- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review A

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Nonadiabatic self-healing of Trotter errors in digitized counterdiabatic dynamics

Mara Vizzuso1,*, Gianluca Passarelli1, Giovanni Cantele1, Procolo Lucignano1, Xi Chen2, and Koushik Paul3,4,†

  • *Contact author: mara.vizzuso@unina.it
  • †Contact author: koushikpal09@gmail.com
PDF

Phys. Rev. A 113, 062432 – Published 11 June, 2026

DOI: https://doi.org/10.1103/5ck5-tp69

Abstract

Trotter errors in digitized quantum dynamics arise from approximating time-ordered evolution under noncommuting Hamiltonian terms with a product formula. In the adiabatic regime, such errors are known to exhibit long-time self-healing [Phys. Rev. Lett. 131, 060602 (2023)], where discretization effects are effectively suppressed. Here we show that self-healing persists at finite evolution times once nonadiabatic errors induced by finite-speed ramps are compensated. Using counterdiabatic driving to cancel diabatic transitions and isolate discretization effects, we study both noninteracting and interacting spin models and characterize the finite-time scaling with the Trotter steps and the total evolution time. In the instantaneous eigenbasis of the driven (gapped) Hamiltonian, the leading digital error maps to an effective harmonic perturbation whose dominant Fourier component yields an analytic upper bound on the finite-time Trotter error and reveals the phase-cancellation mechanism underlying self-healing. Our results establish finite-time self-healing as a generic feature of digitized counterdiabatic protocols, clarify its mechanism beyond the long-time adiabatic limit, and provide practical guidance for high-fidelity state preparation on gate-based quantum processors.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (83)

