head-Marsden","year":"2021","unstructured":"Head-Marsden K, Krastanov S, Mazziotti DA, Narang P (2021) Capturing non-Markovian dynamics on near-term quantum computers. Phys Rev Research 3(1):013182. https:\/\/doi.org\/10.1103\/PhysRevResearch.3.013182","journal-title":"Phys Rev Research"},{"issue":"26","key":"120_CR31","doi-asserted-by":"publisher","first-page":"5022","DOI":"10.1103\/PhysRevLett.78.5022","volume":"78","author":"S Hill","year":"1997","unstructured":"Hill S, Wootters WK (1997) Entanglement of a pair of quantum bits. Phys Rev Lett 78(26):5022. https:\/\/doi.org\/10.1103\/PhysRevLett.78.5022","journal-title":"Phys Rev Lett"},{"key":"120_CR32","unstructured":"Hinton G, Srivastava N, Swersky K (2012) Neural networks for machine learning lecture 6a overview of mini-batch gradient descent. 14:2"},{"key":"120_CR33","doi-asserted-by":"publisher","unstructured":"Johansson J, Nation P, Nori F (2012) QuTIP: an open-source python framework for the dynamics of open quantum systems. Comput Phys Commun 183:1760. https:\/\/doi.org\/10.1016\/j.cpc.2012.02.021","DOI":"10.1016\/j.cpc.2012.02.021"},{"key":"120_CR34","doi-asserted-by":"publisher","first-page":"4620","DOI":"10.1038\/srep04620","volume":"4","author":"E-M Laine","year":"2014","unstructured":"Laine E-M, Breuer H-P, Piilo J (2014) Nonlocal memory effects allow perfect teleportation with mixed states. Sci Rep 4:4620. https:\/\/doi.org\/10.1038\/srep04620","journal-title":"Sci Rep"},{"key":"120_CR35","doi-asserted-by":"publisher","first-page":"30001","DOI":"10.1209\/0295-5075\/128\/30001","volume":"128","author":"C-F Li","year":"2020","unstructured":"Li C-F, Guo G-C, Piilo J (2020) Non-Markovian quantum dynamics: what is it good for? EPL 128:30001. https:\/\/doi.org\/10.1209\/0295-5075\/128\/30001","journal-title":"EPL"},{"issue":"14","key":"120_CR36","doi-asserted-by":"publisher","first-page":"140502","DOI":"10.1103\/PhysRevLett.124.140502","volume":"124","author":"IA Luchnikov","year":"2020","unstructured":"Luchnikov IA, Vintskevich SV, Grigoriev DA, Filippov SN (2020) Machine learning non-Markovian quantum dynamics. Phys Rev Lett 124(14):140502. https:\/\/doi.org\/10.1103\/PhysRevLett.124.140502","journal-title":"Phys Rev Lett"},{"issue":"4","key":"120_CR37","doi-asserted-by":"publisher","first-page":"044101","DOI":"10.1103\/PhysRevA.86.044101","volume":"86","author":"S Luo","year":"2012","unstructured":"Luo S, Fu S, Song H (2012) Quantifying non-Markovianity via correlations. Phys Rev A 86(4):044101. https:\/\/doi.org\/10.1103\/PhysRevA.86.044101","journal-title":"Phys Rev A"},{"key":"120_CR38","doi-asserted-by":"publisher","DOI":"10.1088\/1367-2630\/18\/2\/023023","volume":"18","author":"JR McClean","year":"2016","unstructured":"McClean JR, Romero J, Babbush R, Aspuru-Guzik A (2016) The theory of variational hybrid quantum-classical algorithms. New J Phys 18:023023. https:\/\/doi.org\/10.1088\/1367-2630\/18\/2\/023023","journal-title":"New J Phys"},{"issue":"3","key":"120_CR39","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.98.032309","volume":"98","author":"K Mitarai","year":"2018","unstructured":"Mitarai K, Negoro M, Kitagawa MKF (2018) Quantum circuit learning. Phys Rev A 98(3):032309. https:\/\/doi.org\/10.1103\/PhysRevA.98.032309","journal-title":"Phys Rev A"},{"issue":"2","key":"120_CR40","doi-asserted-by":"publisher","first-page":"2250007","DOI":"10.1142\/S123016122250007X","volume":"29","author":"J