- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Applied

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Energy-participation-ratio analysis for very anharmonic superconducting circuits

Figen Yilmaz1,*, Siddharth Singh1, Martijn F.S. Zwanenburg1, Jinlun Hu1, Taryn V. Stefanski1,2, and Christian Kraglund Andersen1

  • 1QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands
  • 2Quantum Engineering Centre for Doctoral Training, H. H. Wills Physics Laboratory and Department of Electrical and Electronic Engineering, University of Bristol, BS8 1FD Bristol, United Kingdom

  • *Contact author: f.yilmaz@tudelft.nl
PDF

Phys. Rev. Applied 25, 044021 – Published 8 April, 2026

DOI: https://doi.org/10.1103/1rbn-c4xf

Abstract

Superconducting circuits are being used for large-scale quantum devices, and a major challenge is to perform accurate numerical simulations of device parameters. One of the most advanced methods for analyzing superconducting circuit designs is the energy-participation-ratio (EPR) method, which constructs quantum Hamiltonians based on the energy distribution extracted from classical electromagnetic simulations. In the EPR approach, we extract linear terms from finite-element simulations and add nonlinear terms using the energy participation ratio extracted from the classical simulations. However, the EPR method relies on a low-order expansion of nonlinear terms, which is prohibitive for accurately describing highly anharmonic circuits. An example of such a circuit is the fluxonium qubit, which has recently attracted increasing attention due to its high lifetimes and low error rates. In this work, we extend the EPR approach to effectively address highly nonlinear superconducting circuits, and, as a proof of concept, we apply our approach to a fluxonium qubit. Specifically, we design, fabricate, and experimentally measure a fluxonium qubit coupled to a readout resonator. We compare the measured frequencies of both the qubit and the resonator to those extracted from the EPR analysis, and we find an excellent agreement. Furthermore, we compare the dispersive shift as a function of external flux obtained from experiments with our EPR analysis and a simpler lumped-element model. Our findings reveal that the EPR results closely align with the experimental data, providing more accurate estimations compared to the simplified lumped-element simulations.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (58)

