{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T15:16:35Z","timestamp":1783696595790,"version":"3.55.0"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T00:00:00Z","timestamp":1778198400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T00:00:00Z","timestamp":1778198400000},"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 Inf Process"],"DOI":"10.1007\/s11128-026-05197-y","type":"journal-article","created":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T05:04:28Z","timestamp":1778216668000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Investigation of macroscopic resonant tunneling mechanism in superconducting flux qubit"],"prefix":"10.1007","volume":"25","author":[{"given":"Jianxin","family":"Shi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,5,8]]},"reference":[{"issue":"22","key":"5197_CR1","first-page":"15398","volume":"62","author":"DV Averin","year":"2000","unstructured":"Averin, D.V., Friedman, J.R., Lukens, J.E.: Macroscopic resonant tunneling of magnetic flux. Phys. Rev. B 62(22), 15398\u201315407 (2000)","journal-title":"Phys. Rev. B"},{"issue":"19","key":"5197_CR2","volume":"88","author":"JR Friedman","year":"2002","unstructured":"Friedman, J.R., Averin, D.V.: Aharonov-Casher-effect suppression of macroscopic tunneling of magnetic flux. Phys. Rev. Lett. 88(19), 197004 (2002)","journal-title":"Phys. Rev. Lett."},{"issue":"5607","key":"5197_CR3","doi-asserted-by":"publisher","first-page":"1869","DOI":"10.1126\/science.1081045","volume":"299","author":"I Chiorescu","year":"2003","unstructured":"Chiorescu, I., Nakamura, Y., Harmans, C.J.P.M., Mooij, J.E.: Coherent quantum dynamics of a superconducting flux qubit. Science 299(5607), 1869\u20131871 (2003)","journal-title":"Science"},{"issue":"6","key":"5197_CR4","volume":"73","author":"E Zipper","year":"2006","unstructured":"Zipper, E., Kurpas, M., Szelag, M., Dajka, J., Szopa, M.: Flux qubit on a mesoscopic nonsuperconducting ring. Phys. Rev. B 73(6), 064508 (2006)","journal-title":"Phys. Rev. B"},{"key":"5197_CR5","doi-asserted-by":"crossref","unstructured":"Harris, R., Johnson, M.W., Han, S., Berkley, A.J., Johansson, J., Bunyk, P., et al.: Probing Noise In Flux Qubits Via Macroscopic Resonant Tunneling. arXiv:0706.1232 (2007)","DOI":"10.1103\/PhysRevLett.101.117003"},{"issue":"3","key":"5197_CR6","volume":"80","author":"S-K Son","year":"2009","unstructured":"Son, S.-K., Han, S., Chu, S.-I.: Floquet formulation for the investigation of multiphoton quantum interference in a superconducting qubit driven by a strong Ac field. Phys. Rev. A 80(3), 033833 (2009)","journal-title":"Phys. Rev. A"},{"issue":"14","key":"5197_CR7","volume":"81","author":"T Lanting","year":"2010","unstructured":"Lanting, T., Harris, R., Johansson, J., Amin, M.H.S., Berkley, A.J., Gildert, S., et al.: Cotunneling in pairs of coupled flux qubits. Phys. Rev. B 81(14), 144510 (2010)","journal-title":"Phys. Rev. B"},{"issue":"8","key":"5197_CR8","volume":"26","author":"J Shi","year":"2017","unstructured":"Shi, J., Xu, W., Sun, G., Chen, J., Kang, L., Wu, P.: Macroscopic resonant tunneling in an Rf - SQUID flux qubit under a single - cycle sinusoidal driving. Chin. Phys. B 26(8), 080304 (2017)","journal-title":"Chin. Phys. B"},{"issue":"7","key":"5197_CR9","volume":"64","author":"JQ You","year":"2001","unstructured":"You, J.Q., Nori, F.: Coherent quantum dynamics of a superconducting flux qubit coupled to a harmonic oscillator. Phys. Rev. B 64(7), 075312 (2001)","journal-title":"Phys. Rev. B"},{"issue":"5556","key":"5197_CR10","first-page":"102","volume":"295","author":"JE Mooij","year":"2002","unstructured":"Mooij, J.E., et al.: Josephson persistent-current qubit. Science 295(5556), 102\u2013107 (2002)","journal-title":"Science"},{"issue":"6791","key":"5197_CR11","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1038\/35017505","volume":"406","author":"JR Friedman","year":"2000","unstructured":"Friedman, J.R., et al.: Quantum superposition of distinct macroscopic states. Nature 406(6791), 43\u201346 (2000)","journal-title":"Nature"},{"issue":"6730","key":"5197_CR12","doi-asserted-by":"publisher","first-page":"786","DOI":"10.1038\/19718","volume":"398","author":"Y Nakamura","year":"1999","unstructured":"Nakamura, Y., Pashkin, Y.A., Tsai, J.S.: Coherent control of macroscopic quantum states in a single-Cooper-pair box. Nature 398(6730), 786\u2013788 (1999)","journal-title":"Nature"},{"issue":"14","key":"5197_CR13","first-page":"1015","volume":"60","author":"TP Orlando","year":"1999","unstructured":"Orlando, T.P., et al.: Superconducting persistent-current qubit. Phys. Rev. B 60(14), 1015\u20131022 (1999)","journal-title":"Phys. Rev. B"},{"issue":"17","key":"5197_CR14","volume":"67","author":"JM Martinis","year":"2003","unstructured":"Martinis, J.M., et al.: A tunable superconducting flux qubit. Phys. Rev. B 67(17), 174509 (2003)","journal-title":"Phys. Rev. B"},{"issue":"5569","key":"5197_CR15","doi-asserted-by":"publisher","first-page":"886","DOI":"10.1126\/science.1069372","volume":"296","author":"D Vion","year":"2002","unstructured":"Vion, D., et al.: Manipulating the quantum state of an electrical circuit. Science 296(5569), 886\u2013889 (2002)","journal-title":"Science"},{"issue":"22","key":"5197_CR16","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevB.98.220403","volume":"66","author":"F Ishikawa","year":"2002","unstructured":"Ishikawa, F., et al.: Macroscopic quantum tunneling in a superconducting loop with a large- inductance Josephson junction. Phys. Rev. B 66(22), 224512 (2002)","journal-title":"Phys. Rev. B"},{"issue":"1","key":"5197_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1103\/RevModPhys.59.1","volume":"59","author":"AJ Leggett","year":"1987","unstructured":"Leggett, A.J.: The quantum mechanics of macroscopic systems. Rev. Mod. Phys. 59(1), 1\u201341 (1987)","journal-title":"Rev. Mod. Phys."},{"issue":"2","key":"5197_CR18","volume":"69","author":"G Burkard","year":"2004","unstructured":"Burkard, G., Wilhelm, F.K.: Quantum-state engineering of a superconducting flux qubit. Phys. Rev. A 69(2), 022313 (2004)","journal-title":"Phys. Rev. A"},{"issue":"10","key":"5197_CR19","volume":"71","author":"RW Simmonds","year":"2005","unstructured":"Simmonds, R.W., et al.: Non-adiabatic control of a superconducting flux qubit. Phys. Rev. B 71(10), 100503 (2005)","journal-title":"Phys. Rev. B"},{"issue":"7160","key":"5197_CR20","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1038\/nature06184","volume":"449","author":"J Majer","year":"2007","unstructured":"Majer, J., et al.: Coupling superconducting qubits via a cavity bus. Nature 449(7160), 443\u2013447 (2007)","journal-title":"Nature"},{"issue":"7005","key":"5197_CR21","doi-asserted-by":"publisher","first-page":"162","DOI":"10.1038\/nature02851","volume":"431","author":"A Wallraff","year":"2004","unstructured":"Wallraff, A., et al.: Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature 431(7005), 162\u2013167 (2004)","journal-title":"Nature"},{"issue":"6","key":"5197_CR22","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.69.062320","volume":"69","author":"A Blais","year":"2004","unstructured":"Blais, A., et al.: Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation. Phys. Rev. A 69(6), 062320 (2004)","journal-title":"Phys. Rev. A"},{"issue":"10","key":"5197_CR23","volume":"73","author":"DI Schuster","year":"2006","unstructured":"Schuster, D.I., et al.: Resonant interactions between superconducting qubits and electrical resonators. Phys. Rev. B 73(10), 100501 (2006)","journal-title":"Phys. Rev. B"},{"issue":"7252","key":"5197_CR24","doi-asserted-by":"publisher","first-page":"240","DOI":"10.1038\/nature08121","volume":"460","author":"L DiCarlo","year":"2009","unstructured":"DiCarlo, L., et al.: Demonstration of two-qubit algorithms with a superconducting quantum processor. Nature 460(7252), 240\u2013244 (2009)","journal-title":"Nature"},{"issue":"7496","key":"5197_CR25","doi-asserted-by":"publisher","first-page":"500","DOI":"10.1038\/nature13171","volume":"508","author":"R Barends","year":"2014","unstructured":"Barends, R., et al.: Superconducting quantum circuits at the surface - code threshold for fault - tolerant quantum computing. Nature 508(7496), 500\u2013503 (2014)","journal-title":"Nature"},{"issue":"7541","key":"5197_CR26","doi-asserted-by":"publisher","first-page":"66","DOI":"10.1038\/nature14270","volume":"519","author":"J Kelly","year":"2015","unstructured":"Kelly, J., et al.: State and error characterization of a 5-qubit superconducting quantum processor. Nature 519(7541), 66\u201369 (2015)","journal-title":"Nature"},{"issue":"7779","key":"5197_CR27","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1038\/s41586-019-1666-5","volume":"574","author":"F Arute","year":"2019","unstructured":"Arute, F., et al.: Quantum supremacy using a programmable superconducting processor. Nature 574(7779), 505\u2013510 (2019)","journal-title":"Nature"},{"issue":"11","key":"5197_CR28","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1063\/1.2155757","volume":"58","author":"JQ You","year":"2005","unstructured":"You, J.Q., Nori, F.: Quantum computing with superconducting circuits: design, fabrication, and control. Phys. Today 58(11), 42\u201349 (2005)","journal-title":"Phys. Today"},{"issue":"6124","key":"5197_CR29","doi-asserted-by":"publisher","first-page":"1169","DOI":"10.1126\/science.1231930","volume":"339","author":"MH Devoret","year":"2013","unstructured":"Devoret, M.H., Schoelkopf, R.J.: Superconducting circuits for quantum information: an outlook. Science 339(6124), 1169\u20131174 (2013)","journal-title":"Science"},{"issue":"7198","key":"5197_CR30","doi-asserted-by":"publisher","first-page":"1031","DOI":"10.1038\/nature07128","volume":"453","author":"J Clarke","year":"2008","unstructured":"Clarke, J., Wilhelm, F.K.: Superconducting qubits. Nature 453(7198), 1031\u20131042 (2008)","journal-title":"Nature"},{"issue":"2","key":"5197_CR31","doi-asserted-by":"publisher","DOI":"10.1063\/1.5089550","volume":"6","author":"P Krantz","year":"2019","unstructured":"Krantz, P., et al.: A quantum engineer\u2019s guide to superconducting qubits. Appl. Phys. Rev. 6(2), 021318 (2019)","journal-title":"Appl. Phys. Rev."},{"issue":"4","key":"5197_CR32","first-page":"29","volume":"59","author":"I Siddiqi","year":"2006","unstructured":"Siddiqi, I., Glazman, L.I.: Macroscopic quantum coherence in superconducting circuits. Phys. Today 59(4), 29\u201335 (2006)","journal-title":"Phys. Today"},{"issue":"10","key":"5197_CR33","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.105.104515","volume":"77","author":"C-H Yu","year":"2008","unstructured":"Yu, C.-H., et al.: Macroscopic quantum tunneling in a superconducting flux qubit with a large-inductance Josephson junction. Phys. Rev. B 77(10), 104515 (2008)","journal-title":"Phys. Rev. B"},{"issue":"7160","key":"5197_CR34","doi-asserted-by":"publisher","first-page":"328","DOI":"10.1038\/nature06126","volume":"449","author":"AA Houck","year":"2007","unstructured":"Houck, A.A., et al.: Integrated quantum circuits with superconducting transmission-line resonators. Nature 449(7160), 328\u2013331 (2007)","journal-title":"Nature"},{"key":"5197_CR35","volume-title":"Principles of magnetic resonance","author":"CP Slichter","year":"2013","unstructured":"Slichter, C.P.: Principles of magnetic resonance. Springer Science and Business Media, Berlin (2013)"},{"issue":"17","key":"5197_CR36","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevB.46.10650","volume":"70","author":"A Datta","year":"2004","unstructured":"Datta, A., et al.: Macroscopic quantum tunneling and decoherence in a superconducting flux qubit. Phys. Rev. B 70(17), 174511 (2004)","journal-title":"Phys. Rev. B"},{"issue":"1","key":"5197_CR37","first-page":"118","volume":"97","author":"V Bouchiat","year":"2002","unstructured":"Bouchiat, V., Vion, D., Joyez, P., Pothier, H., Esteve, D.: Quantum-non-demolition measurements on a superconducting qubit. Phys. Scr. T. 97(1), 118\u2013122 (2002)","journal-title":"Phys. Scr. T."},{"issue":"18","key":"5197_CR38","volume":"72","author":"BR Johnson","year":"2005","unstructured":"Johnson, B.R., et al.: Fast, high-fidelity single-qubit gates in a superconducting flux qubit. Phys. Rev. B 72(18), 180505 (2005)","journal-title":"Phys. Rev. B"},{"issue":"13","key":"5197_CR39","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevB.61.8647","volume":"68","author":"FK Wilhelm","year":"2003","unstructured":"Wilhelm, F.K., et al.: Dephasing and relaxation in a superconducting flux qubit. Phys. Rev. B 68(13), 134502 (2003)","journal-title":"Phys. Rev. B"},{"key":"5197_CR40","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198509141.001.0001","volume-title":"Exploring the Quantum: Atoms, Cavities, And Photons","author":"S Haroche","year":"2006","unstructured":"Haroche, S., Raimond, J.-M.: Exploring the Quantum: Atoms, Cavities, And Photons. Oxford University Press, Oxford (2006)"},{"issue":"22","key":"5197_CR41","volume":"69","author":"KW Lehnert","year":"2004","unstructured":"Lehnert, K.W., et al.: Coupling a single Cooper-pair box to a microwave cavity mode. Phys. Rev. B 69(22), 220504 (2004)","journal-title":"Phys. Rev. B"}],"updated-by":[{"DOI":"10.1007\/s11128-026-05240-y","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T00:00:00Z","timestamp":1780617600000}}],"container-title":["Quantum Information Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-026-05197-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11128-026-05197-y","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-026-05197-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T14:56:00Z","timestamp":1783695360000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11128-026-05197-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,8]]},"references-count":41,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2026,5]]}},"alternative-id":["5197"],"URL":"https:\/\/doi.org\/10.1007\/s11128-026-05197-y","relation":{},"ISSN":["1573-1332"],"issn-type":[{"value":"1573-1332","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5,8]]},"assertion":[{"value":"11 August 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 April 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 May 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 June 2026","order":5,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Update","order":6,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The article PDF has been corrected.","order":7,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 June 2026","order":8,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":9,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"A Correction to this paper has been published:","order":10,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"https:\/\/doi.org\/10.1007\/s11128-026-05240-y","URL":"https:\/\/doi.org\/10.1007\/s11128-026-05240-y","order":11,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"165"}}