Morris","year":"2022","unstructured":"Morris J, Pollock FA, Modi K (2022) Quantifying non-Markovian memory in a superconducting quantum computer. Open Systems & Information Dynamics 29(2):2250007. https:\/\/doi.org\/10.1142\/S123016122250007X","journal-title":"Open Systems & Information Dynamics"},{"key":"120_CR41","doi-asserted-by":"publisher","first-page":"308","DOI":"10.1093\/comjnl\/7.4.308","volume":"7","author":"JA Nelder","year":"1965","unstructured":"Nelder JA, Mead R (1965) A simple method for function minimization. Comput J 7:308","journal-title":"Comput J"},{"issue":"3","key":"120_CR42","doi-asserted-by":"publisher","first-page":"032105","DOI":"10.1103\/PhysRevA.94.032105","volume":"94","author":"AC Neto","year":"2016","unstructured":"Neto AC, Karpat G, Fanchini FF (2016) Inequivalence of correlation-based measures of non-Markovianity. Phys Rev A 94(3):032105. https:\/\/doi.org\/10.1103\/PhysRevA.94.032105","journal-title":"Phys Rev A"},{"key":"120_CR43","first-page":"032105","volume-title":"Quantum computation and quantum information","author":"MA Nielsen","year":"2000","unstructured":"Nielsen MA, Chuang I (2000) Quantum computation and quantum information. Cambridge University Press, Cambridge, p 032105"},{"issue":"9","key":"120_CR44","doi-asserted-by":"publisher","first-page":"022110","DOI":"10.1103\/PhysRevB.97.094304","volume":"97","author":"A Norambuena","year":"2018","unstructured":"Norambuena A, Mu\u00f1oz E, Dinani HT, Jarmola A, Maletinsky P, Budker D, Maze JR (2018) Spin-lattice relaxation of individual solidstate spins. Phys Rev B 97(9):022110. https:\/\/doi.org\/10.1103\/PhysRevB.97.094304","journal-title":"Phys Rev B"},{"issue":"2","key":"120_CR45","doi-asserted-by":"publisher","first-page":"022110","DOI":"10.1103\/PhysRevA.101.022110","volume":"101","author":"A Norambuena","year":"2020","unstructured":"Norambuena A, Maze JR, Rabl P, Coto R (2020) Quantifying phononinduced non-Markovianity in color centers in diamond. Phys Rev A 101(2):022110. https:\/\/doi.org\/10.1103\/PhysRevA.101.022110","journal-title":"Phys Rev A"},{"issue":"2","key":"120_CR46","doi-asserted-by":"publisher","first-page":"361","DOI":"10.1103\/RevModPhys.86.361","volume":"86","author":"E Paladino","year":"2014","unstructured":"Paladino E, Galperin YM, Falci G, Altshuler BL (2014) 1=f noise: implications for solid-state quantum information. Rev Mod Phys 86(2):361. https:\/\/doi.org\/10.1103\/RevModPhys.86.361","journal-title":"Rev Mod Phys"},{"issue":"5","key":"120_CR47","doi-asserted-by":"publisher","first-page":"052412","DOI":"10.1103\/PhysRevA.104.052412","volume":"104","author":"AM Palmieri","year":"2021","unstructured":"Palmieri AM, Bianchi F, Paris MGA, Benedetti C (2021) Multiclass classification of dephasing channels. Phys Rev A 104(5):052412. https:\/\/doi.org\/10.1103\/PhysRevA.104.052412","journal-title":"Phys Rev A"},{"issue":"2","key":"120_CR48","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.105.022605","volume":"105","author":"M Papi\u010d","year":"2022","unstructured":"Papi\u010d M, de Vega I (2022) Neural-network-based qubit-environment characterization. Phys Rev A 105(2):022605. https:\/\/doi.org\/10.1103\/PhysRevA.105.022605","journal-title":"Phys Rev