  1. R. P. Feynman, in Feynman and Computation: Exploring the Limits of Computers, edited by A. G. H. Hey (CRC, Boca Raton, 2018), pp. 133–153.
  2. P. O. J. Scherer, Computational Physics: Simulation of Classical and Quantum Systems, Graduate Texts in Physics (Springer, Cham, 2017).
  3. I. Kassal, J. D. Whitfield, A. Perdomo-Ortiz, M.-H. Yung, and A. Aspuru-Guzik, Simulating chemistry using quantum computers, Annu. Rev. Phys. Chem. 62, 185 (2011).
  4. F. Tacchino, A. Chiesa, S. Carretta, and D. Gerace, Quantum computers as universal quantum simulators: State-of-the-art and perspectives, Adv. Quantum Technol. 3, 1900052 (2020).
  5. I. M. Georgescu, S. Ashhab, and F. Nori, Quantum simulation, Rev. Mod. Phys. 86, 153 (2014).
  6. I. Buluta and F. Nori, Quantum simulators, Science 326, 108 (2009).
  7. A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Practical quantum advantage in quantum simulation, Nature (London) 607, 667 (2022).
  8. A. Trabesinger, Quantum simulation, Nat. Phys. 8, 263 (2012).
  9. D. Hangleiter, J. Carolan, and K. P. Y. Thébault, Analogue Quantum Simulation: A New Instrument for Scientific Understanding (Springer, Cham, 2022).
  10. A. J. Daley, Twenty-five years of analogue quantum simulation, Nat. Rev. Phys. 5, 702 (2023).
  11. R. J. MacDonell, C. E. Dickerson, C. J. T. Birch, A. Kumar, C. L. Edmunds, M. J. Biercuk, C. Hempel, and I. Kassal, Analog quantum simulation of chemical dynamics, Chem. Sci. 12, 9794 (2021).
  12. S. Lloyd, Universal quantum simulators, Science 273, 1073 (1996).
  13. B. Fauseweh, Quantum many-body simulations on digital quantum computers: State-of-the-art and future challenges, Nat. Commun. 15, 2123 (2024).
  14. B. P. Lanyon, C. Hempel, D. Nigg, M. Müller, R. Gerritsma, F. Zähringer, P. Schindler, J. T. Barreiro, M. Rambach, G. Kirchmair, M. Hennrich, P. Zoller, R. Blatt, and C. F. Roos, Universal digital quantum simulation with trapped ions, Science 334, 57 (2011).
  15. E. Hairer, G. Wanner, and C. Lubich, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics Vol. 31 (Springer, Berlin, 2006), pp. 179–236.
  16. H. De Raedt and B. De Raedt, Applications of the generalized Trotter formula, Phys. Rev. A 28, 3575 (1983).
  17. R. D. Somma, A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation, J. Math. Phys. 57, 062202 (2016).
  18. F. Tacchino, A. Chiesa, M. D. LaHaye, I. Tavernelli, S. Carretta, and D. Gerace, Digital quantum simulations of spin models on hybrid platform and near-term quantum processors, Proceedings 12, 24 (2019).
  19. M. Dalmonte and S. Montangero, Lattice gauge theory simulations in the quantum information era, Contemp. Phys. 57, 388 (2016).
  20. L. Lin and Y. Tong, Near-optimal ground state preparation, Quantum 4, 372 (2020).
  21. J. Wang, J. Surace, I. Frérot, B. Legat, M.-O. Renou, V. Magron, and A. Acín, Certifying ground-state properties of many-body systems, Phys. Rev. X 14, 031006 (2024).
  22. P.-L. Zheng, S.-J. Du, and Y. Zhang, Ground-state properties via machine learning quantum constraints, Phys. Rev. Res. 4, L032043 (2022).
  23. M. Motta, G. O. Jones, J. E. Rice, T. P. Gujarati, R. Sakuma, I. Liepuoniute, J. M. Garcia, and Y.-y. Ohnishi, Quantum chemistry simulation of ground- and excited-state properties of the sulfonium cation on a superconducting quantum processor, Chem. Sci. 14, 2915 (2023).
  24. G. Jaerger, Quantum Information: An Overview (Springer, New York, 2007), pp. 1–27.
  25. V. Vedral and M. B. Plenio, Basics of quantum computation, Prog. Quantum Electron. 22, 1 (1998).
  26. T. Ko and S. Lim, Quantum advantages in ground state preparation, combinatorial optimization, and quantum state preparation, arXiv:2510.01563.
  27. S. Gunther, N. A. Petersson, and J. L. DuBois, Quantum optimal control for pure-state preparation using one initial state, AVS Quantum Sci. 3, 043801 (2021).
  28. D. Poulin and P. Wocjan, Preparing ground states of quantum many-body systems on a quantum computer, Phys. Rev. Lett. 102, 130503 (2009).
  29. Y. Ge, J. Tura, and J. I. Cirac, Faster ground state preparation and high-precision ground energy estimation with fewer qubits, J. Math. Phys. 60, 022202 (2019).
  30. R. Zhang, G. Wang, and P. Johnson, Computing ground state properties with early fault-tolerant quantum computers, Quantum 6, 761 (2022).
