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Parameter optimization for the unimon qubit

Rostislav Duda1,*, Eric Hyyppä2, Olli Mukkula3, Vasilii Vadimov1, and Mikko Möttönen1,4

  • *Contact author: rostislav.duda@aalto.fi
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Phys. Rev. Research 7, 033092 – Published 25 July, 2025

DOI: https://doi.org/10.1103/7yvd-z6hk

Abstract

Inductively shunted superconducting qubits, such as the unimon qubit, combine high anharmonicity with protection from low-frequency charge noise, positioning them as promising candidates for the implementation of fault-tolerant superconducting quantum computers. In this work, we develop accurate closed-form approximations for the frequency and anharmonicity of the unimon qubit that are also applicable to any single-mode superconducting qubits with a single-well potential profile, such as the quarton qubit or the kinemon qubit. We use these results to theoretically explore the single-qubit gate fidelity and coherence times across the parameter space of qubits with a single-well potential. We find that the gate fidelity can be optimized by tuning the Hamiltonian to (1) a high qubit mode impedance of 1–2kΩ, (2) a low qubit frequency of 1 GHz, (3) and a perfect cancellation of the linear inductance and the Josephson inductance attained at a flux bias of half flux quantum. According to our theoretical analysis, the proposed qubit parameters have the potential to enhance the single-qubit gate fidelity of the unimon beyond 99.99%, even without significant improvements to the dielectric quality factor or the flux noise density measured for the first unimon qubits. Furthermore, we compare unimon, transmon, and fluxonium qubits in terms of their energy spectra and qubit coherence subject to dielectric loss and 1/f flux noise in order to highlight the advantages and limitations of each qubit type.

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References (68)

  1. R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys. 21, 467 (1982).
  2. F. Arute et al., Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
  3. H.-S. Zhong et al., Quantum computational advantage using photons, Science 370, 1460 (2020).
  4. Y. Wu et al., Strong quantum computational advantage using a superconducting quantum processor, Phys. Rev. Lett. 127, 180501 (2021).
  5. A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Practical quantum advantage in quantum simulation, Nature 607, 667 (2022).
  6. A. D. King et al., Computational supremacy in quantum simulation, arXiv:2403.00910.
  7. J. I. Cirac and P. Zoller, Quantum computations with cold trapped ions, Phys. Rev. Lett. 74, 4091 (1995).
  8. D. Loss and D. P. DiVincenzo, Quantum computation with quantum dots, Phys. Rev. A 57, 120 (1998).
  9. E. Knill, R. Laflamme, and G. J. Milburn, A scheme for efficient quantum computation with linear optics, Nature 409, 46 (2001).
  10. D. Bluvstein et al., Logical quantum processor based on reconfigurable atom arrays, Nature 626, 58 (2024).
  11. M. H. Devoret and J. M. Martinis, Implementing qubits with superconducting integrated circuits, Quantum Inf. Process. 3, 163 (2004).
  12. P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver, A quantum engineer’s guide to superconducting qubits, Appl. Phys. Rev. 6, 021318 (2019).
  13. M. Kjaergaard, M. E. Schwartz, J. Braumüller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver, Superconducting qubits: Current state of play, Annu. Rev. Condens. Matter Phys. 11, 369 (2020).
  14. A. Blais, A. L. Grimsmo, S. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys. 93, 025005 (2021).
  15. V. Bouchiat, D. Vion, P. Joyez, D. Esteve, and M. H. Devoret, Quantum coherence with a single Cooper pair, Phys. Scr. 1998, 165 (1998).
  16. 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).
  17. M. Tuokkola, Y. Sunada, H. Kivijärvi, J. Albanese, L. Grönberg, J.-P. Kaikkonen, V. Vesterinen, J. Govenius, and M. Möttönen, Methods to achieve near-millisecond energy relaxation and dephasing times for a superconducting transmon qubit, Nat. Commun. 16, 5421 (2025).
  18. Z. Li, P. Liu, P. Zhao, Z. Mi, H. Xu, X. Liang, T. Su, W. Sun, G. Xue, J.-N. Zhang, W. Liu, Y. Jin, and H. Yu, Error per single-qubit gate below 10−4 in a superconducting qubit, npj Quantum Inf. 9, 111 (2023).
  19. B. Chiaro and Y. Zhang, Active leakage cancellation in single qubit gates, arXiv:2503.14731.
  20. R. Li, K. Kubo, Y. Ho, Z. Yan, Y. Nakamura, and H. Goto, Realization of high-fidelity CZ gate based on a double-transmon coupler, Phys. Rev. X 14, 041050 (2024).
  21. R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus, C. Neill, P. O’Malley, P. Roushan, J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, Coherent Josephson qubit suitable for scalable quantum integrated circuits, Phys. Rev. Lett. 111, 080502 (2013).
  22. 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).
  23. D. A. Rower, L. Ding, H. Zhang, M. Hays, J. An, P. M. Harrington, I. T. Rosen, J. M. Gertler, T. M. Hazard, B. M. Niedzielski et al., Suppressing counter-rotating errors for fast single-qubit gates with fluxonium, PRX Quantum 5, 040342 (2024).
  24. H. Zhang, S. Chakram, T. Roy, N. Earnest, Y. Lu, Z. Huang, D. Weiss, J. Koch, and D. I. Schuster, Universal fast-flux control of a coherent, low-frequency qubit, Phys. Rev. X 11, 011010 (2021).
  25. 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).
  26. L. Ding, M. Hays, Y. Sung, B. Kannan, J. An, A. Di Paolo, A. H. Karamlou, T. M. Hazard, K. Azar, D. K. Kim, B. M. Niedzielski, A. Melville, M. E. Schwartz, J. L. Yoder, T. P. Orlando, S. Gustavsson, J. A. Grover, K. Serniak, and W. D. Oliver, High-fidelity, frequency-flexible two-qubit fluxonium gates with a transmon coupler, Phys. Rev. X 13, 031035 (2023).
  27. M. F. Zwanenburg, S. Singh, E. Y. Huang, F. Yilmaz, T. V. Stefanski, J. Hu, P. Kumaravadivel, and C. K. Andersen, Single-qubit gates beyond the rotating-wave approximation for strongly anharmonic low-frequency qubits, arXiv:2503.08238.
  28. 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).
  29. A. Ranni, H. Havir, S. Haldar, and V. F. Maisi, High impedance Josephson junction resonators in the transmission line geometry, Appl. Phys. Lett. 123, 114002 (2023).
  30. E. Hyyppä et al., Unimon qubit, Nat. Commun. 13, 6895 (2022).
  31. S. Tuohino, V. Vadimov, W. Teixeira, T. Malmelin, M. Silveri, and M. Möttönen, Multimode physics of the unimon circuit, Phys. Rev. Res. 6, 033001 (2024).
  32. Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, Coherent control of macroscopic quantum states in a single-cooper-pair box, Nature 398, 786 (1999).
  33. F. Yan, S. Gustavsson, A. Kamal, J. Birenbaum, A. P. Sears, D. Hover, T. J. Gudmundsen, D. Rosenberg, G. Samach, S. Weber, J. L. Yoder, T. P. Orlando, J. Clarke, A. J. Kerman, and W. D. Oliver, The flux qubit revisited to enhance coherence and reproducibility, Nat. Commun. 7, 12964 (2016).
  34. 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).
  35. D. Rieger, S. Günzler, M. Spiecker, P. Paluch, P. Winkel, L. Hahn, J. K. Hohmann, A. Bacher, W. Wernsdorfer, and I. M. Pop, Granular aluminium nanojunction fluxonium qubit, Nat. Mater. 22, 194 (2023).
  36. M. Peruzzo, F. Hassani, G. Szep, A. Trioni, E. Redchenko, M. Žemlička, and J. M. Fink, Geometric superinductance qubits: Controlling phase delocalization across a single Josephson junction, PRX Quantum 2, 040341 (2021).
  37. N. Earnest, S. Chakram, Y. Lu, N. Irons, R. K. Naik, N. Leung, L. Ocola, D. A. Czaplewski, B. Baker, J. Lawrence, J. Koch, and D. I. Schuster, Realization of a Λ system with metastable states of a capacitively shunted fluxonium, Phys. Rev. Lett. 120, 150504 (2018).
  38. I. V. Pechenezhskiy, R. A. Mencia, L. B. Nguyen, Y.-H. Lin, and V. E. Manucharyan, The superconducting quasicharge qubit, Nature 585, 368 (2020).
  39. F. Yan, Y. Sung, P. Krantz, A. Kamal, D. K. Kim, J. L. Yoder, T. P. Orlando, S. Gustavsson, and W. D. Oliver, Engineering framework for optimizing superconducting qubit designs, arXiv:2006.04130.
  40. D. Kalacheva, G. Fedorov, J. Zotova, S. Kadyrmetov, A. Kirkovskii, A. Dmitriev, and O. Astafiev, Kinemon: An inductively shunted transmon artificial atom, Phys. Rev. Appl. 21, 024058 (2024).
  41. Y. Ye, J. B. Kline, A. Yen, G. Cunningham, M. Tan, A. Zang, M. Gingras, B. M. Niedzielski, H. Stickler, K. Serniak, M. E. Schwartz, and K. P. O’Brien, Near-ultrastrong nonlinear light-matter coupling in superconducting circuits, Nat Commun 16, 3799 (2025).
  42. J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed. (Cambridge University Press, Cambridge, 2020).
  43. D. Zwillinger, CRC Standard Mathematical Tables and Formulae, 31st ed. (Chapman and Hall/CRC, Boca Raton, 2002).
  44. N. H. Abel, Beweis der unmöglichkeit, algebraische gleichungen von höheren graden als dem vierten allgemein aufzulösen, J. Reine Angew. Math. (Crelles J.) 1826, 65 (1826).
  45. L. H. Pedersen, N. M. Møller, and K. Mølmer, Fidelity of quantum operations, Phys. Lett. A 367, 47 (2007).
  46. G. Ithier, E. Collin, P. Joyez, P. J. Meeson, D. Vion, D. Esteve, F. Chiarello, A. Shnirman, Y. Makhlin, J. Schriefl, and G. Schön, Decoherence in a superconducting quantum bit circuit, Phys. Rev. B 72, 134519 (2005).
  47. J. Braumüller, L. Ding, A. P. Vepsäläinen, Y. Sung, M. Kjaergaard, T. Menke, R. Winik, D. Kim, B. M. Niedzielski, A. Melville, J. L. Yoder, C. F. Hirjibehedin, T. P. Orlando, S. Gustavsson, and W. D. Oliver, Characterizing and optimizing qubit coherence based on squid geometry, Phys. Rev. Appl. 13, 054079 (2020).
  48. V. Lahtinen and M. Möttönen, Effects of device geometry and material properties on dielectric losses in superconducting coplanar-waveguide resonators, J. Phys.: Condens. Matter 32, 405702 (2020).
  49. F. Motzoi, J. M. Gambetta, P. Rebentrost, and F. K. Wilhelm, Simple pulses for elimination of leakage in weakly nonlinear qubits, Phys. Rev. Lett. 103, 110501 (2009).
  50. L. S. Theis, F. Motzoi, and F. K. Wilhelm, Simultaneous gates in frequency-crowded multilevel systems using fast, robust, analytic control shapes, Phys. Rev. A 93, 012324 (2016).
  51. D. Zhu, T. Jaako, Q. He, and P. Rabl, Quantum computing with superconducting circuits in the picosecond regime, Phys. Rev. Appl. 16, 014024 (2021).
  52. M. Werninghaus, D. J. Egger, F. Roy, S. Machnes, F. K. Wilhelm, and S. Filipp, Leakage reduction in fast superconducting qubit gates via optimal control, npj Quantum Inf. 7, 14 (2021).
  53. E. Hyyppä, A. Vepsäläinen, M. Papič, C. F. Chan, S. Inel, A. Landra, W. Liu, J. Luus, F. Marxer, C. Ockeloen-Korppi, S. Orbell, B. Tarasinski, and J. Heinsoo, Reducing leakage of single-qubit gates for superconducting quantum processors using analytical control pulse envelopes, PRX Quantum 5, 030353 (2024).
  54. Y. Nakamura, Y. A. Pashkin, T. Yamamoto, and J. S. Tsai, Charge echo in a cooper-pair box, Phys. Rev. Lett. 88, 047901 (2002).
  55. Y. Makhlin and A. Shnirman, Dephasing of solid-state qubits at optimal points, Phys. Rev. Lett. 92, 178301 (2004).
  56. O. Mukkula, Optimizing coherence and anharmonicity of superconducting qubit, Master’s thesis, University of Helsinki, 2024.
  57. A. Stockklauser, P. Scarlino, J. V. Koski, S. Gasparinetti, C. K. Andersen, C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, and A. Wallraff, Strong coupling cavity QED with gate-defined double quantum dots enabled by a high impedance resonator, Phys. Rev. X 7, 011030 (2017).
  58. D. Niepce, J. Burnett, and J. Bylander, High kinetic inductance NbN nanowire superinductors, Phys. Rev. Appl. 11, 044014 (2019).
  59. L. Grünhaupt, M. Spiecker, D. Gusenkova, N. Maleeva, S. T. Skacel, I. Takmakov, F. Valenti, P. Winkel, H. Rotzinger, W. Wernsdorfer, A. V. Ustinov, and I. M. Pop, Granular aluminium as a superconducting material for high-impedance quantum circuits, Nat. Mater. 18, 816 (2019).
  60. 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).
  61. N. Maleeva, L. Grünhaupt, T. Klein, F. Levy-Bertrand, O. Dupre, M. Calvo, F. Valenti, P. Winkel, F. Friedrich, W. Wernsdorfer et al., Circuit quantum electrodynamics of granular aluminum resonators, Nat. Commun. 9, 3889 (2018).
  62. F. Marxer, A. Vepsäläinen, S. W. Jolin, J. Tuorila, A. Landra, C. Ockeloen-Korppi, W. Liu, O. Ahonen, A. Auer, L. Belzane et al., Long-distance transmon coupler with CZ-gate fidelity above 99.8%, PRX Quantum 4, 010314 (2023).
  63. C. Wang, C. Axline, Y. Y. Gao, T. Brecht, Y. Chu, L. Frunzio, M. Devoret, and R. J. Schoelkopf, Surface participation and dielectric loss in superconducting qubits, Appl. Phys. Lett. 107, 162601 (2015).
  64. S. Kono, J. Pan, M. Chegnizadeh, X. Wang, A. Youssefi, M. Scigliuzzo, and T. J. Kippenberg, Mechanically induced correlated errors on superconducting qubits with relaxation times exceeding 0.4 ms, Nat. Commun. 15, 3950 (2024).
  65. N. Savola, Design and modelling of long-coherence qubits using energy participation ratios, Master’s thesis, Aalto University, 2023.
  66. M. Kounalakis, C. Dickel, A. Bruno, N. K. Langford, and G. A. Steele, Tuneable hopping and nonlinear cross-kerr interactions in a high-coherence superconducting circuit, npj Quantum Inf. 4, 38 (2018).
  67. H. Zhang, C. Ding, D. Weiss, Z. Huang, Y. Ma, C. Guinn, S. Sussman, S. P. Chitta, D. Chen, A. A. Houck, J. Koch, and D. I. Schuster, Tunable inductive coupler for high-fidelity gates between fluxonium qubits, PRX Quantum 5, 020326 (2024).
  68. R. Duda and E. Hyyppä, Parameter optimization for the unimon qubit: Supplementary data (2025), Zenodo dataset, https://zenodo.org/records/15358210.

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Phys. Rev. Research 7, 033092– Published 25 July, 2025
  • Received 7 May 2025
  • Accepted 7 July 2025
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