A"},{"issue":"4","key":"120_CR49","doi-asserted-by":"publisher","first-page":"040405","DOI":"10.1103\/PhysRevLett.120.040405","volume":"120","author":"FA Pollock","year":"2018","unstructured":"Pollock FA, Rodr\u00edguez-Rosario C, Frauenheim T, Paternostro M, Modi K (2018) Operational Markov condition for quantum processes. Phys Rev Lett 120(4):040405. https:\/\/doi.org\/10.1103\/PhysRevLett.120.040405","journal-title":"Phys Rev Lett"},{"key":"120_CR50","doi-asserted-by":"publisher","unstructured":"Preskill J (2018) Quantum computing in the NISQ era and beyond. Quantum 2:79. https:\/\/doi.org\/10.22331\/q-2018-08-06-79","DOI":"10.22331\/q-2018-08-06-79"},{"key":"120_CR51","doi-asserted-by":"publisher","first-page":"12430","DOI":"10.1038\/srep12430","volume":"5","author":"D Reich","year":"2015","unstructured":"Reich D, Katz N, Koch C (2015) Exploiting non-Markovianity for quantum control. Sci Rep 5:12430. https:\/\/doi.org\/10.1038\/srep12430","journal-title":"Sci Rep"},{"key":"120_CR52","doi-asserted-by":"publisher","unstructured":"Rivas A, Huelga S, Plenio MB Entanglement and non-Markovianity of quantum evolutions. Phys Rev Lett 105(5):050403. https:\/\/doi.org\/10.1103\/PhysRevLett.105.050403","DOI":"10.1103\/PhysRevLett.105.050403"},{"key":"120_CR53","doi-asserted-by":"publisher","first-page":"400","DOI":"10.1214\/aoms\/1177729586","volume":"22","author":"H Robbins","year":"1951","unstructured":"Robbins H, Monro S (1951) A stochastic approximation method Ann Math Stat 22:400. https:\/\/doi.org\/10.1214\/aoms\/1177729586","journal-title":"A stochastic approximation method Ann Math Stat"},{"key":"120_CR54","unstructured":"Ruder S (2016) An overview of gradient descent optimization algorithms. Preprint at arXiv:1609.04747"},{"issue":"3","key":"120_CR55","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1063\/PT.3.4164","volume":"72","author":"D Sarma","year":"2019","unstructured":"Sarma D, Deng DL, Duan LM (2019) Machine learning meets quantum physics. Physics Today 72(3):48. https:\/\/doi.org\/10.1063\/PT.3.4164","journal-title":"Physics Today"},{"key":"120_CR56","doi-asserted-by":"crossref","unstructured":"Schuld, M (2021) \u2019Quantum machine learning models are kernel methods. Preprint at arXiv:2101.11020","DOI":"10.1007\/978-3-030-83098-4_6"},{"issue":"3","key":"120_CR57","doi-asserted-by":"publisher","first-page":"032331","DOI":"10.1103\/PhysRevA.99.032331","volume":"99","author":"M Schuld","year":"2019","unstructured":"Schuld M, Bergholm V, Gogolin C, Izaac J (2019) Killoran, N Evaluating analytic gradients on quantum hardware. Phys Rev A 99(3):032331. https:\/\/doi.org\/10.1103\/PhysRevA.99.032331","journal-title":"Phys Rev A"},{"issue":"2","key":"120_CR58","doi-asserted-by":"publisher","first-page":"172","DOI":"10.1080\/00107514.2014.964942","volume":"56","author":"M Schuld","year":"2015","unstructured":"Schuld M, Sinayskiy I, Petruccione F (2015) An introduction to quantum machine learning. Contemp Phys 56(2):172\u2013185. https:\/\/doi.org\/10.1080\/00107514.2014.964942","journal-title":"Contemp Phys"},{"issue":"08","key":"120_CR59","doi-asserted-by":"publisher","first-page":"1840010","DOI":"10.1142\/S0219749918400105","volume":"16","author":"S Shrapnel","year":"2018","unstructured":"Shrapnel S, Costa F, Milburn G (2018) Quantum Markovianity as a supervised learning task. International Journal of Quantum Information 