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Superconducting-qubit gates robust to parameter fluctuations

E.M. Wright1,2,3,*, L. Van Damme1,4, N.J. Glaser1,2, A. Devra1,4, F.A. Roy2,5, J. Englhardt1,2, N. Bruckmoser1,2, L. Koch1,2, A. Marx1,2 et al.

J. Schirk1,2, C.M.F. Schneider1,2, L. Södergren1,2, I. Tsitsilin1,2, F. Wallner1,2, S.J. Glaser1,4, M. Werninghaus1,2,†, and S. Filipp1,2,4

  • *Contact author: emily.wright@wmi.badw.de
  • †Contact author: max.werninghaus@wmi.badw.de
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Phys. Rev. Applied 25, 054037 – Published 14 May, 2026

DOI: https://doi.org/10.1103/tf39-n6s3

Abstract

State-of-the-art single-qubit gates on superconducting transmon qubits can achieve the fidelities required for error-corrected computations; however, parameter fluctuations due to qubit instabilities, environmental changes, and control inaccuracies make it difficult to maintain this performance. To mitigate the effects of these parameter variations, we numerically derive gates that are robust to amplitude and frequency errors using gradient-ascent pulse engineering. We analyze how fluctuations in qubit frequency, drive amplitude, and coherence affect gate performance over time. The robust pulses suppress coherent errors from drive-amplitude drifts over 15 times more than a Gaussian pulse with derivative removal by adiabatic gate (DRAG) corrections. Furthermore, the robust gates, originally designed to compensate for quasistatic errors, also demonstrate resilience to stochastic, time-dependent noise, which is reflected in the dephasing time. They suppress added errors during increases in dephasing by up to 1.7 times more than DRAG.

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References (89)

  1. 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).
  2. Google Quantum AI, Suppressing quantum errors by scaling a surface code logical qubit, Nature 614, 676 (2023).
  3. E. Knill, R. Laflamme, and W. H. Zurek, Resilient quantum computation: Error models and thresholds, Proc. R. Soc. Lond. A 454, 365 (1998).
  4. R. Acharya, D. A. Abanin, L. Aghababaie-Beni, I. Aleiner, T. I. Andersen, M. Ansmann, F. Arute, K. Arya, A. Asfaw, N. Astrakhantsev et al., Quantum error correction below the surface code threshold, Nature 638, 920 (2025).
  5. V. Sivak, A. Morvan, M. Broughton, M. Neeley, A. Eickbusch, D. Abanin, A. Abbas, R. Acharya, L. A. Beni, G. Aigeldinger et al., Reinforcement learning control of quantum error correction, ArXiv:2511.08493.
  6. V. Tripathi, H. Chen, M. Khezri, K.-W. Yip, E. M. Levenson-Falk, and D. A. Lidar, Suppression of crosstalk in superconducting qubits using dynamical decoupling, Phys. Rev. Appl. 18, 024068 (2022).
  7. J. J. Burnett, A. Bengtsson, M. Scigliuzzo, D. Niepce, M. Kudra, P. Delsing, and J. Bylander, Decoherence benchmarking of superconducting qubits, npj Quantum Inf. 5, 1 (2019).
  8. J. P. G. van Dijk, E. Kawakami, R. N. Schouten, M. Veldhorst, L. M. K. Vandersypen, M. Babaie, E. Charbon, and F. Sebastiano, Impact of classical control electronics on qubit fidelity, Phys. Rev. Appl. 12, 044054 (2019).
  9. R. Kueng, D. M. Long, A. C. Doherty, and S. T. Flammia, Comparing experiments to the fault-tolerance threshold, Phys. Rev. Lett. 117, 170502 (2016).
  10. Á. Márton and J. K. Asbóth, Coherent errors and readout errors in the surface code, Quantum 7, 1116 (2023).
  11. J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Decoherence in Josephson qubits from dielectric loss, Phys. Rev. Lett. 95, 210503 (2005).
  12. A. Agarwal, L. P. Lindoy, D. Lall, F. Jamet, and I. Rungger, Modelling non-Markovian noise in driven superconducting qubits, QST 9, 035017 (2024).
  13. X. Li, J. Wang, Y.-Y. Jiang, G.-M. Xue, X. Cai, J. Zhou, M. Gong, Z.-F. Liu, S.-Y. Zheng, D.-K. Ma et al., Cosmic-ray-induced correlated errors in superconducting qubit array, Nat. Commun. 16, 4677 (2025).
  14. D. Ristè, C. C. Bultink, M. J. Tiggelman, R. N. Schouten, K. W. Lehnert, and L. DiCarlo, Millisecond charge-parity fluctuations and induced decoherence in a superconducting transmon qubit, Nat. Commun. 4, 1913 (2013).
  15. C. Müller, J. Lisenfeld, A. Shnirman, and S. Poletto, Interacting two-level defects as sources of fluctuating high-frequency noise in superconducting circuits, Phys. Rev. B 92, 035442 (2015).
  16. M. Carroll, S. Rosenblatt, P. Jurcevic, I. Lauer, and A. Kandala, Dynamics of superconducting qubit relaxation times, npj Quantum Inf. 8, 1 (2022).
  17. F. Berritta, J. Benestad, J. A. Krzywda, O. Krause, M. A. Marciniak, S. Krøjer, C. W. Warren, E. Hogedal, A. Nylander, I. Ahmad et al., Real-time adaptive tracking of fluctuating relaxation rates in superconducting qubits, Phys. Rev. X 16, 011025 (2026).
  18. T. Thorbeck, A. Eddins, I. Lauer, D. T. McClure, and M. Carroll, Two-level-system dynamics in a superconducting qubit due to background ionizing radiation, PRX Quantum 4, 020356 (2023).
  19. C. D. Wilen, S. Abdullah, N. A. Kurinsky, C. Stanford, L. Cardani, G. D’Imperio, C. Tomei, L. Faoro, L. B. Ioffe, C. H. Liu et al., Correlated charge noise and relaxation errors in superconducting qubits, Nature 594, 369 (2021).
  20. R. T. Gordon, C. E. Murray, C. Kurter, M. Sandberg, S. A. Hall, K. Balakrishnan, R. Shelby, B. Wacaser, A. A. Stabile, J. W. Sleight et al., Environmental radiation impact on lifetimes and quasiparticle tunneling rates of fixed-frequency transmon qubits, Appl. Phys. Lett. 120, 074002 (2022).
  21. S. Schlör, J. Lisenfeld, C. Müller, A. Bilmes, A. Schneider, D. P. Pappas, A. V. Ustinov, and M. Weides, Correlating decoherence in transmon qubits: Low frequency noise by single fluctuators, Phys. Rev. Lett. 123, 190502 (2019).
  22. Z. Chen, K. J. Satzinger, J. Atalaya, A. N. Korotkov, A. Dunsworth, D. Sank, C. Quintana, M. McEwen, R. Barends, P. V. Klimov et al., Exponential suppression of bit or phase errors with cyclic error correction, Nature 595, 383 (2021).
  23. P. V. Klimov, A. Bengtsson, C. Quintana, A. Bourassa, S. Hong, A. Dunsworth, K. J. Satzinger, W. P. Livingston, V. Sivak, M. Y. Niu et al., Optimizing quantum gates towards the scale of logical qubits, Nat. Commun. 15, 2442 (2024).
  24. B. Shao, X. Yang, R. Liu, Y. Zhai, D. Lu, T. Xin, and J. Li, Multiple classical noise mitigation by multiobjective robust quantum optimal control, Phys. Rev. Appl. 21, 034042 (2024).
  25. C. L. Edmunds, C. Hempel, R. J. Harris, V. Frey, T. M. Stace, and M. J. Biercuk, Dynamically corrected gates suppressing spatiotemporal error correlations as measured by randomized benchmarking, Phys. Rev. Res. 2, 013156 (2020).
  26. Andre R. R. Carvalho, H. Ball, M. J. Biercuk, M. R. Hush, and F. Thomsen, Error-robust quantum logic optimization using a cloud quantum computer interface, Phys. Rev. Appl. 15, 064054 (2021).
  27. Z. Zhang, Z. Miao, and X.-H. Deng, Smolyak-algorithm-assisted robust control of quantum systems with uncertainties, Phys. Rev. A 111, 042624 (2025).
  28. C.-H. Huang and H.-S. Goan, Robust quantum gates for stochastic time-varying noise, Phys. Rev. A 95, 062325 (2017).
  29. N. H. Le, M. Cykiert, and E. Ginossar, Scalable and robust quantum computing on qubit arrays with fixed coupling, npj Quantum Inf. 9, 1 (2023).
  30. H. Xue and X.-H. Deng, Traversing quantum control robustness landscapes: A new paradigm for quantum gate engineering, ArXiv:2412.19473.
  31. M. Harutyunyan, F. Holweck, D. Sugny, and S. Guérin, Digital optimal robust control, Phys. Rev. Lett. 131, 200801 (2023).
  32. K. Yi, Y.-J. Hai, K. Luo, J. Chu, L. Zhang, Y. Zhou, Y. Song, S. Liu, T. Yan, X.-H. Deng, Y. Chen, and D. Yu, Robust quantum gates against correlated noise in integrated quantum chips, Phys. Rev. Lett. 132, 250604 (2024).
  33. H. Amer, E. Piliouras, E. Barnes, and S. E. Economou, Implementing and benchmarking dynamically corrected gates on superconducting devices using space curve quantum control, ArXiv:2504.09767.
  34. G. Dridi, K. Liu, and S. Guérin, Optimal robust quantum control by inverse geometric optimization, Phys. Rev. Lett. 125, 250403 (2020).
  35. H. G. Tonchev, B. T. Torosov, and N. V. Vitanov, Robust, fast, and high-fidelity composite single-qubit gates for superconducting transmon qubits, Phys. Rev. A 112, 012605 (2025).
  36. H. K. Cummins, G. Llewellyn, and J. A. Jones, Tackling systematic errors in quantum logic gates with composite rotations, Phys. Rev. A 67, 042308 (2003).
  37. Z.-C. Shi, J.-T. Ding, Y.-H. Chen, J. Song, Y. Xia, X. Yi, and F. Nori, Supervised learning for robust quantum control in composite-pulse systems, Phys. Rev. Appl. 21, 044012 (2024).
  38. C. Kabytayev, T. J. Green, K. Khodjasteh, M. J. Biercuk, L. Viola, and K. R. Brown, Robustness of composite pulses to time-dependent control noise, Phys. Rev. A 90, 012316 (2014).
  39. D. Dong, M. A. Mabrok, I. R. Petersen, B. Qi, C. Chen, and H. Rabitz, Sampling-based learning control for quantum systems with uncertainties, IEEE Trans. Control Syst. Technol. 23, 2155 (2015).
  40. D. Dong, C. Wu, C. Chen, B. Qi, I. R. Petersen, and F. Nori, Learning robust pulses for generating universal quantum gates, Sci. Rep. 6, 36090 (2016).
  41. M. Cykiert and E. Ginossar, Robust optimal control for a systematic error in the control amplitude of transmon qubits, Phys. Scr. 99, 105059 (2024).
  42. M. Kuzmanović, I. Björkman, J. J. McCord, S. Dogra, and G. S. Paraoanu, High-fidelity robust qubit control by phase-modulated pulses, Phys. Rev. Res. 6, 013188 (2024).
  43. E. Berger, V. Maurya, Z. M. McIntyre, K. X. Wei, H. Haas, and D. Puzzuoli, Dimensionality reduction for closed-loop quantum gate calibration, ArXiv:2412.05230.
  44. S. Watanabe, Y. Tabuchi, K. Heya, S. Tamate, and Y. Nakamura, ZZ-interaction-free single-qubit-gate optimization in superconducting qubits, Phys. Rev. A 109, 012616 (2024).
  45. N. J. Glaser, F. A. Roy, I. Tsitsilin, L. Koch, N. Bruckmoser, J. Schirk, J. H. Romeiro, G. B. P. Huber, F. Wallner, M. Singh et al., Closed-loop optimization for high-fidelity controlled-z gates in superconducting qubits, Phys. Rev. Appl. 24, 024048 (2025).
  46. 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).
  47. C. P. Koch, U. Boscain, T. Calarco, G. Dirr, S. Filipp, S. J. Glaser, R. Kosloff, S. Montangero, T. Schulte-Herbrüggen, D. Sugny, and F. K. Wilhelm, Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe, EPJ Quantum Technol. 9, 1 (2022).
  48. N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, and S. J. Glaser, Optimal control of coupled spin dynamics: Design of NMR pulse sequences by gradient ascent algorithms, J. Magn. Res. 172, 296 (2005).
  49. S. Machnes, U. Sander, S. J. Glaser, P. de Fouquières, A. Gruslys, S. Schirmer, and T. Schulte-Herbrüggen, Comparing, optimizing, and benchmarking quantum-control algorithms in a unifying programming framework, Phys. Rev. A 84, 022305 (2011).
  50. 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).
  51. J. M. Gambetta, F. Motzoi, S. T. Merkel, and F. K. Wilhelm, Analytic control methods for high-fidelity unitary operations in a weakly nonlinear oscillator, Phys. Rev. A 83, 012308 (2011).
  52. F. Motzoi and F. K. Wilhelm, Improving frequency selection of driven pulses using derivative-based transition suppression, Phys. Rev. A 88, 062318 (2013).
  53. 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).
  54. E. Hyyppä, A. Vepsäläinen, M. Papič, C. F. Chan, S. Inel, A. Landra, W. Liu, J. Luus, F. Marxer, C. Ockeloen-Korppi et al., Reducing leakage of single-qubit gates for superconducting quantum processors using analytical control pulse envelopes, PRX Quantum 5, 030353 (2024).
  55. T. E. Skinner and N. I. Gershenzon, Optimal control design of pulse shapes as analytic functions, J. Magn. Res. 204, 248 (2010).
  56. D. C. McKay, C. J. Wood, S. Sheldon, J. M. Chow, and J. M. Gambetta, Efficient Z gates for quantum computing, Phys. Rev. A 96, 022330 (2017).
  57. L. Van Damme, A. Devra, E. Malvetti, and S. J. Glaser, GRAPE_SQC, https://github.com/LeoVanDamme/GRAPE_SCQ (2025).
  58. E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, Randomized benchmarking of quantum gates, Phys. Rev. A 77, 012307 (2008).
  59. E. Magesan, J. M. Gambetta, and J. Emerson, Scalable and robust randomized benchmarking of quantum processes, Phys. Rev. Lett. 106, 180504 (2011).
  60. C. J. Wood and J. M. Gambetta, Quantification and characterization of leakage errors, Phys. Rev. A 97, 032306 (2018).
  61. J. Kelly, R. Barends, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, I.-C. Hoi, E. Jeffrey et al., Optimal quantum control using randomized benchmarking, Phys. Rev. Lett. 112, 240504 (2014).
  62. E. Magesan, J. M. Gambetta, B. R. Johnson, C. A. Ryan, J. M. Chow, S. T. Merkel, M. P. da Silva, G. A. Keefe, M. B. Rothwell, T. A. Ohki et al., Efficient measurement of quantum gate error by interleaved randomized benchmarking, Phys. Rev. Lett. 109, 080505 (2012).