  1. F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A. Buell et al., Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
  2. Y. Wu, W.-S. Bao, S. Cao, F. Chen, M.-C. Chen, X. Chen, T.-H. Chung, H. Deng, Y. Du, D. Fan et al., Strong quantum computational advantage using a superconducting quantum processor, Phys. Rev. Lett. 127, 180501 (2021).
  3. S. Krinner, N. Lacroix, A. Remm, A. Di Paolo, E. Genois, C. Leroux, C. Hellings, S. Lazar, F. Swiadek, J. Herrmann et al., Realizing repeated quantum error correction in a distance-three surface code, Nature 605, 669 (2022).
  4. Y. Zhao, Y. Ye, H.-L. Huang, Y. Zhang, D. Wu, H. Guan, Q. Zhu, Z. Wei, T. He, S. Cao et al., Realization of an error-correcting surface code with superconducting qubits, Phys. Rev. Lett. 129, 030501 (2022).
  5. Q. Zhu, S. Cao, F. Chen, M.-C. Chen, X. Chen, T.-H. Chung, H. Deng, Y. Du, D. Fan, M. Gong et al., Quantum computational advantage via 60-qubit 24-cycle random circuit sampling, Sci. Bull. 67, 240 (2022).
  6. Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. van sen Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme et al., Evidence for the utility of quantum computing before fault tolerance, Nature 618, 500 (2023).
  7. A. Asfaw, A. Megrant, C. Jones, C. Gidney, D. Bacon, D. Debroy, D. Kafri, E. Lucero, H. Neven, J. Hilton et al., Suppressing quantum errors by scaling a surface code logical qubit, Nature 614, 676 (2023).
  8. R. Acharya, L. Aghababaie-Beni, I. Aleiner, T. I. Andersen, M. Ansmann, F. Arute, K. Arya, A. Asfaw, N. Astrakhantsev, J. Atalaya et al., Quantum error correction below the surface code threshold, Nature 638, 920 (2025).
  9. H. Ali, J. Marques, O. Crawford, J. Majaniemi, M. Serra-Peralta, D. Byfield, B. Varbanov, B. M. Terhal, L. DiCarlo, and E. T. Campbell, Reducing the error rate of a superconducting logical qubit using analog readout information, Phys. Rev. Appl. 22, 044031 (2024).
  10. M. H. Devoret et al., Quantum fluctuations in electrical circuits, Les Houches, Session LXIII 7, 133 (1995).
  11. A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation, Phys. Rev. A 69, 062320 (2004).
  12. J. Bourassa, F. Beaudoin, J. M. Gambetta, and A. Blais, Josephson-junction-embedded transmission-line resonators: From Kerr medium to in-line transmon, Phys. Rev. A 86, 013814 (2012).
  13. M. Leib, F. Deppe, A. Marx, R. Gross, and M. J. Hartmann, Networks of nonlinear superconducting transmission line resonators, New J. Phys. 14, 075024 (2012).
  14. H. L. Mortensen, K. Mølmer, and C. K. Andersen, Normal modes of a superconducting transmission-line resonator with embedded lumped element circuit components, Phys. Rev. A 94, 053817 (2016).
  15. W. C. Smith, A. Kou, U. Vool, I. M. Pop, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, Quantization of inductively shunted superconducting circuits, Phys. Rev. B 94, 144507 (2016).
  16. A. Parra-Rodriguez, E. Rico, E. Solano, and I. L. Egusquiza, Quantum networks in divergence-free circuit QED, Quantum Sci. Technol. 3, 024012 (2018).
  17. Z. K. Minev, T. G. McConkey, M. Takita, A. D. Corcoles, and J. M. Gambetta, Circuit quantum electrodynamics (CQED) with modular quasi-lumped models, arXiv:2103.10344.
  18. A. Di Paolo, T. E. Baker, A. Foley, D. Sénéchal, and A. Blais, Efficient modeling of superconducting quantum circuits with tensor networks, npj Quantum Inf. 7, 11 (2021).
  19. I. L. Egusquiza and A. Parra-Rodriguez, Algebraic canonical quantization of lumped superconducting networks, Phys. Rev. B 106, 024510 (2022).
  20. A. Parra-Rodriguez and I. L. Egusquiza, Geometrical description and Faddeev-Jackiw quantization of electrical networks, Quantum 8, 1466 (2024).
  21. A. Osborne, T. Larson, S. G. Jones, R. W. Simmonds, A. Gyenis, and A. Lucas, Symplectic geometry and circuit quantization, PRX Quantum 5, 020309 (2024).
  22. A. Parra-Rodriguez and I. Egusquiza, Faddeev-Jackiw quantisation of nonreciprocal quasi-lumped electrical networks, arXiv:2401.09120.
  23. S. E. Nigg, H. Paik, B. Vlastakis, G. Kirchmair, S. Shankar, L. Frunzio, M. H. Devoret, R. J. Schoelkopf, and S. M. Girvin, Black-box superconducting circuit quantization, Phys. Rev. Lett. 108, 240502 (2012).
  24. F. Solgun, D. W. Abraham, and D. P. DiVincenzo, Blackbox quantization of superconducting circuits using exact impedance synthesis, Phys. Rev. B 90, 134504 (2014).
  25. Z. K. Minev, Z. Leghtas, S. O. Mundhada, L. Christakis, I. M. Pop, and M. H. Devoret, Energy-participation quantization of Josephson circuits, npj Quantum Inf. 7, 131 (2021).
  26. J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Charge-insensitive qubit design derived from the Cooper pair box, Phys. Rev. A 76, 042319 (2007).
  27. K.-H. Yu, X.-Y. Jiao, and L.-J. Jin, Using the inductive-energy participation ratio to characterize a superconducting quantum chip, Phys. Rev. Appl. 21, 034027 (2024).
  28. V. E. Manucharyan, J. Koch, L. I. Glazman, and M. H. Devoret, Fluxonium: Single Cooper-pair circuit free of charge offsets, Science 326, 113 (2009).
  29. L. B. Nguyen, Y.-H. Lin, A. Somoroff, R. Mencia, N. Grabon, and V. E. Manucharyan, High-coherence fluxonium qubit, Phys. Rev. X 9, 041041 (2019).
  30. L. Ding, M. Hays, Y. Sung, B. Kannan, J. An, A. Di Paolo, A. H. Karamlou, T. M. Hazard, K. Azar, D. K. Kim et al., High-fidelity, frequency-flexible two-qubit fluxonium gates with a transmon coupler, Phys. Rev. X 13, 031035 (2023).
  31. A. Somoroff, Q. Ficheux, R. A. Mencia, H. Xiong, R. Kuzmin, and V. E. Manucharyan, Millisecond coherence in a superconducting qubit, Phys. Rev. Lett. 130, 267001 (2023).
  32. H. Zhang, C. Ding, D. Weiss, Z. Huang, Y. Ma, C. Guinn, S. Sussman, S. P. Chitta, D. Chen, A. A. Houck et al., Tunable inductive coupler for high-fidelity gates between fluxonium qubits, PRX Quantum 5, 020326 (2024).
  33. F. Wang, K. Lu, H. Zhan, L. Ma, F. Wu, H. Sun, H. Deng, Y. Bai, F. Bao, X. Chang et al., Achieving millisecond coherence fluxonium through overlap Josephson junctions, arXiv:2405.05481.
  34. Y.-H. Lin, L. B. Nguyen, N. Grabon, J. San Miguel, N. Pankratova, and V. E. Manucharyan, Demonstration of protection of a superconducting qubit from energy decay, Phys. Rev. Lett. 120, 150503 (2018).
  35. N. Earnest, S. Chakram, Y. Lu, N. Irons, R. K. Naik, N. Leung, L. Ocola, D. A. Czaplewski, B. Baker, J. Lawrence et al., Realization of a λ system with metastable states of a capacitively shunted fluxonium, Phys. Rev. Lett. 120, 150504 (2018).
  36. B. D. Josephson, Possible new effects in superconductive tunnelling, Phys. Lett. 1, 251 (1962).
  37. B. D. Josephson, Supercurrents through barriers, Advan. Phys. 14, 419 (1965).
  38. N. A. Masluk, I. M. Pop, A. Kamal, Z. K. Minev, and M. H. Devoret, Microwave characterization of Josephson junction arrays: Implementing a low loss superinductance, Phys. Rev. Lett. 109, 137002 (2012).
  39. J. Puertas Martínez, S. Léger, N. Gheeraert, R. Dassonneville, L. Planat, F. Foroughi, Y. Krupko, O. Buisson, C. Naud, W. Hasch-Guichard et al., A tunable Josephson platform to explore many-body quantum optics in circuit-QED, npj Quantum Inf. 5, 19 (2019).
  40. T. M. Hazard, A. Gyenis, A. Di Paolo, A. T. Asfaw, S. A. Lyon, A. Blais, and A. A. Houck, Nanowire superinductance fluxonium qubit, Phys. Rev. Lett. 122, 010504 (2019).
  41. D. Niepce, J. Burnett, and J. Bylander, High kinetic inductance NbN nanowire superinductors, Phys. Rev. Appl. 11, 044014 (2019).
  42. L. Grünhaupt, M. Spiecker, D. Gusenkova, N. Maleeva, S. T. Skacel, I. Takmakov, F. Valenti, P. Winkel, H. Rotzinger, W. Wernsdorfer et al., Granular aluminium as a superconducting material for high-impedance quantum circuits, Nat. Mater. 18, 816 (2019).
  43. D. Rieger, S. Günzler, M. Spiecker, P. Paluch, P. Winkel, L. Hahn, J. Hohmann, A. Bacher, W. Wernsdorfer, and I. Pop, Granular aluminium nanojunction fluxonium qubit, Nat. Mater. 22, 194 (2023).
  44. T. V. Stefanski and C. K. Andersen, Flux-pulse-assisted readout of a fluxonium qubit, Phys. Rev. Appl. 22, 014079 (2024).
  45. W.-J. Lin, H. Cho, Y. Chen, M. G. Vavilov, C. Wang, and V. E. Manucharyan, 24 days-stable CNOT-gate on fluxonium qubits with over 99.9% fidelity, PRX Quantum 6, 010349 (2025).
  46. G. Zhu, D. G. Ferguson, V. E. Manucharyan, and J. Koch, Circuit QED with fluxonium qubits: Theory of the dispersive regime, Phys. Rev. B 87, 024510 (2013).
  47. B. Chiaro, A. Megrant, A. Dunsworth, Z. Chen, R. Barends, B. Campbell, Y. Chen, A. Fowler, I. C. Hoi, E. Jeffrey et al., Dielectric surface loss in superconducting resonators with flux-trapping holes, Supercond. Sci. Technol. 29, 104006 (2016).
  48. A. Potts, P. R. Routley, G. J. Parker, J. J. Baumberg, and P. A. J. de Groot, Novel fabrication methods for submicrometer Josephson junction qubits, J. Mater. Sci.: Mater. Electron. 12, 289 (2001).
  49. A. Dunsworth, A. Megrant, C. Quintana, Z. Chen, R. Barends, B. Burkett, B. Foxen, Y. Chen, B. Chiaro, A. Fowler et al., Characterization and reduction of capacitive loss induced by sub-micron Josephson junction fabrication in superconducting qubits, Appl. Phys. Lett. 111, 022601 (2017).
  50. F. Deppe, S. Saito, H. Tanaka, and H. Takayanagi, Determination of the capacitance of nm scale Josephson junctions, J. Appl. Phys. 95, 2607 (2004).
  51. I. N. Moskalenko, I. A. Simakov, N. N. Abramov, A. A. Grigorev, D. O. Moskalev, A. A. Pishchimova, N. S. Smirnov, E. V. Zikiy, I. A. Rodionov, and I. S. Besedin, High fidelity two-qubit gates on fluxoniums using a tunable coupler, npj Quantum Inf. 8, 130 (2022).
  52. J. G. Kroll, F. Borsoi, K. Van Der Enden, W. Uilhoorn, D. De Jong, M. Quintero-Pérez, D. Van Woerkom, A. Bruno, S. Plissard, D. Car et al., Magnetic-field-resilient superconducting coplanar-waveguide resonators for hybrid circuit quantum electrodynamics experiments, Phys. Rev. Appl. 11, 064053 (2019).
  53. S. P. Chitta, T. Zhao, Z. Huang, I. Mondragon-Shem, and J. Koch, Computer-aided quantization and numerical analysis of superconducting circuits, New J. Phys. 24, 103020 (2022).
  54. P. Groszkowski and J. Koch, Scqubits: A Python package for superconducting qubits, Quantum 5, 583 (2021).
  55. The extended EPR Repository: https://github.com/AndersenQubitLab/andersen-lab-pyEPR.git.
  56. Qiskit-Metal Repository: https://github.com/AndersenQubitLab/andersen-lab-qiskit-metal.
  57. Measurement Data Repository: https://doi.org/10.4121/833e20fb-9863-4d0d-828a-d2ec44108e12.
  58. A. Bruno, G. de Lange, S. Asaad, K. L. van der Enden, N. K. Langford, and L. DiCarlo, Reducing intrinsic loss in superconducting resonators by surface treatment and deep etching of silicon substrates, Appl. Phys. Lett. 106, 182601 (2015).

Outline

Information

Phys. Rev. Applied 25, 044021– Published 8 April, 2026
  • Received 9 January 2026
  • Accepted 5 March 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation