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Noisy quantum simulation: Performance and resource considerations for the Tavis-Cummings and Heisenberg models

Alisa Haukisalmi1, Daniel Paz Ramos1, Matti Raasakka1,*, Andrea Marchesin1, Lauri Ylinen2, and Ilkka Tittonen1

  • 1Micro and Quantum Systems Research Group, Department of Electronics and Nanoengineering, Aalto University, Espoo, Finland
  • 2Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland

  • *Contact author: matti.raasakka@aalto.fi
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Phys. Rev. Research 7, 043254 – Published 5 December, 2025

DOI: https://doi.org/10.1103/x9t3-64wm

Abstract

Fault-tolerant quantum computers promise the simulation of complex quantum systems beyond the reach of classical computation. In contrast, current noisy intermediate-scale quantum devices are constrained by hardware noise. Consequently, quantum simulation methods remain limited in their near-term applicability. Two prominent techniques addressing these challenges are zero-noise extrapolation (ZNE) and incremental structural learning (ISL). In this work, ZNE and ISL are benchmarked for simulating the Trotterized time evolution of two models: the Tavis-Cummings model (TCM) and the Heisenberg spin chain (HSC), using a classically simulated noisy hardware backend. The methods are evaluated on the basis of the accuracy of expectation values relative to noiseless simulations and their resource demands such as circuit depths and shot counts. The impact of noise on optimization routines in ISL, previously underexplored, is also investigated. Results indicate that ISL performs more favorably in HSC systems, consistently surpassing ZNE in expectation value accuracy. Conversely, for the TCM, ISL generally yields lower accuracies despite reduced Trotter circuit depths, with weak interactions often leading to pronounced phase lags or flat expectation curves. Notably, when performing ISL optimization under noiseless conditions, the protocol is generally able to reduce dephasing errors, but average accuracies still vary on the simulated Hamiltonian. Our findings highlight the sensitivity of quantum simulation protocols to the structure of the Hamiltonian encoding system dynamics. Trends across systems suggest that ISL optimization benefits from Trotter circuits with stronger interactions, and that ansatz construction favors isotropic couplings. Moreover, although ISL introduces approximation errors, it demonstrates greater robustness than ZNE in systems with deeper Trotter circuits.

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References (109)

  1. M. A. Shafique, A. Munir, and I. Latif, Quantum computing: Circuits, algorithms, and applications, IEEE Access 12, 22296 (2024).
  2. National Academies of Sciences, Engineering, and Medicine, Quantum Computing: Progress and Prospects, edited by E. Grumbling and M. Horowitz (The National Academies Press, Washington, DC, 2019).
  3. A. Montanaro, Quantum algorithms: An overview, npj Quantum Inf. 2, 15023 (2016).
  4. S. B. Ramezani, A. Sommers, H. K. Manchukonda, S. Rahimi, and A. Amirlatifi, Machine learning algorithms in quantum computing: A survey, in 2020 International Joint Conference on Neural Networks (IJCNN) (IEEE, Glasgow, UK, 2020), pp. 1–8.
  5. A. Abbas et al., Challenges and opportunities in quantum optimization, Nat. Rev. Phys. 6, 718 (2024).
  6. L. Bassman Oftelie, M. Urbanek, M. Metcalf, J. Carter, A. F. Kemper, and W. A. d. Jong, Simulating quantum materials with digital quantum computers, Quantum Sci. Technol. 6, 043002 (2021).
  7. T. Byrnes and Y. Yamamoto, Simulating lattice gauge theories on a quantum computer, Phys. Rev. A 73, 022328 (2006).
  8. B. Bauer, S. Bravyi, M. Motta, and G. K.-L. Chan, Quantum algorithms for quantum chemistry and quantum materials science, Chem. Rev. 120, 12685 (2020).
  9. A. Smith, M. S. Kim, F. Pollmann, and J. Knolle, Simulating quantum many-body dynamics on a current digital quantum computer, npj Quantum Inf. 5, 106 (2019).
  10. R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys. 21, 467 (1982).
  11. I. Georgescu, S. Ashhab, and F. Nori, Quantum simulation, Rev. Mod. Phys. 86, 153 (2014).
  12. S. Lloyd, Universal quantum simulators, Science 273, 1073 (1996).
  13. H. F. Trotter, On the product of semi-groups of operators, Proc. Am. Math. Soc. 10, 545 (1959).
  14. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th ed. (Cambridge University Press, Cambridge, 2010).
  15. Y. Alexeev et al., Quantum computer systems for scientific discovery, PRX Quantum 2, 017001 (2021).
  16. A. D. Córcoles, A. Kandala, A. Javadi-Abhari, D. T. McClure, A. W. Cross, K. Temme, P. D. Nation, M. Steffen, and J. M. Gambetta, Challenges and opportunities of near-term quantum computing systems, Proc. IEEE 108, 1338 (2020).
  17. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  18. 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).
  19. C. Burdine and E. P. Blair, Trotterless simulation of open quantum systems for NISQ quantum devices, Adv. Quantum Technol. 8, 2400240 (2025).
  20. Z. Cai, R. Babbush, S. C. Benjamin, S. Endo, W. J. Huggins, Y. Li, J. R. McClean, and T. E. O’Brien, Quantum error mitigation, Rev. Mod. Phys. 95, 045005 (2023).
  21. K. Temme, S. Bravyi, and J. M. Gambetta, Error mitigation for short-depth quantum circuits, Phys. Rev. Lett. 119, 180509 (2017).
  22. S. Endo, S. C. Benjamin, and Y. Li, Practical quantum error mitigation for near-future applications, Phys. Rev. X 8, 031027 (2018).
  23. S. Endo, Z. Cai, S. C. Benjamin, and X. Yuan, Hybrid quantum-classical algorithms and quantum error mitigation, J. Phys. Soc. Jpn. 90, 032001 (2021).
  24. E. Fontana, N. Fitzpatrick, D. M. Ramo, R. Duncan, and I. Rungger, Evaluating the noise resilience of variational quantum algorithms, Phys. Rev. A 104, 022403 (2021).
  25. B. Khanal and P. Rivas, Evaluating the impact of noise on variational quantum circuits in NISQ era devices, in 2023 Congress in Computer Science, Computer Engineering, & Applied Computing (CSCE) (IEEE, Las Vegas, USA, 2023), pp. 1658–1664.
  26. I. Kolotouros, Designing and improving quantum algorithms for the NISQ era, Ph.D. dissertation, University of Edinburgh, 2024, https://era.ed.ac.uk/handle/1842/42877.
  27. C. Cîrstoiu, Z. Holmes, J. Iosue, L. Cincio, P. J. Coles, and A. Sornborger, Variational fast forwarding for quantum simulation beyond the coherence time, npj Quantum Inf. 6, 82 (2020).
  28. B. Jaderberg, A. Agarwal, K. Leonhardt, M. Kiffner, and D. Jaksch, Minimum hardware requirements for hybrid quantum–classical DMFT, Quantum Sci. Technol. 5, 034015 (2020).
  29. 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 (NISQ) algorithms, Rev. Mod. Phys. 94, 015004 (2022).
  30. Y. Li and S. C. Benjamin, Efficient variational quantum simulator incorporating active error minimization, Phys. Rev. X 7, 021050 (2017).
  31. X. Yuan, S. Endo, Q. Zhao, Y. Li, and S. C. Benjamin, Theory of variational quantum simulation, Quantum 3, 191 (2019).
  32. N. Fitzpatrick, H. Apel, and D. M. Ramo, Evaluating low-depth quantum algorithms for time evolution on fermion-boson systems, arXiv:2106.03985.
  33. A. Miessen, P. J. Ollitrault, and I. Tavernelli, Quantum algorithms for quantum dynamics: A performance study on the spin-boson model, Phys. Rev. Res. 3, 043212 (2021).
  34. M. Tavis and F. W. Cummings, Exact solution for an N-molecule–radiation-field Hamiltonian, Phys. Rev. 170, 379 (1968).
  35. W. Heisenberg, Zur theorie des ferromagnetismus, Z. Phys. 49, 619 (1928).
  36. Natural units are used, where ℏ=1.
  37. P. C. Kluber, Trotterization in quantum theory, arXiv:2310.13296.
  38. M. Steudtner and S. Wehner, Fermion-to-qubit mappings with varying resource requirements for quantum simulation, New J. Phys. 20, 063010 (2018).
  39. R. S. Sarkar, S. Chakraborty, and B. Adhikari, Scalable quantum circuits for exponential of Pauli strings and Hamiltonian simulations, arXiv:2405.13605.
  40. M. Dion, T. Belabbas, and N. Bastien, Efficiently manipulating Pauli strings with PauliArray, arXiv:2405.19287.
  41. X. Yang, X. Nie, Y. Ji, T. Xin, D. Lu, and J. Li, Improved quantum computing with higher-order trotter decomposition, Phys. Rev. A 106, 042401 (2022).
  42. S. Zhang, Y. Lu, K. Zhang, W. Chen, Y. Li, J.-N. Zhang, and K. Kim, Error-mitigated quantum gates exceeding physical fidelities in a trapped-ion system, Nat. Commun. 11, 587 (2020).
  43. E. van den Berg, Z. K. Minev, A. Kandala, and K. Temme, Probabilistic error cancellation with sparse Pauli–Lindblad models on noisy quantum processors, Nat. Phys. 19, 1116 (2023).
  44. F. B. Maciejewski, Z. Zimborás, and M. Oszmaniec, Mitigation of readout noise in near-term quantum devices by classical post-processing based on detector tomography, Quantum 4, 257 (2020).
  45. S. Bravyi, S. Sheldon, A. Kandala, D. C. Mckay, and J. M. Gambetta, Mitigating measurement errors in multiqubit experiments, Phys. Rev. A 103, 042605 (2021).
  46. P. Czarnik, A. Arrasmith, P. J. Coles, and L. Cincio, Error mitigation with Clifford quantum-circuit data, Quantum 5, 592 (2021).
  47. A. Lowe, M. H. Gordon, P. Czarnik, A. Arrasmith, P. J. Coles, and L. Cincio, Unified approach to data-driven quantum error mitigation, Phys. Rev. Res. 3, 033098 (2021).
  48. J. Pérez-Guijarro, A. Pagès-Zamora, and J. R. Fonollosa, Extension of Clifford data regression methods for quantum error mitigation, in ICASSP 2024-2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (IEEE, Seoul, Korea, 2024), pp. 9691–9695.
  49. S. Filippov, M. Leahy, M. A. C. Rossi, and G. García-Pérez, Scalable tensor-network error mitigation for near-term quantum computing, arXiv:2307.11740.
  50. C. Song, J. Cui, H. Wang, J. Hao, H. Feng, and Y. Li, Quantum computation with universal error mitigation on a superconducting quantum processor, Sci. Adv. 5, eaaw5686 (2019).
  51. R. LaRose, A. Mari, V. Russo, D. Strano, and W. J. Zeng, Error mitigation increases the effective quantum volume of quantum computers, arXiv:2203.05489.
  52. V. Russo, A. Mari, N. Shammah, R. LaRose, and W. J. Zeng, Testing platform-independent quantum error mitigation on noisy quantum computers, IEEE Trans. Quantum Eng. 4, 1 (2023).
  53. S. Moradi, C. Spielvogel, D. Krajnc, C. Brandner, S. Hillmich, R. Wille, T. Traub-Weidinger, X. Li, M. Hacker, W. Drexler, and L. Papp, Error mitigation enables pet radiomic cancer characterization on quantum computers, Eur. J. Nucl. Med. Mol. Imaging 50, 3826 (2023).
  54. T. Giurgica-Tiron, Y. Hindy, R. LaRose, A. Mari, and W. J. Zeng, Digital zero noise extrapolation for quantum error mitigation, in 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, Denver, USA, 2020), pp. 306–316.
  55. Shot noise refers to the statistical fluctuations arising from the finite number of circuit executions (shots) used to estimate expectation values.
  56. J. B. Altepeter, D. F. James, and P. G. Kwiat, Qubit quantum state tomography, in Quantum State Estimation, Lecture Notes in Physics, edited by M. Paris and J. Řeháček (Springer, Berlin, 2004), pp. 113–145.
  57. M. Ostaszewski, E. Grant, and M. Benedetti, Structure optimization for parameterized quantum circuits, Quantum 5, 391 (2021).
  58. In this work, this subprotocol is referred to as ISL transpilation, to avoid confusion with (backend) transpilation of quantum circuits to a specific hardware.
  59. Rotosolve minimizes the angle for each rotation gate around a fixed axis (x, y, or z), whereas Rotoselect performs all three and chooses the optimal configuration.
  60. B. Jaderberg, A. Eisfeld, D. Jaksch, and S. Mostame, Recompilation-enhanced simulation of electron-phonon dynamics on IBM quantum computers, New J. Phys. 24, 093017 (2022).
  61. B. Jaderberg, L. W. Anderson, W. Xie, S. Albanie, M. Kiffner, and D. Jaksch, Quantum self-supervised learning, Quantum Sci. Technol. 7, 035005 (2022).
  62. A. Di Paolo, P. K. Barkoutsos, I. Tavernelli, and A. Blais, Variational quantum simulation of ultrastrong light-matter coupling, Phys. Rev. Res. 2, 033364 (2020).
  63. W. Li, J. Ren, S. Huai, T. Cai, Z. Shuai, and S. Zhang, Efficient quantum simulation of electron-phonon systems by variational basis state encoder, Phys. Rev. Res. 5, 023046 (2023).
  64. E. Jaynes and F. Cummings, Comparison of quantum and semiclassical radiation theories with application to the beam maser, Proc. IEEE 51, 89 (1963).
  65. R. H. Dicke, Coherence in spontaneous radiation processes, Phys. Rev. 93, 99 (1954).
  66. Z. Dong, G. Zhang, A.-G. Wu, and R.-B. Wu, On the dynamics of the Tavis–Cummings model, IEEE Trans. Auto. Control 68, 2048 (2023).
  67. G.-W. Deng, D. Wei, S.-X. Li, J. R. Johansson, W.-C. Kong, H.-O. Li, G. Cao, M. Xiao, G.-C. Guo, F. Nori, H.-W. Jiang, and G.-P. Guo, Coupling two distant double quantum dots with a microwave resonator, Nano Lett. 15, 6620 (2015).
  68. M. A. Sillanpää, J. I. Park, and R. W. Simmonds, Coherent quantum state storage and transfer between two phase qubits via a resonant cavity, Nature (London) 449, 438 (2007).
  69. D. J. van Woerkom, P. Scarlino, J. H. Ungerer, C. Müller, J. V. Koski, A. J. Landig, C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, and A. Wallraff, Microwave photon-mediated interactions between semiconductor qubits, Phys. Rev. X 8, 041018 (2018).
  70. J. Larson and T. Mavrogordatos, The Jaynes–Cummings Model and its Descendants (IOP Publishing, Bristol, UK, 2021), pp. 2053–2563.
  71. J. A. Campos-Gonzalez-Angulo and J. Yuen-Zhou, Generalization of the Tavis–Cummings model for multi-level anharmonic systems: Insights on the second excitation manifold, J. Chem. Phys. 156, 194308 (2022).
  72. N. P. D. Sawaya, T. Menke, T. H. Kyaw, S. Johri, A. Aspuru-Guzik, and G. G. Guerreschi, Resource-efficient digital quantum simulation of d-level systems for photonic, vibrational, and spin-s Hamiltonians, npj Quantum Inf. 6, 49 (2020).
  73. C. Candu and M. de Leeuw, Spin chains, in Introduction to Integrability – FS 2013, Lecture Notes (ETH Zurich, 2013).
  74. F. A. Zwanenburg, A. S. Dzurak, A. Morello, M. Y. Simmons, L. C. L. Hollenberg, G. Klimeck, S. Rogge, S. N. Coppersmith, and M. A. Eriksson, Silicon Quantum electronics, Rev. Mod. Phys. 85, 961 (2013).
  75. R. Fazio, Quantum phase transitions and vortex dynamics in superconducting networks, Phys. Rep. 355, 235 (2001).
  76. R. Blatt and C. F. Roos, Quantum simulations with trapped ions, Nat. Phys. 8, 277 (2012).
  77. J. Lu, L. Zhou, L.-M. Kuang, and C. P. Sun, Controlling soliton excitations in Heisenberg spin chains through the magic angle, Phys. Rev. E 79, 016606 (2009).
  78. O. V. Marchukov, A. G. Volosniev, M. Valiente, D. Petrosyan, and N. T. Zinner, Quantum spin transistor with a Heisenberg spin chain, Nat. Commun. 7, 13070 (2016).
  79. Y. Jiang, Heisenberg spin chain and Bethe ansatz, Lecture Notes from “An Introduction to Quantum Integrable Systems”, Chinese Academy of Sciences, Fall 2022, https://www.koushare.com/courseTopicInfo/i/AIQIS/0.
  80. Qiskit Development Team, Qiskit: An open-source framework for quantum computing, 2024, https://www.ibm.com/quantum/qiskit.
  81. While the IBM Nairobi processor was publicly available at the time of selecting this noise model, it was retired in November 2023 [82].
  82. IBM Quantum Documentation, Retired systems, 2023, https://docs.quantum.ibm.com/run/retired-systems/.
  83. IBM Quantum Platform, 2023, https://quantum-computing.ibm.com/.