  63. A. M. Stephens, Fault-tolerant thresholds for quantum error correction with the surface code, Phys. Rev. A 89, 022321 (2014).
  64. S. Sheldon, L. S. Bishop, E. Magesan, S. Filipp, J. M. Chow, and J. M. Gambetta, Characterizing errors on qubit operations via iterative randomized benchmarking, Phys. Rev. A 93, 012301 (2016).
  65. H. Ball, T. M. Stace, S. T. Flammia, and M. J. Biercuk, Effect of noise correlations on randomized benchmarking, Phys. Rev. A 93, 022303 (2016).
  66. P. V. Klimov, J. Kelly, Z. Chen, M. Neeley, A. Megrant, B. Burkett, R. Barends, K. Arya, B. Chiaro, Y. Chen et al., Fluctuations of energy-relaxation times in superconducting qubits, Phys. Rev. Lett. 121, 090502 (2018).
  67. A. P. M. Place, L. V. H. Rodgers, P. Mundada, B. M. Smitham, M. Fitzpatrick, Z. Leng, A. Premkumar, J. Bryon, A. Vrajitoarea, S. Sussman et al., New material platform for superconducting transmon qubits with coherence times exceeding 0.3 ms, Nat. Commun. 12, 1779 (2021).
  68. C. Wang, X. Li, H. Xu, Z. Li, J. Wang, Z. Yang, Z. Mi, X. Liang, T. Su, C. Yang et al., Towards practical quantum computers: Transmon qubit with a lifetime approaching 0.5 ms, npj Quantum Inf. 8, 1 (2022).
  69. H. Deng, Z. Song, R. Gao, T. Xia, F. Bao, X. Jiang, H.-S. Ku, Z. Li, X. Ma, J. Qin et al., Titanium nitride film on sapphire substrate with low dielectric loss for superconducting qubits, Phys. Rev. Appl. 19, 024013 (2023).
  70. M. Bal, A. A. Murthy, S. Zhu, F. Crisa, X. You, Z. Huang, T. Roy, J. Lee, D. v. Zanten, R. Pilipenko et al., Systematic improvements in transmon qubit coherence enabled by niobium surface encapsulation, npj Quantum Inf. 10, 1 (2024).
  71. S. Ganjam, Y. Wang, Y. Lu, A. Banerjee, C. U. Lei, L. Krayzman, K. Kisslinger, C. Zhou, R. Li, Y. Jia et al., Surpassing millisecond coherence in on chip superconducting quantum memories by optimizing materials and circuit design, Nat. Commun. 15, 3687 (2024).
  72. M. Mergenthaler, C. Müller, M. Ganzhorn, S. Paredes, P. Müller, G. Salis, V. P. Adiga, M. Brink, M. Sandberg, J. B. Hertzberg, S. Filipp, and A. Fuhrer, Effects of surface treatments on flux tunable transmon qubits, npj Quantum Inf. 7, 157 (2021a).
  73. M. Mergenthaler, S. Paredes, P. Müller, C. Müller, S. Filipp, M. Sandberg, J. B. Hertzberg, V. P. Adiga, M. Brink, and A. Fuhrer, Ultrahigh vacuum packaging and surface cleaning for quantum devices, Rev. Sci. Instrum. 92, 025121 (2021b).
  74. O. Dial, D. T. McClure, S. Poletto, G. A. Keefe, M. B. Rothwell, J. M. Gambetta, D. W. Abraham, J. M. Chow, and M. Steffen, Bulk and surface loss in superconducting transmon qubits, Supercond. Sci. Technol. 29, 044001 (2016).
  75. C. Wang, C. Axline, Y. Y. Gao, T. Brecht, Y. Chu, L. Frunzio, M. H. Devoret, and R. J. Schoelkopf, Surface participation and dielectric loss in superconducting qubits, Appl. Phys. Lett. 107, 162601 (2015).
  76. J. Lisenfeld, A. Bilmes, A. Megrant, R. Barends, J. Kelly, P. Klimov, G. Weiss, J. M. Martinis, and A. V. Ustinov, Electric field spectroscopy of material defects in transmon qubits, npj Quantum Inf. 5, 105 (2019).
  77. A. Bilmes, A. Megrant, P. Klimov, G. Weiss, J. M. Martinis, A. V. Ustinov, and J. Lisenfeld, Resolving the positions of defects in superconducting quantum bits, Sci. Rep. 10, 3090 (2020).
  78. A. Bilmes, S. Volosheniuk, J. D. Brehm, A. V. Ustinov, and J. Lisenfeld, Quantum sensors for microscopic tunneling systems, npj Quantum Inf. 7, 27 (2021).
  79. Z. T. Wang, P. Zhao, Z. H. Yang, Y. Tian, H. F. Yu, and S. P. Zhao, Escaping detrimental interactions with microwave-dressed transmon qubits, Chin. Phys. Lett. 40, 070304 (2023).
  80. J. Lisenfeld, A. Bilmes, and A. V. Ustinov, Enhancing the coherence of superconducting quantum bits with electric fields, npj Quantum Inf. 9, 8 (2023).
  81. P. Zhao, T. Ma, Y. Jin, and H. Yu, Combating fluctuations in relaxation times of fixed-frequency transmon qubits with microwave-dressed states, Phys. Rev. A 105, 062605 (2022).
  82. Y. Kim, L. C. G. Govia, A. Dane, E. v. d. Berg, D. M. Zajac, B. Mitchell, Y. Liu, K. Balakrishnan, G. Keefe, A. Stabile et al., Error mitigation with stabilized noise in superconducting quantum processors, Nat. Commun. 16, 8439 (2025).
  83. L. Chen, K.-H. Lee, C.-H. Liu, B. Marinelli, R. K. Naik, Z. Kang, N. Goss, H. Kim, D. I. Santiago, and I. Siddiqi, Scalable and site-specific frequency tuning of two-level system defects in superconducting qubit arrays, ArXiv:2503.04702.
  84. M. Steffen, M. Sandberg, and S. Srinivasan, Recent research trends for high coherence quantum circuits, Supercond. Sci. Technol. 30, 030301 (2017).
  85. M. Mohseni, A. Scherer, K. G. Johnson, O. Wertheim, M. Otten, N. A. Aadit, Y. Alexeev, K. M. Bresniker, K. Y. Camsari, B. Chapman et al., How to build a quantum supercomputer: Scaling from hundreds to millions of qubits, ArXiv:2411.10406.
  86. S. Gustavsson, O. Zwier, J. Bylander, F. Yan, F. Yoshihara, Y. Nakamura, T. P. Orlando, and W. D. Oliver, Improving quantum gate fidelities by using a qubit to measure microwave pulse distortions, Phys. Rev. Lett. 110, 040502 (2013).
  87. D. C. McKay, S. Filipp, A. Mezzacapo, E. Magesan, J. M. Chow, and J. M. Gambetta, Universal gate for fixed-frequency qubits via a tunable bus, Phys. Rev. Appl. 6, 064007 (2016).
  88. M. Roth, M. Ganzhorn, N. Moll, S. Filipp, G. Salis, and S. Schmidt, Analysis of a parametrically driven exchange-type gate and a two-photon excitation gate between superconducting qubits, Phys. Rev. A 96, 062323 (2017).
  89. P. de Fouquieres, S. Schirmer, S. Glaser, and I. Kuprov, Second order gradient ascent pulse engineering, J. Magn. Res. 212, 412 (2011).

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Phys. Rev. Applied 25, 054037– Published 14 May, 2026
  • Received 9 January 2026
  • Revised 12 March 2026
  • Accepted 2 April 2026
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