- Physical Review Journals
All JournalsPhysics Magazine

PRX Quantum

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
  • Tutorial
  • Open Access

Pauli Propagation: A Computational Framework for Simulating Quantum Systems

Manuel S. Rudolph1,2,*, Tyson Jones1,2, Yanting Teng1,2, Armando Angrisani1,2, and Zoë Holmes1,2,3

  • *Contact author: manuel.rudolph@epfl.ch
PDF

PRX Quantum 7, 032001 – Published 11 August, 2026

DOI: https://doi.org/10.1103/6vd7-l9bn

Abstract

Classical methods to simulate quantum systems are not only a key element of the physicist’s toolkit for studying many-body models but are also increasingly important for verifying and challenging upcoming quantum computers. Pauli propagation has recently emerged as a promising new family of classical algorithms for simulating digital quantum systems. Here we provide a comprehensive account of Pauli propagation, tracing its algorithmic structure from its bit-level implementation and formulation as a tree-search problem, all the way to its high-level user applications for simulating quantum circuits and dynamics. Utilizing these observations, we present PauliPropagation.jl, a Julia software package that can perform rapid Pauli propagation simulation straight out-of-the-box and can be used more generally as a building block for novel simulation algorithms.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Simulating quantum computers on classical machines is essential for verifying their performance, and no single method is optimal for all kinds of simulation tasks. We provide a complete framework for “Pauli propagation”: a technique that efficiently tracks how quantum properties evolve and spread over time, with a focus on the target measurements, not the full quantum states. The approach is not natively limited by quantum entanglement and is particularly suited for simulating non-local quantum computations. This work presents a unified framework connecting the theoretical guarantees and challenges with the low-level implementation details that are essential for practical success. It is accompanied by a user-friendly yet high-performance tool that enables researchers to simulate and study large quantum circuits straight out of the box.

Article Text

References (92)

  1. R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. (Amsterdam) 349, 117 (2014).
  2. F. Verstraete, V. Murg, and J. I. Cirac, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Adv. Phys. 57, 143 (2008).
  3. Y.-Y. Shi, L.-M. Duan, and G. Vidal, Classical simulation of quantum many-body systems with a tree tensor network, Phys. Rev. A 74, 022320 (2006).
  4. F. Verstraete and J. I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, arXiv:cond-mat/0407066.
  5. J. Biamonte and V. Bergholm, Tensor networks in a nutshell, arXiv:1708.00006.
  6. J. Tindall, M. Fishman, E. M. Stoudenmire, and D. Sels, Efficient tensor network simulation of IBM’s Eagle kicked Ising experiment, PRX Quantum 5, 010308 (2024).
  7. J. Tindall, A. Mello, M. Fishman, M. Stoudenmire, and D. Sels, Dynamics of disordered quantum systems with two- and three-dimensional tensor networksarXiv:2503.05693.
  8. S. Patra, S. S. Jahromi, S. Singh, and R. Orus, Efficient tensor network simulation of IBM’s largest quantum processors, arXiv:2309.15642.
  9. G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017).
  10. D. Wu, R. Rossi, F. Vicentini, N. Astrakhantsev, F. Becca, X. Cao, J. Carrasquilla, F. Ferrari, A. Georges, M. Hibat-Allah et al., Variational benchmarks for quantum many-body problemsarXiv:2302.04919.
  11. T. Begušić, J. Gray, and G. K.-L. Chan, Fast and converged classical simulations of evidence for the utility of quantum computing before fault tolerance, Sci. Adv. 10 (2024).
  12. M. S. Rudolph, E. Fontana, Z. Holmes, and L. Cincio, Classical surrogate simulation of quantum systems with LOWESA, arXiv:2308.09109.
  13. Y. Shao, F. Wei, S. Cheng, and Z. Liu, Simulating noisy variational quantum algorithms: A polynomial approach, Phys. Rev. Lett. 133, 120603 (2024).
  14. A. Angrisani, A. Schmidhuber, M. S. Rudolph, M. Cerezo, Z. Holmes, and H.-Y. Huang, Classically estimating observables of noiseless quantum circuits, Phys. Rev. Lett. 135, 170602 (2025).
  15. E. Fontana, M. S. Rudolph, R. Duncan, I. Rungger, and C. Cîrstoiu, Classical simulations of noisy variational quantum circuits, npj Quantum Inf. 11, 1 (2025).
  16. T. Begušić and G. K.-L. Chan, Real-time operator evolution in two and three dimensions via sparse Pauli dynamics, PRX Quantum 6, 020302 (2025).
  17. S. Lerch, R. Puig, M. Rudolph, A. Angrisani, T. Jones, M. Cerezo, S. Thanasilp, and Z. Holmes, Efficient quantum-enhanced classical simulation for patches of quantum landscapes, arXiv:2411.19896.
  18. P. Bermejo, B. Aizpurua, and R. Orús, Improving gradient methods via coordinate transformations: Applications to quantum machine learning, Phys. Rev. Res. 6, 023069 (2024).
  19. M. Cerezo, M. Larocca, D. García-Martín, N. L. Diaz, P. Braccia, E. Fontana, M. S. Rudolph, P. Bermejo, A. Ijaz, S. Thanasilp et al., Does provable absence of barren plateaus imply classical simulability?, Nat. Commun. 16, 7907 (2025).
  20. B. Fuller, M. C. Tran, D. Lykov, C. Johnson, M. Rossmannek, K. X. Wei, A. He, Y. Kim, D. Vu, K. Sharma et al., Improved quantum computation using operator backpropagationarXiv:2502.01897.
  21. P. Rall, D. Liang, J. Cook, and W. Kretschmer, Simulation of qubit quantum circuits via Pauli propagation, Phys. Rev. A 99, 062337 (2019).
  22. L. Broers and L. Mathey, Exclusive-or encoded algebraic structure for efficient quantum dynamicsarXiv:2404.09312.
  23. L. Broers, R.-Y. Sun, and S. Yunoki, Scalable simulation of quantum many-body dynamics with or-represented quantum algebraarXiv:2506.13241.
  24. T. Begušić, K. Hejazi, and G. K.-L. Chan, Simulating quantum circuit expectation values by Clifford perturbation theory, J. Chem. Phys. 162, 154110 (2025).
  25. A. Miller, J. Favre, Z. Holmes, Ö. Salehi, R. Chakraborty, A. Nykänen, Z. Zimboras, A. Glos, and G. García-Pérez, Simulation of fermionic circuits using Majorana propagation, arXiv:2503.18939.
  26. M. D’Anna, J. Nys, and J. Carrasquilla, Majorana string simulation of nonequilibrium dynamics in two-dimensional lattice fermion systems, arXiv:2511.02809.
  27. F. Alam, J. L. Bosse, I. Čepaitė, A. Chapman, L. Clinton, M. Crichigno, E. Crosson, T. Cubitt, C. Derby, O. Dowinton et al., Fermionic dynamics on a trapped-ion quantum computer beyond exact classical simulation, arXiv:2510.26300.
  28. F. Alam, J. L. Bosse, I. Čepaitė, A. Chapman, L. Clinton, M. Crichigno, E. Crosson, T. Cubitt, C. Derby, O. Dowinton et al., Programmable digital quantum simulation of 2D Fermi-Hubbard dynamics using 72 superconducting qubits, arXiv:2510.26845.
  29. D. Aharonov, X. Gao, Z. Landau, Y. Liu, and U. Vazirani, A polynomial-time classical algorithm for noisy random circuit sampling, in Proceedings of the 55th Annual ACM Symposium on Theory of Computing (2023), Vol. 945.
  30. T. Schuster, C. Yin, X. Gao, and N. Y. Yao, A polynomial-time classical algorithm for noisy quantum circuits, Phys. Rev. X 15, 041018 (2025).
  31. A. Angrisani, A. A. Mele, M. S. Rudolph, M. Cerezo, and Z. Holmes, Simulating quantum circuits with arbitrary local noise using Pauli propagation, PRX Quantum 7, 020313 (2026).
  32. P. Bermejo, P. Braccia, M. S. Rudolph, Z. Holmes, L. Cincio, and M. Cerezo, Quantum convolutional neural networks are effectively classically simulable, PRX Quantum 7, 020304 (2026).
  33. PauliPropagation.jl, GitHub Repository (2026), https://github.com/SparqleSim/PauliPropagation.jl.
  34. M. S. Rudolph, A. Angrisani, A. Wright, I. Sanderski, R. Puig, and Z. Holmes, Thermal state simulation with Pauli and Majorana propagation, arXiv:2602.04878.
  35. J. M. Chow, J. M. Gambetta, A. D. Córcoles, S. T. Merkel, J. A. Smolin, C. Rigetti, S. Poletto, G. A. Keefe, M. B. Rothwell, J. R. Rozen, M. B. Ketchen, and M. Steffen, Universal quantum gate set approaching fault-tolerant thresholds with superconducting qubits, Phys. Rev. Lett. 109, 060501 (2012).
  36. J. Emerson, M. Silva, O. Moussa, C. Ryan, M. Laforest, J. Baugh, D. G. Cory, and R. Laflamme, Symmetrized characterization of noisy quantum processes, Science 317, 1893 (2007).
  37. C. King and M. B. Ruskai, Minimal entropy of states emerging from noisy quantum channels, IEEE Trans. Inf. Theor. 47, 192 (2001).
  38. M. Ben-Or, D. Gottesman, and A. Hassidim, Quantum refrigerator, arXiv:1301.1995.
  39. A. A. Mele, A. Angrisani, S. Ghosh, S. Khatri, J. Eisert, D. Stilck França, and Y. Quek, Noise-induced shallow circuits and the absence of barren plateaus, Nat. Phys. 1 (2026).
  40. A. Deshpande, M. Hinsche, S. Najafi, K. Sharma, R. Sweke, and C. Zoufal, Dynamic parameterized quantum circuits: Expressive and barren-plateau free, arXiv:2411.05760.
  41. T. Jones and S. Benjamin, Questlink—mathematica embiggened by a hardware-optimised quantum emulator, Quantum Sci. Technol. 5, 034012 (2020).
  42. B. Barak, C.-N. Chou, and X. Gao, Spoofing linear cross-entropy benchmarking in shallow quantum circuits, arXiv:2005.02421.
  43. A. Tanggara, M. Gu, and K. Bharti, Classically spoofing system linear cross entropy score benchmarking, arXiv:2405.00789.
  44. Y. Shao, F. Wei, S. Cheng, and Z. Liu, Simulating noisy variational quantum algorithms: A polynomial approach, Phys. Rev. Lett. 133, 120603 (2024).
  45. G. González-García, J. I. Cirac, and R. Trivedi, Pauli path simulations of noisy quantum circuits beyond average case, Quantum 9, 1730 (2025).
  46. N. A. Nemkov, E. O. Kiktenko, and A. K. Fedorov, Fourier expansion in variational quantum algorithms, Phys. Rev. A 108, 032406 (2023).
  47. C. Cirstoiu, A Fourier analysis framework for approximate classical simulations of quantum circuits, arXiv:2410.13856.
  48. S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. A 70, 052328 (2004).
  49. S. Bravyi and D. Gosset, Improved classical simulation of quantum circuits dominated by Clifford gates, Phys. Rev. Lett. 116, 250501 (2016).
  50. D. Gottesman, Stabilizer codes and quantum error correction, Ph.D. thesis, California Institute of Technology, 1997, arXiv:quant-ph/9705052.
  51. C. Gidney, Stim: A fast stabilizer circuit simulator, Quantum 5, 497 (2021).
  52. H. Pashayan, O. Reardon-Smith, K. Korzekwa, and S. D. Bartlett, Fast estimation of outcome probabilities for quantum circuits, PRX Quantum 3, 020361 (2022).
  53. S. Bravyi, D. Browne, P. Calpin, E. Campbell, D. Gosset, and M. Howard, Simulation of quantum circuits by low-rank stabilizer decompositions, Quantum 3, 181 (2019).
  54. M. D’Anna, Y. Zhang, R. Wiersema, M. S. Rudolph, and J. Carrasquilla, Circuit compression for 2D quantum dynamics, arXiv:2507.01883.
  55. Y. Teng, S. Y. Chang, M. S. Rudolph, and Z. Holmes, Leveraging symmetry merging in Pauli propagation, arXiv:2512.12094.
  56. MajoranaPropagation.jl, GitHub Repository (2025), https://github.com/SparqleSim/MajoranaPropagation.jl.
  57. N. Loizeau, J. C. Peacock, and D. Sels, Quantum many-body simulations with paulistrings. jl, SciPost Phys. Codebases 54, r1.5 (2025).
  58. F. Uskov and O. Lychkovskiy, Quantum dynamics in one and two dimensions via the recursion method, Phys. Rev. B 109, L140301 (2024).
  59. I. Ermakov, O. Lychkovskiy, and T. Byrnes, Unified framework for efficiently computable quantum circuits, arXiv:2401.08187.
  60. A. Teretenkov, F. Uskov, and O. Lychkovskiy, Pseudomode expansion of many-body correlation functions, Phys. Rev. B 111, 174308 (2025).
  61. I. Ermakov, Operator growth in many-body systems of higher spins, arXiv:2504.07833.
  62. I. Shirokov, V. Khrushchev, F. Uskov, I. Dudinets, I. Ermakov, and O. Lychkovskiy, Quench dynamics via recursion method and dynamical quantum phase transitions, arXiv:2503.24362.
  63. T. Rakovszky, C. Von Keyserlingk, and F. Pollmann, Dissipation-assisted operator evolution method for capturing hydrodynamic transport, Phys. Rev. B 105, 075131 (2022).10.1103/PhysRevB.105.075131
  64. N. Dowling, K. Modi, and G. A. White, Bridging entanglement and magic resources through operator space, arXiv:2501.18679.
  65. F. Verstraete, J. J. Garcia-Ripoll, and J. I. Cirac, Matrix product density operators: Simulation of finite-temperature and dissipative systems, Phys. Rev. Lett. 93, 207204 (2004).
  66. M. J. Hartmann, J. Prior, S. R. Clark, and M. B. Plenio, Density matrix renormalization group in the Heisenberg picture, Phys. Rev. Lett. 102, 057202 (2009).
  67. H.-J. Liao, K. Wang, Z.-S. Zhou, P. Zhang, and T. Xiang, Simulation of IBM’s kicked Ising experiment with projected entangled pair operator, arXiv:2308.03082.
  68. V. Martinez, A. Angrisani, E. Pankovets, O. Fawzi, and D. Stilck França, Efficient simulation of parametrized quantum circuits under nonunital noise through Pauli backpropagation, Phys. Rev. Lett. 134, 250602 (2025).
  69. Anonymous, quantum disadvantage, in Sigbovik 2024 (Sigbovik, 2024), pp. 199–205.
  70. M. M. Rams, P. Czarnik, and L. Cincio, Precise extrapolation of the correlation function asymptotics in uniform tensor network states with application to the Bose-Hubbard and XXZ models, Phys. Rev. X 8, 041033 (2018).
  71. T. Jones, A. Brown, I. Bush, and S. C. Benjamin, Quest and high performance simulation of quantum computers, Sci. Rep. 9, 10736 (2019).
  72. E. Knill, Quantum computing with realistically noisy devices, Nature (London) 434, 39 (2005).
  73. O. Koska, M. Baboulin, and A. Gazda, A tree-approach Pauli decomposition algorithm with application to quantum computing, in ISC High Performance 2024 Research Paper Proceedings (39th International Conference) (Prometeus GmbH, 2024), pp. 1–11.
  74. L. Hantzko, L. Binkowski, and S. Gupta, Pauli transfer matrices, arXiv:2411.00526.
  75. N. Dowling, P. Kos, and X. Turkeshi, Magic resources of the Heisenberg picture, Phys. Rev. Lett. 135, 050401 (2025).
  76. E. Chitambar and G. Gour, Quantum resource theories, Rev. Mod. Phys. 91, 025001 (2019).
  77. A. Nietner, M. Ioannou, R. Sweke, R. Kueng, J. Eisert, M. Hinsche, and J. Haferkamp, On the average-case complexity of learning output distributions of quantum circuits, arXiv:2305.05765.
  78. D. A. Roberts and B. Yoshida, Chaos and complexity by design, J. High Energy Phys. 04 (2017) 121.
  79. R. Belyansky, P. Bienias, Y. A. Kharkov, A. V. Gorshkov, and B. Swingle, Minimal model for fast scrambling, Phys. Rev. Lett. 125, 130601 (2020).
  80. M. R. Geller, A. Arrasmith, Z. Holmes, B. Yan, P. J. Coles, and A. Sornborger, Quantum simulation of operator spreading in the chaotic Ising model, Phys. Rev. E 105, 035302 (2022).
  81. A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum entanglement growth under random unitary dynamics, Phys. Rev. X 7, 031016 (2017).
  82. C. W. von Keyserlingk, T. Rakovszky, F. Pollmann, and S. L. Sondhi, Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws, Phys. Rev. X 8, 021013 (2018).
  83. X. Mi, P. Roushan, C. Quintana, S. Mandra, J. Marshall, C. Neill, F. Arute, K. Arya, J. Atalaya, R. Babbush et al., Information scrambling in quantum circuits, Science 374, 1479 (2021).
  84. 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).
  85. M. K. Joshi, A. Elben, B. Vermersch, T. Brydges, C. Maier, P. Zoller, R. Blatt, and C. F. Roos, Quantum information scrambling in a trapped-ion quantum simulator with tunable range interactions, Phys. Rev. Lett. 124, 240505 (2020).
  86. D. Bluvstein, A. Omran, H. Levine, A. Keesling, G. Semeghini, S. Ebadi, T. T. Wang, A. A. Michailidis, N. Maskara, W. W. Ho, S. Choi, M. Serbyn, M. Greiner, V. Vuletić, and M. D. Lukin, Controlling quantum many-body dynamics in driven Rydberg atom arrays, Science 371, 1355 (2021).
  87. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  88. Z. Zimboras, B. Koczor, Z. Holmes, E.-M. Borrelli, A. Gilyén, H.-Y. Huang, Z. Cai, A. Acín, L. Aolita, L. Banchi et al., Myths around quantum computation before full fault tolerance: What no-go theorems rule out and what they don’t, arXiv:2501.05694.
  89. Z.-L. Li and S.-X. Zhang, The dual role of low-weight Pauli propagation: A flawed simulator but a powerful initializer for variational quantum algorithms, arXiv:2508.06358.
  90. C.-J. Lin, H. Gharibyan, and V. P. Su, Utility-scale quantum state preparation: Classical training using Pauli path simulation, arXiv:2510.02428.
  91. C. Shrikhande, A. Bachhar, A. R. Jimenez, and N. J. Mayhall, Rapid ground state energy estimation with a sparse Pauli dynamics-enabled variational double bracket flow, arXiv:2511.21651.
  92. P. J. Roache, Verification and Validation in Computational Science and Engineering (Hermosa Albuquerque, NM, 1998), Vol. 895.

Outline

Information

PRX Quantum 7, 032001– Published 11 August, 2026
  • Received 13 January 2026
  • Revised 16 April 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