  84. IBM Quantum Documentation, 2024, https://docs.quantum.ibm.com/api/qiskit/transpiler.
  85. R. LaRose, A. Mari, S. Kaiser, P. J. Karalekas, A. A. Alves, P. Czarnik, M. E. Mandouh, M. H. Gordon, Y. Hindy, A. Robertson, P. Thakre, M. Wahl, D. Samuel, R. Mistri, M. Tremblay, N. Gardner, N. T. Stemen, N. Shammah, and W. J. Zeng, Mitiq: A software package for error mitigation on noisy quantum computers, Quantum 6, 774 (2022).
  86. B. Jaderberg, A. Agarwal, K. Leonhardt, M. Kiffner, D. Jaksch, A. Haukisalmi, and D. Paz, ISL, GitHub repository, 2025, https://github.com/Dpazramos/isl/.
  87. A. Haukisalmi and D. Paz, Spin-boson simulation, GitHub repository, 2025, https://github.com/Dpazramos/spin-boson-sim.
  88. Aalto Scientific Computing, Triton – Aalto University computing cluster, 2024, https://scicomp.aalto.fi/triton/.
  89. P. Virtanen et al., SciPy 1.0: Fundamental algorithms for scientific computing in Python, Nat. Methods 17, 261 (2020).
  90. G. De Palma, M. Marvian, C. Rouzé, and D. S. França, Limitations of variational quantum algorithms: A quantum optimal transport approach, PRX Quantum 4, 010309 (2023).
  91. R. Takagi, S. Endo, S. Minagawa, and M. Gu, Fundamental limits of quantum error mitigation, npj Quantum Inf. 8, 114 (2022).
  92. M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, Efficient quantum state tomography, Nat. Commun. 1, 149 (2010).
  93. G. García-Pérez, M. A. C. Rossi, B. Sokolov, E.-M. Borrelli, and S. Maniscalco, Pairwise tomography networks for many-body quantum systems, Phys. Rev. Res. 2, 023393 (2020).
  94. Recall that for ZNE and ISL, k characterizes measurement resolution, but is not equal to the total circuit shots per time step.
  95. T. Ayral, F.-M. L. Régent, Z. Saleem, Y. Alexeev, and M. Suchara, Quantum divide and compute: Exploring the effect of different noise sources, SN Comput. Sci. 2, 132 (2021).
  96. Here, ansatz layer is used interchangeably with thinned cnot layer (see Sec. 2c).
  97. That is, of equal magnitude but opposite sign.
  98. While maintaining the qubit-pair connectivity dictated by the QPU topology of ibm_nairobi.
  99. The discrepancy arises from five configurations being performed for aggregate statistics, while an additional generic run was considered to provide a randomly selected cost history; this convention was extended to the other parametrizations for consistency in noisy ISL results.
  100. The 18 cases refer to four TCM systems plus five HSC systems, for each of which two cost thresholds are considered.
  101. H. C. Watanabe, R. Raymond, Y. ya Ohnishi, E. Kaminishi, and M. Sugawara, Optimizing parameterized quantum circuits with Free-Axis selection, in 2021 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, Broomfield, CO, USA, 2021), pp. 100–111.
  102. J. V. Pankkonen, L. Ylinen, M. Raasakka, and I. Tittonen, Improving variational quantum circuit optimization via hybrid algorithms and random axis initialization, arXiv:2503.20728.
  103. J. V. Pankkonen, L. Ylinen, M. Raasakka, A. Marchesin, and I. Tittonen, Gate freezing method for gradient-free variational quantum algorithms in circuit optimization, arXiv:2507.07742.
  104. E. Fontana, M. Cerezo, A. Arrasmith, I. Rungger, and P. J. Coles, Non-trivial symmetries in quantum landscapes and their resilience to quantum noise, Quantum 6, 804 (2022).
  105. Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. van den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme, and A. Kandala, Evidence for the utility of quantum computing before fault tolerance, Nature (London) 618, 500 (2023).
  106. Z. Cai, X. Xu, and S. C. Benjamin, Mitigating coherent noise using Pauli conjugation, npj Quantum Inf. 6, 17 (2020).
  107. For example, exploiting the isotropic XX/YY couplings in the TCM or the cyclic structure of the HSC to minimize computational overhead in the ansatz construction.
  108. Note that the same considerations as in Sec. 2c regarding noisy recompilation hold.
  109. W. K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett. 80, 2245 (1998).

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Phys. Rev. Research 7, 043254– Published 5 December, 2025
  • Received 28 February 2024
  • Accepted 10 November 2025
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