{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T12:33:52Z","timestamp":1769517232766,"version":"3.49.0"},"reference-count":86,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T00:00:00Z","timestamp":1765756800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000183","name":"Army Research Office","doi-asserted-by":"crossref","award":["W911NF-22-1-0258"],"award-info":[{"award-number":["W911NF-22-1-0258"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"\n Effective Hamiltonian calculations for large quantum systems can be both analytically intractable and numerically expensive using standard techniques. In this manuscript, we present numerical techniques inspired by Nonperturbative Analytical Diagonalization (NPAD) and the Magnus expansion for the efficient calculation of effective Hamiltonians. While these tools are appropriate for a wide array of applications, we here demonstrate their utility for models that can be realized in circuit-QED settings. Our numerical techniques are available as an open-source Python package,\n \n \n qCH<\/mml:mtext>\n eff<\/mml:mtext>\n <\/mml:msub>\n <\/mml:math>\n , which is available on GitHub and PyPI. We use the CuPy library for GPU-acceleration and report up to 15x speedup on GPU over CPU for NPAD, and up to 42x speedup for the Magnus expansion (compared to QuTiP), for large system sizes.\n <\/jats:p>","DOI":"10.22331\/q-2025-12-15-1946","type":"journal-article","created":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T10:21:01Z","timestamp":1765794061000},"page":"1946","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":["GPU-accelerated Effective Hamiltonian Calculator"],"prefix":"10.22331","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-9483-3394","authenticated-orcid":false,"given":"Abhishek","family":"Chakraborty","sequence":"first","affiliation":[{"name":"NVIDIA, Santa Clara, California 95051, USA"},{"name":"Center for Coherence and Quantum Science (CCQS), University of Rochester, Rochester, NY 14627, USA"},{"name":"Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA"},{"name":"Institute for Quantum Studies (IQS), Chapman University, Orange, CA 92866, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4242-6072","authenticated-orcid":false,"given":"Taylor L.","family":"Patti","sequence":"additional","affiliation":[{"name":"NVIDIA, Santa Clara, California 95051, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7584-3489","authenticated-orcid":false,"given":"Brucek","family":"Khailany","sequence":"additional","affiliation":[{"name":"NVIDIA, Santa Clara, California 95051, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9646-7013","authenticated-orcid":false,"given":"Andrew N.","family":"Jordan","sequence":"additional","affiliation":[{"name":"Center for Coherence and Quantum Science (CCQS), University of Rochester, Rochester, NY 14627, USA"},{"name":"Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA"},{"name":"Institute for Quantum Studies (IQS), Chapman University, Orange, CA 92866, USA"},{"name":"The Kennedy Chair in Physics, Chapman University, Orange, CA 92866, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6974-6797","authenticated-orcid":false,"given":"Anima","family":"Anandkumar","sequence":"additional","affiliation":[{"name":"Department of Computing + Mathematical Sciences (CMS), California Institute of Technology (Caltech), Pasadena, CA 91125, USA"}]}],"member":"9598","published-online":{"date-parts":[[2025,12,15]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Flemming J\u00f8rgensen. ``Effective hamiltonians''. Molecular Physics 29, 1137\u20131164 (1975).","DOI":"10.1080\/00268977500100971"},{"key":"1","doi-asserted-by":"publisher","unstructured":"Georges Jolicard. ``Effective Hamiltonian Theory and Molecular Dynamics''. Annual Review of Physical Chemistry 46, 83\u2013108 (1995).","DOI":"10.1146\/annurev.pc.46.100195.000503"},{"key":"2","doi-asserted-by":"publisher","unstructured":"Daniel Zeuch, Fabian Hassler, Jesse J. Slim, and David P. DiVincenzo. ``Exact rotating wave approximation''. Annals of Physics 423, 168327 (2020).","DOI":"10.1016\/j.aop.2020.168327"},{"key":"3","doi-asserted-by":"publisher","unstructured":"Easwar Magesan and Jay M. Gambetta. ``Effective Hamiltonian models of the cross-resonance gate''. Physical Review A 101, 052308 (2020).","DOI":"10.1103\/PhysRevA.101.052308"},{"key":"4","doi-asserted-by":"publisher","unstructured":"Gioele Consani and Paul A Warburton. ``Effective Hamiltonians for interacting superconducting qubits: Local basis reduction and the Schrieffer\u2013Wolff transformation''. New Journal of Physics 22, 053040 (2020).","DOI":"10.1088\/1367-2630\/ab83d1"},{"key":"5","doi-asserted-by":"publisher","unstructured":"Jayameenakshi Venkatraman, Xu Xiao, Rodrigo G. Corti\u00f1as, Alec Eickbusch, and Michel H. Devoret. ``Static Effective Hamiltonian of a Rapidly Driven Nonlinear System''. Physical Review Letters 129, 100601 (2022). arxiv:2108.02861.","DOI":"10.1103\/PhysRevLett.129.100601"},{"key":"6","doi-asserted-by":"publisher","unstructured":"Juan Carlos Sandoval-Santana, Victor Guadalupe Ibarra-Sierra, Jos\u00e9 Luis Cardoso, Alejandro Kunold, Pedro Roman-Taboada, and Gerardo Naumis. ``Method for Finding the Exact Effective Hamiltonian of Time-Driven Quantum Systems''. Annalen der Physik 531, 1900035 (2019).","DOI":"10.1002\/andp.201900035"},{"key":"7","doi-asserted-by":"publisher","unstructured":"Holger Haas, Daniel Puzzuoli, Feihao Zhang, and David G. Cory. ``Engineering effective Hamiltonians''. New Journal of Physics 21, 103011 (2019).","DOI":"10.1088\/1367-2630\/ab4525"},{"key":"8","doi-asserted-by":"publisher","unstructured":"Nicola Macr\u00ec, Luigi Giannelli, Elisabetta Paladino, and Giuseppe Falci. ``Coarse-Grained Effective Hamiltonian via the Magnus Expansion for a Three-Level System''. Entropy 25, 234 (2023).","DOI":"10.3390\/e25020234"},{"key":"9","doi-asserted-by":"publisher","unstructured":"Omar Gamel and Daniel F. V. James. ``Time-averaged quantum dynamics and the validity of the effective Hamiltonian model''. Physical Review A 82, 052106 (2010).","DOI":"10.1103\/PhysRevA.82.052106"},{"key":"10","doi-asserted-by":"publisher","unstructured":"Andreas Brinkmann. ``Introduction to average Hamiltonian theory. I. Basics''. Concepts in Magnetic Resonance Part A 45A, e21414 (2016).","DOI":"10.1002\/cmr.a.21414"},{"key":"11","doi-asserted-by":"publisher","unstructured":"Ben J. Powell. ``Introduction to effective low-energy hamiltonians in condensed matter physics and chemistry''. Chapter 10, pages 309\u2013366. John Wiley & Sons, Ltd. (2011). arXiv:0906.1640.","DOI":"10.1002\/9780470930779.ch10"},{"key":"12","doi-asserted-by":"publisher","unstructured":"A. B. Klimov, L. L. S\u00e1nchez-Soto, A. Navarro, and E. C. Yustas. ``Effective Hamiltonians in quantum optics: a systematic approach''. Journal of Modern Optics 49, 2211\u20132226 (2002).","DOI":"10.1080\/09500340210134675"},{"key":"13","doi-asserted-by":"publisher","unstructured":"D F James and J Jerke. ``Effective Hamiltonian theory and its applications in quantum information''. Canadian Journal of Physics 85, 625\u2013632 (2007).","DOI":"10.1139\/p07-060"},{"key":"14","doi-asserted-by":"publisher","unstructured":"D. F. V. James. ``Quantum Computation with Hot and Cold Ions: An Assessment of Proposed Schemes''. Scalable Quantum Computers: Paving the Way to Realization 48, 53\u201367 (2000).","DOI":"10.1002\/3527603182.ch5"},{"key":"15","doi-asserted-by":"publisher","unstructured":"Nicholas Anto-Sztrikacs, Ahsan Nazir, and Dvira Segal. ``Effective-Hamiltonian Theory of Open Quantum Systems at Strong Coupling''. PRX Quantum 4, 020307 (2023). arxiv:2211.05701.","DOI":"10.1103\/PRXQuantum.4.020307"},{"key":"16","doi-asserted-by":"publisher","unstructured":"Sergey Bravyi, David P. DiVincenzo, Daniel Loss, and Barbara M. Terhal. ``Simulation of Many-Body Hamiltonians using Perturbation Theory with Bounded-Strength Interactions''. Physical Review Letters 101, 070503 (2008). arxiv:0803.2686.","DOI":"10.1103\/PhysRevLett.101.070503"},{"key":"17","doi-asserted-by":"publisher","unstructured":"Bo Peng, Yuan Su, Daniel Claudino, Karol Kowalski, Guang Hao Low, and Martin Roetteler. ``Quantum Simulation of Boson-Related Hamiltonians: Techniques, Effective Hamiltonian Construction, and Error Analysis''. Quantum Science and Technology 10, 023002 (2025). arXiv:2307.06580.","DOI":"10.1088\/2058-9565\/adbf42"},{"key":"18","doi-asserted-by":"publisher","unstructured":"Alexandre Blais, Ren-Shou Huang, Andreas Wallraff, S. M. Girvin, and R. J. Schoelkopf. ``Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation''. Physical Review A 69, 062320 (2004).","DOI":"10.1103\/PhysRevA.69.062320"},{"key":"19","doi-asserted-by":"publisher","unstructured":"Alexandre Blais, Jay Gambetta, A. Wallraff, D. I. Schuster, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf. ``Quantum-information processing with circuit quantum electrodynamics''. Physical Review A 75, 032329 (2007).","DOI":"10.1103\/PhysRevA.75.032329"},{"key":"20","doi-asserted-by":"publisher","unstructured":"Andrew D. Greentree, Charles Tahan, Jared H. Cole, and Lloyd C. L. Hollenberg. ``Quantum phase transitions of light''. Nature Physics 2, 856\u2013861 (2006).","DOI":"10.1038\/nphys466"},{"key":"21","doi-asserted-by":"publisher","unstructured":"Wallace Givens. ``Computation of Plain Unitary Rotations Transforming a General Matrix to Triangular Form''. Journal of the Society for Industrial and Applied Mathematics 6, 26\u201350 (1958).","DOI":"10.1137\/0106004"},{"key":"22","doi-asserted-by":"publisher","unstructured":"Jens Koch, Terri M. Yu, Jay Gambetta, A. A. Houck, D. I. Schuster, J. Majer, Alexandre Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf. ``Charge-insensitive qubit design derived from the Cooper pair box''. Physical Review A 76, 042319 (2007).","DOI":"10.1103\/PhysRevA.76.042319"},{"key":"23","doi-asserted-by":"publisher","unstructured":"Alexandre Blais, Arne L. Grimsmo, S. M. Girvin, and Andreas Wallraff. ``Circuit Quantum Electrodynamics''. Reviews of Modern Physics 93, 025005 (2021). arxiv:2005.12667.","DOI":"10.1103\/RevModPhys.93.025005"},{"key":"24","doi-asserted-by":"publisher","unstructured":"Zihao Wang, Rayleigh W. Parker, Elizabeth Champion, and Machiel S. Blok. ``Systematic study of High ${E}_J\/{E}_C$ transmon qudits up to $d = 12$''. Physical Review Applied 23, 034046 (2025). arXiv:2407.17407.","DOI":"10.1103\/physrevapplied.23.034046"},{"key":"25","doi-asserted-by":"publisher","unstructured":"Sergey Bravyi, David P. DiVincenzo, and Daniel Loss. ``Schrieffer\u2013Wolff transformation for quantum many-body systems''. Annals of Physics 326, 2793\u20132826 (2011).","DOI":"10.1016\/j.aop.2011.06.004"},{"key":"26","doi-asserted-by":"publisher","unstructured":"Boxi Li, Tommaso Calarco, and Felix Motzoi. ``Nonperturbative Analytical Diagonalization of Hamiltonians with Application to Circuit QED''. PRX Quantum 3, 030313 (2022).","DOI":"10.1103\/PRXQuantum.3.030313"},{"key":"27","doi-asserted-by":"publisher","unstructured":"Isidora Araya Day, Sebastian Miles, Hugo Kerstens, Daniel Varjas, and Anton R. Akhmerov. ``Pymablock: An algorithm and a package for quasi-degenerate perturbation theory''. SciPost Physics Codebases (2025). arXiv:2404.03728. code: 10.21468\/SciPostPhysCodeb.50-r2.1.","DOI":"10.21468\/scipostphyscodeb.50"},{"key":"28","doi-asserted-by":"publisher","unstructured":"S. Blanes, F. Casas, J. A. Oteo, and J. Ros. ``The Magnus expansion and some of its applications''. Physics Reports 470, 151\u2013238 (2009).","DOI":"10.1016\/j.physrep.2008.11.001"},{"key":"29","unstructured":"Ryosuke Okuta, Yuya Unno, Daisuke Nishino, Shohei Hido, and Crissman Loomis. ``Cupy: A numpy-compatible library for nvidia gpu calculations''. In Proceedings of Workshop on Machine Learning Systems (LearningSys) in The Thirty-first Annual Conference on Neural Information Processing Systems (NIPS). (2017). url: http:\/\/learningsys.org\/nips17\/assets\/papers\/paper_16.pdf. code: cupy\/cupy."},{"key":"30","doi-asserted-by":"publisher","unstructured":"Claude Cohen-Tannoudji, Jacques Dupont-Roc, and Gilbert Grynberg. ``Atom-Photon Interactions: Basic Processes and Applications''. John Wiley & Sons. (1998).","DOI":"10.1002\/9783527617197"},{"key":"31","doi-asserted-by":"publisher","unstructured":"Guanyu Zhu, David G. Ferguson, Vladimir E. Manucharyan, and Jens Koch. ``Circuit QED with fluxonium qubits: Theory of the dispersive regime''. Physical Review B 87, 024510 (2013). arXiv:1210.1605.","DOI":"10.1103\/PhysRevB.87.024510"},{"key":"32","doi-asserted-by":"publisher","unstructured":"Vladimir E. Manucharyan, Jens Koch, Leonid I. Glazman, and Michel H. Devoret. ``Fluxonium: Single cooper-pair circuit free of charge offsets''. Science 326, 113\u2013116 (2009).","DOI":"10.1126\/science.1175552"},{"key":"33","doi-asserted-by":"publisher","unstructured":"Vanessa Paulisch, Han Rui, Hui Khoon Ng, and Berthold-Georg Englert. ``Beyond adiabatic elimination: A hierarchy of approximations for multi-photon processes''. The European Physical Journal Plus 129, 12 (2014).","DOI":"10.1140\/epjp\/i2014-14012-8"},{"key":"34","unstructured":"William H. Press. ``Numerical Recipes 3rd Edition: The Art of Scientific Computing''. Cambridge University Press. Cambridge [u.a.] (2007). 3. ed. edition. url: https:\/\/numerical.recipes\/."},{"key":"35","doi-asserted-by":"publisher","unstructured":"Jens Koch and Karyn Le Hur. ``Superfluid\u2013Mott-insulator transition of light in the Jaynes-Cummings lattice''. Physical Review A 80, 023811 (2009).","DOI":"10.1103\/PhysRevA.80.023811"},{"key":"36","doi-asserted-by":"publisher","unstructured":"Matthew P. A. Fisher, Peter B. Weichman, G. Grinstein, and Daniel S. Fisher. ``Boson localization and the superfluid-insulator transition''. Physical Review B 40, 546\u2013570 (1989).","DOI":"10.1103\/PhysRevB.40.546"},{"key":"37","doi-asserted-by":"publisher","unstructured":"Dimitris G. Angelakis, Marcelo Franca Santos, and Sougato Bose. ``Photon-blockade-induced Mott transitions and $XY$ spin models in coupled cavity arrays''. Physical Review A 76, 031805 (2007).","DOI":"10.1103\/PhysRevA.76.031805"},{"key":"38","doi-asserted-by":"publisher","unstructured":"M.J. Hartmann, F.G.S.L. Brand\u00e3o, and M.B. Plenio. ``Quantum many-body phenomena in coupled cavity arrays''. Laser & Photonics Reviews 2, 527\u2013556 (2008). arXiv:0808.2557.","DOI":"10.1002\/lpor.200810046"},{"key":"39","doi-asserted-by":"publisher","unstructured":"E. K. Twyeffort Irish, C. D. Ogden, and M. S. Kim. ``Polaritonic characteristics of insulator and superfluid states in a coupled-cavity array''. Physical Review A 77, 033801 (2008).","DOI":"10.1103\/PhysRevA.77.033801"},{"key":"40","doi-asserted-by":"publisher","unstructured":"E.T. Jaynes and F.W. Cummings. ``Comparison of quantum and semiclassical radiation theories with application to the beam maser''. Proceedings of the IEEE 51, 89\u2013109 (1963).","DOI":"10.1109\/PROC.1963.1664"},{"key":"41","doi-asserted-by":"publisher","unstructured":"Peter Groszkowski and Jens Koch. ``Scqubits: a Python package for superconducting qubits''. Quantum 5, 583 (2021).","DOI":"10.22331\/q-2021-11-17-583"},{"key":"42","doi-asserted-by":"publisher","unstructured":"Sai Pavan Chitta, Tianpu Zhao, Ziwen Huang, Ian Mondragon-Shem, and Jens Koch. ``Computer-aided quantization and numerical analysis of superconducting circuits''. New Journal of Physics 24, 103020 (2022). arXiv:2206.08320.","DOI":"10.1088\/1367-2630\/ac94f2"},{"key":"43","doi-asserted-by":"publisher","unstructured":"J. R. Johansson, P. D. Nation, and Franco Nori. ``QuTiP: An open-source Python framework for the dynamics of open quantum systems''. Computer Physics Communications 183, 1760\u20131772 (2012).","DOI":"10.1016\/j.cpc.2012.02.021"},{"key":"44","doi-asserted-by":"publisher","unstructured":"J. R. Johansson, P. D. Nation, and Franco Nori. ``QuTiP 2: A Python framework for the dynamics of open quantum systems''. Computer Physics Communications 184, 1234\u20131240 (2013).","DOI":"10.1016\/j.cpc.2012.11.019"},{"key":"45","doi-asserted-by":"publisher","unstructured":"Mogens Dalgaard and Felix Motzoi. ``Fast, high precision dynamics in quantum optimal control theory''. Journal of Physics B: Atomic, Molecular and Optical Physics 55, 085501 (2022). arxiv:2110.06187.","DOI":"10.1088\/1361-6455\/ac6366"},{"key":"46","doi-asserted-by":"publisher","unstructured":"Richard R. Ernst, Geoffrey Bodenhausen, and and Alexander Wokaun. ``Principles of Nuclear Magnetic Resonance in One and Two Dimensions''. International Series of Monographs on Chemistry. Oxford University Press. Oxford, New York (1990).","DOI":"10.1093\/oso\/9780198556473.001.0001"},{"key":"47","doi-asserted-by":"publisher","unstructured":"Z. Xiao, E. Doucet, T. Noh, L. Ranzani, R.W. Simmonds, L.C.G. Govia, and A. Kamal. ``Perturbative Diagonalization for Time-Dependent Strong Interactions''. Physical Review Applied 18, 024009 (2022). arxiv:2103.09260.","DOI":"10.1103\/PhysRevApplied.18.024009"},{"key":"48","doi-asserted-by":"publisher","unstructured":"Shai Machnes, Elie Ass\u00e9mat, David Tannor, and Frank K. Wilhelm. ``Tunable, flexible, and efficient optimization of control pulses for practical qubits''. Physical Review Letters 120, 150401 (2018). arXiv:1507.04261v2.","DOI":"10.1103\/PhysRevLett.120.150401"},{"key":"49","doi-asserted-by":"publisher","unstructured":"Philip Krantz, Morten Kjaergaard, Fei Yan, Terry P. Orlando, Simon Gustavsson, and William D. Oliver. ``A Quantum Engineer's Guide to Superconducting Qubits''. Applied Physics Reviews 6, 021318 (2019). arxiv:1904.06560.","DOI":"10.1063\/1.5089550"},{"key":"50","doi-asserted-by":"publisher","unstructured":"S.E. Rasmussen, K.S. Christensen, S.P. Pedersen, L.B. Kristensen, T. B\u00e6kkegaard, N.J.S. Loft, and N.T. Zinner. ``Superconducting Circuit Companion\u2014an Introduction with Worked Examples''. PRX Quantum 2, 040204 (2021).","DOI":"10.1103\/PRXQuantum.2.040204"},{"key":"51","doi-asserted-by":"publisher","unstructured":"Christiane P. Koch, Ugo Boscain, Tommaso Calarco, Gunther Dirr, Stefan Filipp, Steffen J. Glaser, Ronnie Kosloff, Simone Montangero, Thomas Schulte-Herbr\u00fcggen, Dominique Sugny, and Frank K. Wilhelm. ``Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe''. EPJ Quantum Technology 9, 1\u201360 (2022).","DOI":"10.1140\/epjqt\/s40507-022-00138-x"},{"key":"52","doi-asserted-by":"publisher","unstructured":"Joel Howard, Alexander Lidiak, Casey Jameson, Bora Basyildiz, Kyle Clark, Tongyu Zhao, Mustafa Bal, Junling Long, David P. Pappas, Meenakshi Singh, and Zhexuan Gong. ``Implementing two-qubit gates at the quantum speed limit''. Physical Review Research 5, 043194 (2023).","DOI":"10.1103\/PhysRevResearch.5.043194"},{"key":"53","doi-asserted-by":"publisher","unstructured":"Denys I. Bondar, Lloren\u00e7 Balada Gaggioli, Georgios Korpas, Jakub Marecek, Jiri Vala, and Kurt Jacobs. ``Globally optimal control of quantum dynamics''. Physical Review Research 7 (2025). arXiv:2209.05790.","DOI":"10.1103\/g4fb-xm13"},{"key":"54","doi-asserted-by":"publisher","unstructured":"Yuquan Chen, Yajie Hao, Ze Wu, Bi-Ying Wang, Ran Liu, Yanjun Hou, Jiangyu Cui, Man-Hong Yung, and Xinhua Peng. ``Accelerating quantum optimal control through iterative gradient-ascent pulse engineering''. Physical Review A 108, 052603 (2023).","DOI":"10.1103\/PhysRevA.108.052603"},{"key":"55","doi-asserted-by":"publisher","unstructured":"T. S. Mahesh, Priya Batra, and M. Harshanth Ram. ``Quantum optimal control: practical aspects and diverse methods.''. J. Indian Inst. Sci. 103, 591\u2013607 (2023).","DOI":"10.1007\/s41745-022-00311-2"},{"key":"56","doi-asserted-by":"publisher","unstructured":"J. J. Sakurai and Jim Napolitano. ``Modern quantum mechanics''. Cambridge University Press. Cambridge (2020). Third edition.","DOI":"10.1017\/9781108587280"},{"key":"57","doi-asserted-by":"publisher","unstructured":"F. Bloch and A. Siegert. ``Magnetic Resonance for Nonrotating Fields''. Physical Review 57, 522\u2013527 (1940).","DOI":"10.1103\/PhysRev.57.522"},{"key":"58","doi-asserted-by":"publisher","unstructured":"P. W. Milonni, D. F. V. James, and H. Fearn. ``Photodetection and causality in quantum optics''. Physical Review A 52, 1525\u20131537 (1995).","DOI":"10.1103\/PhysRevA.52.1525"},{"key":"59","doi-asserted-by":"publisher","unstructured":"Brian Baker, Andy C. Y. Li, Nicholas Irons, Nathan Earnest, and Jens Koch. ``Adaptive rotating-wave approximation for driven open quantum systems''. Physical Review A 98, 052111 (2018).","DOI":"10.1103\/PhysRevA.98.052111"},{"key":"60","doi-asserted-by":"publisher","unstructured":"Alexandru Petrescu, Camille Le Calonnec, Catherine Leroux, Agustin Di Paolo, Pranav Mundada, Sara Sussman, Andrei Vrajitoarea, Andrew A. Houck, and Alexandre Blais. ``Accurate methods for the analysis of strong-drive effects in parametric gates''. Physical Review Applied 19, 044003 (2023). arxiv:2107.02343.","DOI":"10.1103\/PhysRevApplied.19.044003"},{"key":"61","doi-asserted-by":"publisher","unstructured":"Pranav S. Mundada, Andr\u00e1s Gyenis, Ziwen Huang, Jens Koch, and Andrew A. Houck. ``Floquet-Engineered Enhancement of Coherence Times in a Driven Fluxonium Qubit''. Physical Review Applied 14, 054033 (2020).","DOI":"10.1103\/PhysRevApplied.14.054033"},{"key":"62","doi-asserted-by":"publisher","unstructured":"Anne M. Forney, Steven R. Jackson, and Frederick W. Strauch. ``Multi-frequency control pulses for multi-level superconducting quantum circuits''. Physical Review A 81, 012306 (2010). arxiv:0909.1577.","DOI":"10.1103\/PhysRevA.81.012306"},{"key":"63","doi-asserted-by":"publisher","unstructured":"Sang-Kil Son, Siyuan Han, and Shih-I Chu. ``Floquet formulation for the investigation of multiphoton quantum interference in a superconducting qubit driven by a strong ac field''. Physical Review A 79, 032301 (2009).","DOI":"10.1103\/PhysRevA.79.032301"},{"key":"64","doi-asserted-by":"publisher","unstructured":"Jon H. Shirley. ``Solution of the schr\u00f6dinger equation with a hamiltonian periodic in time''. Physical Review 138, B979\u2013B987 (1965).","DOI":"10.1103\/PhysRev.138.B979"},{"key":"65","doi-asserted-by":"crossref","unstructured":"Daniel Zeuch and David P. DiVincenzo. ``Refuting a Proposed Axiom for Defining the Exact Rotating Wave Approximation'' (2020). arXiv:2010.02751.","DOI":"10.1016\/j.aop.2020.168327"},{"key":"66","doi-asserted-by":"publisher","unstructured":"L. S. Theis and F. K. Wilhelm. ``Nonadiabatic corrections to fast dispersive multiqubit gates involving $Z$ control''. Physical Review A 95, 022314 (2017).","DOI":"10.1103\/physreva.95.022314"},{"key":"67","doi-asserted-by":"publisher","unstructured":"Moein Malekakhlagh, Alexandru Petrescu, and Hakan E. T\u00fcreci. ``Lifetime renormalization of weakly anharmonic superconducting qubits. I. Role of number nonconserving terms''. Physical Review B 101, 134509 (2020).","DOI":"10.1103\/PhysRevB.101.134509"},{"key":"68","doi-asserted-by":"publisher","unstructured":"Alexandru Petrescu, Moein Malekakhlagh, and Hakan E. T\u00fcreci. ``Lifetime renormalization of driven weakly anharmonic superconducting qubits. II. The readout problem''. Physical Review B 101, 134510 (2020).","DOI":"10.1103\/PhysRevB.101.134510"},{"key":"69","doi-asserted-by":"publisher","unstructured":"Moein Malekakhlagh, Easwar Magesan, and David C. McKay. ``First-principles analysis of cross-resonance gate operation''. Physical Review A 102, 042605 (2020).","DOI":"10.1103\/PhysRevA.102.042605"},{"key":"70","doi-asserted-by":"publisher","unstructured":"Ross Shillito, Jonathan A. Gross, Agustin Di Paolo, \u00c9lie Genois, and Alexandre Blais. ``Fast and differentiable simulation of driven quantum systems''. Physical Review Research 3, 033266 (2021).","DOI":"10.1103\/physrevresearch.3.033266"},{"key":"71","doi-asserted-by":"publisher","unstructured":"Daniel Puzzuoli, Sophia Fuhui Lin, Moein Malekakhlagh, Emily Pritchett, Benjamin Rosand, and Christopher J. Wood. ``Algorithms for perturbative analysis and simulation of quantum dynamics''. Journal of Computational Physics 489, 112262 (2023).","DOI":"10.1016\/j.jcp.2023.112262"},{"key":"72","unstructured":"Jakub Marecek and Jiri Vala. ``Quantum Optimal Control via Magnus Expansion and Non-Commutative Polynomial Optimization'' (2020). arXiv:2001.06464."},{"key":"73","doi-asserted-by":"publisher","unstructured":"S. S\u00e1nchez, F. Casas, and A. Fern\u00e1ndez. ``New analytic approximations based on the Magnus expansion''. Journal of Mathematical Chemistry 49, 1741\u20131758 (2011).","DOI":"10.1007\/s10910-011-9855-y"},{"key":"74","doi-asserted-by":"publisher","unstructured":"Eric A. Butcher, Ma\u2019en Sari, Ed Bueler, and Tim Carlson. ``Magnus\u2019 expansion for time-periodic systems: Parameter-dependent approximations''. Communications in Nonlinear Science and Numerical Simulation 14, 4226\u20134245 (2009).","DOI":"10.1016\/j.cnsns.2009.02.030"},{"key":"75","doi-asserted-by":"publisher","unstructured":"Nikita Kopylov. ``Magnus-based geometric integrators for dynamical systems with time-dependent potentials''. PhD thesis. Universitat Politecnica de Valencia. Valencia (Spain) (2019).","DOI":"10.4995\/thesis\/10251\/118798"},{"key":"76","unstructured":"Juergen Geiser and Vahid Yaghoubi. ``Effective Simulation Methods for Structures with Local Nonlinearity: Magnus integrator and Successive Approximations'' (2014). arXiv:1411.7134."},{"key":"77","doi-asserted-by":"publisher","unstructured":"Jamie Bamber and Will Handley. ``Beyond the Runge-Kutta-Wentzel-Kramers-Brillouin method''. Physical Review D 101, 043517 (2020).","DOI":"10.1103\/PhysRevD.101.043517"},{"key":"78","doi-asserted-by":"publisher","unstructured":"Ana Arnal, Fernando Casas, Cristina Chiralt, and Jos\u00e9 Angel Oteo. ``A Unifying Framework for Perturbative Exponential Factorizations''. Mathematics 9, 637 (2021).","DOI":"10.3390\/math9060637"},{"key":"79","doi-asserted-by":"publisher","unstructured":"Michael Vogl, Pontus Laurell, Aaron D. Barr, and Gregory A. Fiete. ``Analog of Hamilton-Jacobi theory for the time-evolution operator''. Physical Review A 100, 012132 (2019).","DOI":"10.1103\/PhysRevA.100.012132"},{"key":"80","doi-asserted-by":"publisher","unstructured":"F. J. Dyson. ``The ${S}$ Matrix in Quantum Electrodynamics''. Physical Review 75, 1736\u20131755 (1949).","DOI":"10.1103\/PhysRev.75.1736"},{"key":"81","doi-asserted-by":"publisher","unstructured":"Boxi Li, Tommaso Calarco, and Felix Motzoi. ``Experimental error suppression in cross-resonance gates via multi-derivative pulse shaping''. npj Quantum Inf 10, 66 (2024) 10 (2023). arxiv:2303.01427.","DOI":"10.48550\/ARXIV.2303.01427"},{"key":"82","doi-asserted-by":"publisher","unstructured":"I. N. Moskalenko, I. S. Besedin, I. A. Simakov, and A. V. Ustinov. ``Tunable coupling scheme for implementing two-qubit gates on fluxonium qubits''. Applied Physics Letters 119, 194001 (2021). arxiv:2107.11550.","DOI":"10.1063\/5.0064800"},{"key":"83","doi-asserted-by":"publisher","unstructured":"D.K. Weiss, Helin Zhang, Chunyang Ding, Yuwei Ma, David I. Schuster, and Jens Koch. ``Fast High-Fidelity Gates for Galvanically-Coupled Fluxonium Qubits Using Strong Flux Modulation''. PRX Quantum 3, 040336 (2022).","DOI":"10.1103\/PRXQuantum.3.040336"},{"key":"84","doi-asserted-by":"publisher","unstructured":"Leon Ding, Max Hays, Youngkyu Sung, Bharath Kannan, Junyoung An, Agustin Di Paolo, Amir H. Karamlou, Thomas M. Hazard, Kate Azar, David K. Kim, Bethany M. Niedzielski, Alexander Melville, Mollie E. Schwartz, Jonilyn L. Yoder, Terry P. Orlando, Simon Gustavsson, Jeffrey A. Grover, Kyle Serniak, and William D. Oliver. ``High-Fidelity, Frequency-Flexible Two-Qubit Fluxonium Gates with a Transmon Coupler''. Physical Review X 13, 031035 (2023).","DOI":"10.1103\/PhysRevX.13.031035"},{"key":"85","doi-asserted-by":"publisher","unstructured":"Emma L. Rosenfeld, Connor T. Hann, David I. Schuster, Matthew H. Matheny, and Aashish A. Clerk. ``Designing high-fidelity two-qubit gates between fluxonium qubits''. PRX Quantum 5, 040317 (2024). arXiv:2403.07242.","DOI":"10.1103\/prxquantum.5.040317"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2025-12-15-1946\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T10:23:16Z","timestamp":1765794196000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2025-12-15-1946\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,15]]},"references-count":86,"URL":"https:\/\/doi.org\/10.22331\/q-2025-12-15-1946","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,15]]},"article-number":"1946"}}