  31. M.-Q. He, D.-B. Zhang, and Z. D. Wang, Quantum Gaussian filter for exploring ground-state properties, Phys. Rev. A 106, 032420 (2022).
  32. J. Sólyom and T. A. L. Ziman, Ground-state properties of axially anisotropic quantum Heisenberg chains, Phys. Rev. B 30, 3980 (1984).
  33. S. Stanisic, J. L. Bosse, F. M. Gambetta, R. A. Santos, W. Mruczkiewicz, T. E. O'Brien, E. Ostby, and A. Montanaro, Observing ground-state properties of the Fermi-Hubbard model using a scalable algorithm on a quantum computer, Nat. Commun. 13, 5743 (2022).
  34. D. A. Puente, F. Motzoi, T. Calarco, G. Morigi, and M. Rizzi, Quantum state preparation via engineered ancilla resetting, Quantum 8, 1299 (2024).
  35. R. Finkelstein, R. B.-S. Tsai, X. Sun, P. Scholl, S. Direkci, T. Gefen, J. Choi, A. L. Shaw, and M. Endres, Universal quantum operations and ancilla-based read-out for tweezer clocks, Nature (London) 634, 321 (2024).
  36. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  37. J. W. Z. Lau, K. H. Lim, H. Shrotriya, and L. C. Kwek, NISQ computing: Where are we and where do we go? AAPPS Bull. 32, 27 (2022).
  38. B. Bonde, P. Patil, and B. Choubey, in High Performance Computing for Drug Discovery and Biomedicine, edited by A. Heifetz (Springer, New York, 2024), pp. 153–179.
  39. M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational quantum algorithms, Nat. Rev. Phys. 3, 625 (2021).
  40. K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik, Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys. 94, 015004 (2022).
  41. T. Albash and D. A. Lidar, Adiabatic quantum computation, Rev. Mod. Phys. 90, 015002 (2018).
  42. D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd, and O. Regev, Adiabatic quantum computation is equivalent to standard quantum computation, SIAM Rev. 50, 755 (2008).
  43. A. Hamma and D. A. Lidar, Adiabatic preparation of topological order, Phys. Rev. Lett. 100, 030502 (2008).
  44. Y. Subaşı, R. D. Somma, and D. Orsucci, Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing, Phys. Rev. Lett. 122, 060504 (2019).
  45. P. C. S. Costa, D. An, Y. R. Sanders, Y. Su, R. Babbush, and D. W. Berry, Optimal scaling quantum linear-systems solver via discrete adiabatic theorem, PRX Quantum 3, 040303 (2022).
  46. D. An and L. Lin, Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm, ACM Trans. Quantum Comput. 3, 1 (2022).
  47. E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, Quantum computation by adiabatic evolution, arXiv:quant-ph/0001106.
  48. D. Cugini, D. Nigro, M. Bruno, and D. Gerace, Exponential optimization of adiabatic quantum-state preparation, Phys. Rev. Res. 7, L012074 (2025).
  49. S. P. Jordan, K. S. M. Lee, and J. Preskill, Quantum algorithms for quantum field theories, Science 336, 1130 (2012).
  50. A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, and M. Head-Gordon, Simulated quantum computation of molecular energies, Science 309, 1704 (2005).
  51. S. Lee, J. Lee, H. Zhai, Y. Tong, A. M. Dalzell, A. Kumar, P. Helms, J. Gray, Z.-H. Cui, W. Liu, M. Kastoryano, R. Babbush, J. Preskill, D. R. Reichman, E. T. Campbell, E. F. Valeev, L. Lin, and G. K.-L. Chan, Evaluating the evidence for exponential quantum advantage in ground-state quantum chemistry, Nat. Commun. 14, 1952 (2023).
  52. L. C. Venuti, T. Albash, M. Marvian, D. Lidar, and P. Zanardi, Relaxation versus adiabatic quantum steady-state preparation, Phys. Rev. A 95, 042302 (2017).
  53. T. Kato, On the adiabatic theorem of quantum mechanics, J. Phys. Soc. Jpn. 5, 435 (1950).
  54. S. Jansen, M.-B. Ruskai, and R. Seiler, Bounds for the adiabatic approximation with applications to quantum computation, J. Math. Phys. 48, 102111 (2007).
  55. C. Zener, Non-adiabatic crossing of energy levels, Proc. R. Soc. London Ser. A 137, 696 (1932).
  56. H. F. Trotter, On the product of semi-groups of operators, Proc. Am. Math. Soc. 10, 545 (1959).
  57. M. Suzuki, Generalized Trotter's formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems, Commun. Math. Phys. 51, 183 (1976).
  58. Y. Lu, Y. Huang, D. An, Q. Zhao, D. Lv, and X. Yuan, Digital adiabatic evolution is universally accurate, Nature Commun. (2026).
  59. A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Theory of Trotter error with commutator scaling, Phys. Rev. X 11, 011020 (2021).
  60. L. K. Kovalsky, F. A. Calderon-Vargas, M. D. Grace, A. B. Magann, J. B. Larsen, A. D. Baczewski, and M. Sarovar, Self-healing of Trotter error in digital adiabatic state preparation, Phys. Rev. Lett. 131, 060602 (2023).
  61. D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, and J. G. Muga, Shortcuts to adiabaticity: Concepts, methods, and applications, Rev. Mod. Phys. 91, 045001 (2019).
  62. E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, in Advances in Atomic, Molecular, and Optical Physics, edited by E. Arimondo, P. R. Berman, and C. C. Lin (Academic, New York, 2013), Vol. 62, Chap. 2, pp. 117–169.
  63. M. Demirplak and S. A. Rice, Adiabatic population transfer with control fields, J. Phys. Chem. A 107, 9937 (2003).
  64. M. Demirplak and S. A. Rice, Assisted adiabatic passage revisited, J. Phys. Chem. B 109, 6838 (2005).
  65. M. V. Berry, Transitionless quantum driving, J. Phys. A: Math. Theor. 42, 365303 (2009).
  66. A. del Campo, Shortcuts to adiabaticity by counterdiabatic driving, Phys. Rev. Lett. 111, 100502 (2013).
  67. F. Petiziol, F. Mintert, and S. Wimberger, Quantum control by effective counterdiabatic driving, Europhys. Lett. 145, 15001 (2024).
  68. S. Iram, E. Dolson, J. Chiel, J. Pelesko, N. Krishnan, O. Güngör, B. Kuznets-Speck, S. Deffner, E. Ilker, J. G. Scott, and M. Hinczewski, Controlling the speed and trajectory of evolution with counterdiabatic driving, Nat. Phys. 17, 135 (2021).
  69. Y.-C. Li and X. Chen, Shortcut to adiabatic population transfer in quantum three-level systems: Effective two-level problems and feasible counterdiabatic driving, Phys. Rev. A 94, 063411 (2016).
  70. Z.-B. Feng, X.-J. Lu, M. Li, R.-Y. Yan, and Y.-Q. Zhou, Speeding up adiabatic population transfer in a Josephson qutrit via counter-diabatic driving, New J. Phys. 19, 123023 (2017).
  71. M. Vizzuso, G. Passarelli, G. Cantele, and P. Lucignano, Convergence of digitized-counterdiabatic QAOA: Circuit depth versus free parameters, New J. Phys. 26, 013002 (2024).
  72. M. Vizzuso, G. Passarelli, G. Cantele, and P. Lucignano, Role of gaps in the digitized counterdiabatic quantum approximate optimization algorithm for fully connected spin models, Phys. Rev. A 110, 062606 (2024).
  73. N. N. Hegade, K. Paul, Y. Ding, M. Sanz, F. Albarrán-Arriagada, E. Solano, and X. Chen, Shortcuts to adiabaticity in digitized adiabatic quantum computing, Phys. Rev. Appl. 15, 024038 (2021).
  74. P. Chandarana, N. N. Hegade, K. Paul, F. Albarrán-Arriagada, E. Solano, A. del Campo, and X. Chen, Digitized-counterdiabatic quantum approximate optimization algorithm, Phys. Rev. Res. 4, 013141 (2022).
  75. D. Sels and A. Polkovnikov, Minimizing irreversible losses in quantum systems by local counterdiabatic driving, Proc. Natl. Acad. Sci. USA 114, E3909 (2017).
  76. P. W. Claeys, M. Pandey, D. Sels, and A. Polkovnikov, Floquet-engineering counterdiabatic protocols in quantum many-body systems, Phys. Rev. Lett. 123, 090602 (2019).
  77. D. Aharonov and A. Ta‐Shma, Adiabatic quantum state generation, SIAM J. Comput. 37, 47 (2007).
  78. G. Passarelli, V. Cataudella, R. Fazio, and P. Lucignano, Counterdiabatic driving in the quantum annealing of the p-spin model: A variational approach, Phys. Rev. Res. 2, 013283 (2020).
  79. Y.-A. Chen, A. M. Childs, M. Hafezi, Z. Jiang, H. Kim, and Y. Xu, Efficient product formulas for commutators and applications to quantum simulation, Phys. Rev. Res. 4, 013191 (2022).
  80. K. Mizuta, T. N. Ikeda, and K. Fujii, Explicit error bounds with commutator scaling for time-dependent product and multi-product formulas, arXiv:2410.14243.
  81. A. Pitchford, C. Grenade, A. Grimsmo, P. D. Nation, and J. R. Johansson, Sesolve: Schrödinger equation solver, https://qutip.org/docs/4.0.2/modules/qutip/sesolve.html (2017).
  82. D. Comparat, General conditions for quantum adiabatic evolution, Phys. Rev. A 80, 012106 (2009).
  83. M. Vizzuso, Nonadiabatic self-healing of Trotter errors in digitized counterdiabatic dynamics arXiv:2512.22636, Zenodo, Geneva, 2026, doi: 10.5281/zenodo.18243426.

Outline

Information

Phys. Rev. A 113, 062432– Published 11 June, 2026
  • Received 14 January 2026
  • Accepted 26 May 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation