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Recently, the general mechanism of fractional resonances has been proposed that leads to slowing the many-body dynamics in systems with both U(1) and parity symmetry. Here, we show that fractional resonance is stable under local noise models. To corroborate our finding, we numerically study the dynamics of a small-scale Bose\u2013Hubbard model that can readily be implemented in existing noisy intermediate-scale quantum (NISQ) devices. Our findings suggest a possible pathway toward a stable nonequilibrium state of matter, with potential applications of quantum memories for quantum information processing.<\/jats:p>","DOI":"10.3390\/sym14122562","type":"journal-article","created":{"date-parts":[[2022,12,5]],"date-time":"2022-12-05T03:38:38Z","timestamp":1670211518000},"page":"2562","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Stable Many-Body Resonances in Open Quantum Systems"],"prefix":"10.3390","volume":"14","author":[{"given":"Rub\u00e9n","family":"Pe\u00f1a","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica, Universidad de Santiago de Chile, Avenida V\u00edctor Jara 3493, Santiago 9170124, Chile"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3557-2709","authenticated-orcid":false,"given":"Thi Ha","family":"Kyaw","sequence":"additional","affiliation":[{"name":"LG Electronics Toronto AI Lab, Toronto, ON M5V 1M3, Canada"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2917-7814","authenticated-orcid":false,"given":"Guillermo","family":"Romero","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, CEDENNA, Universidad de Santiago de Chile, Avenida V\u00edctor Jara 3493, Santiago 9170124, Chile"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"017003","DOI":"10.1103\/PRXQuantum.2.017003","article-title":"Quantum Simulators: Architectures and Opportunities","volume":"2","author":"Altman","year":"2021","journal-title":"PRX Quantum"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"885","DOI":"10.1103\/RevModPhys.80.885","article-title":"Many-body physics with ultracold gases","volume":"80","author":"Bloch","year":"2008","journal-title":"Rev. Mod. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1038\/nature10748","article-title":"Light-cone-like spreading of correlations in a quantum many-body system","volume":"481","author":"Cheneau","year":"2012","journal-title":"Nature"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1038\/nature24622","article-title":"Probing many-body dynamics on a 51-atom quantum simulator","volume":"551","author":"Bernien","year":"2017","journal-title":"Nature"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1547","DOI":"10.1126\/science.aaf8834","article-title":"Exploring the many-body localization transition in two dimensions","volume":"352","author":"Choi","year":"2016","journal-title":"Science"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1175","DOI":"10.1126\/science.aao1401","article-title":"Spectroscopic signatures of localization with interacting photons in superconducting qubits","volume":"358","author":"Roushan","year":"2017","journal-title":"Science"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1038\/s41586-019-0897-9","article-title":"A dissipatively stabilized Mott insulator of photons","volume":"566","author":"Ma","year":"2019","journal-title":"Nature"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"050502","DOI":"10.1103\/PhysRevLett.123.050502","article-title":"Propagation and Localization of Collective Excitations on a 24-Qubit Superconducting Processor","volume":"123","author":"Ye","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"170503","DOI":"10.1103\/PhysRevLett.125.170503","article-title":"Ergodic-Localized Junctions in a Periodically Driven Spin Chain","volume":"125","author":"Zha","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"033043","DOI":"10.1103\/PhysRevResearch.3.033043","article-title":"Experimental characterization of the quantum many-body localization transition","volume":"3","author":"Gong","year":"2021","journal-title":"Phys. Rev. Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1126\/science.aao4309","article-title":"A blueprint for demonstrating quantum supremacy with superconducting qubits","volume":"360","author":"Neill","year":"2018","journal-title":"Science"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"033617","DOI":"10.1103\/PhysRevA.91.033617","article-title":"Modeling spontaneous breaking of time-translation symmetry","volume":"91","author":"Sacha","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"090402","DOI":"10.1103\/PhysRevLett.117.090402","article-title":"Floquet Time Crystals","volume":"117","author":"Else","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"030401","DOI":"10.1103\/PhysRevLett.118.030401","article-title":"Discrete Time Crystals: Rigidity, Criticality, and Realizations","volume":"118","author":"Yao","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"016401","DOI":"10.1088\/1361-6633\/aa8b38","article-title":"Time crystals: A review","volume":"81","author":"Sacha","year":"2017","journal-title":"Rep. Prog. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"150601","DOI":"10.1103\/PhysRevLett.123.150601","article-title":"Period-n Discrete Time Crystals and Quasicrystals with Ultracold Bosons","volume":"123","author":"Pizzi","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Sacha, K. (2020). Time Crystals, Springer. [1st ed.].","DOI":"10.1007\/978-3-030-52523-1"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2341","DOI":"10.1038\/s41467-021-22583-5","article-title":"Higher-order and fractional discrete time crystals in clean long-range interacting systems","volume":"12","author":"Pizzi","year":"2021","journal-title":"Nat. Commun."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"140602","DOI":"10.1103\/PhysRevLett.127.140602","article-title":"Classical Prethermal Phases of Matter","volume":"127","author":"Pizzi","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"094308","DOI":"10.1103\/PhysRevB.104.094308","article-title":"Classical approaches to prethermal discrete time crystals in one, two, and three dimensions","volume":"104","author":"Pizzi","year":"2021","journal-title":"Phys. Rev. B"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"L042023","DOI":"10.1103\/PhysRevResearch.3.L042023","article-title":"Emergent parametric resonances and time-crystal phases in driven Bardeen-Cooper-Schrieffer systems","volume":"3","author":"Usaj","year":"2021","journal-title":"Phys. Rev. Res."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"256803","DOI":"10.1103\/PhysRevLett.115.256803","article-title":"Exponentially Slow Heating in Periodically Driven Many-Body Systems","volume":"115","author":"Abanin","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"014112","DOI":"10.1103\/PhysRevB.95.014112","article-title":"Effective Hamiltonians, prethermalization, and slow energy absorption in periodically driven many-body systems","volume":"95","author":"Abanin","year":"2017","journal-title":"Phys. Rev. B"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1007\/s00220-017-2930-x","article-title":"A Rigorous Theory of Many-Body Prethermalization for Periodically Driven and Closed Quantum Systems","volume":"354","author":"Abanin","year":"2017","journal-title":"Commun. Math. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"120401","DOI":"10.1103\/PhysRevLett.116.120401","article-title":"Rigorous Bound on Energy Absorption and Generic Relaxation in Periodically Driven Quantum Systems","volume":"116","author":"Mori","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.aop.2016.01.012","article-title":"Floquet\u2013Magnus theory and generic transient dynamics in periodically driven many-body quantum systems","volume":"367","author":"Kuwahara","year":"2016","journal-title":"Ann. Phys."},{"key":"ref_27","first-page":"021044","article-title":"Floquet Prethermalization in a Bose-Hubbard System","volume":"10","author":"Ippoliti","year":"2020","journal-title":"Phys. Rev. X"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"012418","DOI":"10.1103\/PhysRevA.105.012418","article-title":"Floquet prethermal phase protected by U(1) symmetry on a superconducting quantum processor","volume":"105","author":"Ying","year":"2022","journal-title":"Phys. Rev. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"40","DOI":"10.21468\/SciPostPhys.11.2.040","article-title":"Statistical Floquet prethermalization of the Bose-Hubbard model","volume":"11","author":"Torre","year":"2021","journal-title":"SciPost Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1063","DOI":"10.1080\/00018732.2010.514702","article-title":"Dynamics of a quantum phase transition and relaxation to a steady state","volume":"59","author":"Dziarmaga","year":"2010","journal-title":"Adv. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"863","DOI":"10.1103\/RevModPhys.83.863","article-title":"Colloquium: Nonequilibrium dynamics of closed interacting quantum systems","volume":"83","author":"Polkovnikov","year":"2011","journal-title":"Rev. Mod. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1038\/nphys3215","article-title":"Quantum many-body systems out of equilibrium","volume":"11","author":"Eisert","year":"2015","journal-title":"Nat. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1146\/annurev-conmatphys-031016-025451","article-title":"Quantum Quench Dynamics","volume":"9","author":"Mitra","year":"2018","journal-title":"Annu. Rev. Condens. Matter Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"054001","DOI":"10.1088\/1361-6633\/aaaf9a","article-title":"Dynamical quantum phase transitions: A review","volume":"81","author":"Heyl","year":"2018","journal-title":"Rep. Prog. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"538","DOI":"10.1038\/s42254-019-0086-7","article-title":"Tensor networks for complex quantum systems","volume":"1","year":"2019","journal-title":"Nat. Rev. Phys."},{"key":"ref_36","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","year":"2011","journal-title":"Ann. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Silvi, P., Tschirsich, F., Gerster, M., J\u00fcnemann, J., Jaschke, D., Rizzi, M., and Montangero, S. (2019). The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems. SciPost Phys. Lect. Notes, 8.","DOI":"10.21468\/SciPostPhysLectNotes.8"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"064307","DOI":"10.1103\/PhysRevB.106.064307","article-title":"Fractional resonances and prethermal states in Floquet systems","volume":"106","author":"Bastidas","year":"2022","journal-title":"Phys. Rev. B"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"546","DOI":"10.1103\/PhysRevB.40.546","article-title":"Boson localization and the superfluid-insulator transition","volume":"40","author":"Fisher","year":"1989","journal-title":"Phys. Rev. B"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"3108","DOI":"10.1103\/PhysRevLett.81.3108","article-title":"Cold Bosonic Atoms in Optical Lattices","volume":"81","author":"Jaksch","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"104401","DOI":"10.1103\/PhysRevB.67.104401","article-title":"Ground-state phase diagram of S = 1 XXZ chains with uniaxial single-ion-type anisotropy","volume":"67","author":"Chen","year":"2003","journal-title":"Phys. Rev. B"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"163203","DOI":"10.1103\/PhysRevLett.126.163203","article-title":"Tunable Single-Ion Anisotropy in Spin-1 Models Realized with Ultracold Atoms","volume":"126","author":"Chung","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"856","DOI":"10.1038\/nphys466","article-title":"Quantum phase transitions of light","volume":"2","author":"Greentree","year":"2006","journal-title":"Nat. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"849","DOI":"10.1038\/nphys462","article-title":"Strongly interacting polaritons in coupled arrays of cavities","volume":"2","author":"Hartmann","year":"2006","journal-title":"Nat. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"031805","DOI":"10.1103\/PhysRevA.76.031805","article-title":"Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays","volume":"76","author":"Angelakis","year":"2007","journal-title":"Phys. Rev. A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"9269","DOI":"10.1073\/pnas.1819316116","article-title":"From Bloch oscillations to many-body localization in clean interacting systems","volume":"116","author":"Baum","year":"2019","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"080504","DOI":"10.1103\/PhysRevLett.127.080504","article-title":"Driving Enhanced Quantum Sensing in Partially Accessible Many-Body Systems","volume":"127","author":"Mishra","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_48","unstructured":"Nielsen, M.A., and Chuang, I.L. (2000). Quantum Computation and Quantum Information, Cambridge University Press."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Wittek, P. (2014). Quantum Machine Learning: What Quantum Computing Means to Data Mining, Academic Press.","DOI":"10.1016\/B978-0-12-800953-6.00004-9"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"045002","DOI":"10.1103\/RevModPhys.91.045002","article-title":"Machine learning and the physical sciences","volume":"91","author":"Carleo","year":"2019","journal-title":"Rev. Mod. Phys."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Schuld, M., and Petruccione, F. (2021). Machine Learning with Quantum Computers, Springer International Publishing. Quantum Science and Technology.","DOI":"10.1007\/978-3-030-83098-4"},{"key":"ref_52","unstructured":"Shor, P. (1994, January 20\u201322). Algorithms for quantum computation: Discrete logarithms and factoring. Proceedings of the 35th Annual Symposium on Foundations of Computer Science, Santa Fe, NM, USA."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1704","DOI":"10.1126\/science.1113479","article-title":"Simulated Quantum Computation of Molecular Energies","volume":"309","author":"Dutoi","year":"2005","journal-title":"Science"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1103\/RevModPhys.86.153","article-title":"Quantum simulation","volume":"86","author":"Georgescu","year":"2014","journal-title":"Rev. Mod. Phys."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"10856","DOI":"10.1021\/acs.chemrev.8b00803","article-title":"Quantum Chemistry in the Age of Quantum Computing","volume":"119","author":"Cao","year":"2019","journal-title":"Chem. Rev."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"015003","DOI":"10.1103\/RevModPhys.92.015003","article-title":"Quantum computational chemistry","volume":"92","author":"McArdle","year":"2020","journal-title":"Rev. Mod. Phys."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"79","DOI":"10.22331\/q-2018-08-06-79","article-title":"Quantum Computing in the NISQ era and beyond","volume":"2","author":"Preskill","year":"2018","journal-title":"Quantum"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1038\/s41586-019-1666-5","article-title":"Quantum supremacy using a programmable superconducting processor","volume":"574","author":"Arute","year":"2019","journal-title":"Nature"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"1460","DOI":"10.1126\/science.abe8770","article-title":"Quantum computational advantage using photons","volume":"370","author":"Zhong","year":"2020","journal-title":"Science"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"180501","DOI":"10.1103\/PhysRevLett.127.180501","article-title":"Strong Quantum Computational Advantage Using a Superconducting Quantum Processor","volume":"127","author":"Wu","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1038\/s41586-022-04725-x","article-title":"Quantum computational advantage with a programmable photonic processor","volume":"606","author":"Madsen","year":"2022","journal-title":"Nature"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"015004","DOI":"10.1103\/RevModPhys.94.015004","article-title":"Noisy intermediate-scale quantum algorithms","volume":"94","author":"Bharti","year":"2022","journal-title":"Rev. Mod. Phys."},{"key":"ref_63","unstructured":"Chen, S., Cotler, J., Huang, H.Y., and Li, J. (2021). Exponential separations between learning with and without quantum memory. arXiv."},{"key":"ref_64","doi-asserted-by":"crossref","unstructured":"Huang, H.Y., Broughton, M., Cotler, J., Chen, S., Li, J., Mohseni, M., Neven, H., Babbush, R., Kueng, R., and Preskill, J. (2021). Quantum advantage in learning from experiments. arXiv.","DOI":"10.1126\/science.abn7293"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1038\/s41467-021-27922-0","article-title":"Quantum algorithmic measurement","volume":"13","author":"Aharonov","year":"2022","journal-title":"Nat. Commun."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"150502","DOI":"10.1103\/PhysRevLett.103.150502","article-title":"Quantum Algorithm for Linear Systems of Equations","volume":"103","author":"Harrow","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1038\/nphys3029","article-title":"Quantum principal component analysis","volume":"10","author":"Lloyd","year":"2014","journal-title":"Nat. Phys."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"160501","DOI":"10.1103\/PhysRevLett.100.160501","article-title":"Quantum Random Access Memory","volume":"100","author":"Giovannetti","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"8621","DOI":"10.1038\/srep08621","article-title":"Scalable quantum memory in the ultrastrong coupling regime","volume":"5","author":"Kyaw","year":"2015","journal-title":"Sci. Rep."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1146\/annurev-conmatphys-031214-014726","article-title":"Many-Body Localization and Thermalization in Quantum Statistical Mechanics","volume":"6","author":"Nandkishore","year":"2015","journal-title":"Annu. Rev. Condens. Matter Phys."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"021001","DOI":"10.1103\/RevModPhys.91.021001","article-title":"Colloquium: Many-body localization, thermalization, and entanglement","volume":"91","author":"Abanin","year":"2019","journal-title":"Rev. Mod. Phys."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"498","DOI":"10.1016\/j.crhy.2018.03.003","article-title":"Many-body localization: An introduction and selected topics","volume":"19","author":"Alet","year":"2018","journal-title":"C. R. Phys."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"205133","DOI":"10.1103\/PhysRevB.105.205133","article-title":"Information-theoretic memory scaling in the many-body localization transition","volume":"105","author":"Bayat","year":"2022","journal-title":"Phys. Rev. B"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/0370-1573(79)90023-1","article-title":"A universal instability of many-dimensional oscillator systems","volume":"52","author":"Chirikov","year":"1979","journal-title":"Phys. Rep."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1038\/nphys4106","article-title":"Equilibration and order in quantum Floquet matter","volume":"13","author":"Moessner","year":"2017","journal-title":"Nat. Phys."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1080\/00018732.2015.1055918","article-title":"Universal high-frequency behavior of periodically driven systems: From dynamical stabilization to Floquet engineering","volume":"64","author":"Bukov","year":"2015","journal-title":"Adv. Phys."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"907","DOI":"10.1088\/0143-0807\/31\/4\/020","article-title":"A pedagogical approach to the Magnus expansion","volume":"31","author":"Blanes","year":"2010","journal-title":"Eur. J. Phys."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"12474","DOI":"10.1103\/PhysRevB.61.12474","article-title":"One-dimensional Bose-Hubbard model with nearest-neighbor interaction","volume":"61","author":"White","year":"2000","journal-title":"Phys. Rev. B"},{"key":"ref_79","first-page":"031043","article-title":"Observation of a Dissipation-Induced Classical to Quantum Transition","volume":"4","author":"Raftery","year":"2014","journal-title":"Phys. Rev. X"},{"key":"ref_80","first-page":"021044","article-title":"Imaging Photon Lattice States by Scanning Defect Microscopy","volume":"6","author":"Underwood","year":"2016","journal-title":"Phys. Rev. X"},{"key":"ref_81","first-page":"011016","article-title":"Observation of a Dissipative Phase Transition in a One-Dimensional Circuit QED Lattice","volume":"7","author":"Fitzpatrick","year":"2017","journal-title":"Phys. Rev. X"},{"key":"ref_82","doi-asserted-by":"crossref","unstructured":"Breuer, H.P., and Petruccione, F. (2007). The Theory of Open Quantum Systems, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780199213900.001.0001"},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"010501","DOI":"10.1103\/PhysRevLett.114.010501","article-title":"Coherence and Decay of Higher Energy Levels of a Superconducting Transmon Qubit","volume":"114","author":"Peterer","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"040502","DOI":"10.1103\/PhysRevLett.93.040502","article-title":"Efficient Simulation of One-Dimensional Quantum Many-Body Systems","volume":"93","author":"Vidal","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_85","doi-asserted-by":"crossref","unstructured":"Hauschild, J., and Pollmann, F. (2018). Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy). SciPost Phys. Lect. Notes, 5.","DOI":"10.21468\/SciPostPhysLectNotes.5"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/12\/2562\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:33:46Z","timestamp":1760146426000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/12\/2562"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,4]]},"references-count":85,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["sym14122562"],"URL":"https:\/\/doi.org\/10.3390\/sym14122562","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,12,4]]}}}