16(08):1840010. https:\/\/doi.org\/10.1142\/S0219749918400105","journal-title":"International Journal of Quantum Information"},{"key":"120_CR60","doi-asserted-by":"publisher","first-page":"042321","DOI":"10.1103\/PhysRevA.83.042321","volume":"83","author":"R Vasile","year":"2011","unstructured":"Vasile R, Olivares S, Paris MGA, Maniscalco S (2011) Continuous-variable quantum key distribution in non-Markovian channels. Phys Rev A 83:042321. https:\/\/doi.org\/10.1103\/PhysRevA.83.042321","journal-title":"Phys Rev A"},{"key":"120_CR61","doi-asserted-by":"publisher","unstructured":"Whalen SJ (2016) Carmichael HJ (2016) Time-local Heisenberg-Langevin equations and the driven qubit. Phys Rev A 93(6):063820. https:\/\/doi.org\/10.1103\/PhysRevA.93.063820","DOI":"10.1103\/PhysRevA.93.063820"},{"key":"120_CR62","doi-asserted-by":"publisher","first-page":"6301","DOI":"10.1038\/s41467-020-20113-3","volume":"11","author":"GAL White","year":"2020","unstructured":"White GAL, Hill CD, Pollock FA, Hollenberg LCL, Modi K (2020) Demonstration of non-Markovian process characterisation and control on a quantum processor. Nat. Commun. 11:6301. https:\/\/doi.org\/10.1038\/s41467-020-20113-3","journal-title":"Nat. Commun."},{"key":"120_CR63","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139525343","volume-title":"Quantum information theory","author":"MM Wilde","year":"2013","unstructured":"Wilde MM (2013) Quantum information theory. Cambridge University Press, Cambridge"},{"key":"120_CR64","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.125.170501","volume":"125","author":"YH Zhang","year":"2020","unstructured":"Zhang YH, Zheng PL, Zhang Y, Deng DL (2020) Topological quantum compiling with reinforcement learning. Phys Rev Lett 125:170501. https:\/\/doi.org\/10.1103\/PhysRevLett.125.170501","journal-title":"Phys Rev Lett"},{"key":"120_CR65","doi-asserted-by":"publisher","first-page":"9918","DOI":"10.1126\/sciadv.aaw9918","volume":"5","author":"D Zhu","year":"2019","unstructured":"Zhu D et al (2019) Training of quantum circuits on a hybrid quantum computer. Sci Adv 5:9918. https:\/\/doi.org\/10.1126\/sciadv.aaw9918","journal-title":"Sci Adv"}],"container-title":["Quantum Machine Intelligence"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s42484-023-00120-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s42484-023-00120-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s42484-023-00120-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,18]],"date-time":"2023-12-18T05:27:21Z","timestamp":1702877241000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s42484-023-00120-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,17]]},"references-count":65,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,12]]}},"alternative-id":["120"],"URL":"https:\/\/doi.org\/10.1007\/s42484-023-00120-5","relation":{"has-preprint":[{"id-type":"doi","id":"10.21203\/rs.3.rs-2493079\/v1","asserted-by":"object"}]},"ISSN":["2524-4906","2524-4914"],"issn-type":[{"value":"2524-4906","type":"print"},{"value":"2524-4914","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,17]]},"assertion":[{"value":"18 January 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 June 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 July 2023","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 declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"29"}}