- 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
  • Open Access

Tensor Network Enhanced Dynamic Multiproduct Formulas

Niall F. Robertson1,*,†, Bibek Pokharel2,*, Bryce Fuller3, Eric Switzer4,5,6, Oles Shtanko2, Mirko Amico3, Adam Byrne1, Andrea D’Urbano1, Salome Hayes-Shuptar1 et al.

Albert Akhriev1, Nathan Keenan1, Sergey Bravyi3, and Sergiy Zhuk1

  • *Co-first author. These two authors contributed equally to this work
  • †Contact author: niall.robertson@ibm.com
PDF

PRX Quantum 6, 020360 – Published 27 June, 2025

DOI: https://doi.org/10.1103/8bzc-dlgt

Abstract

Tensor networks and quantum computation are two of the most powerful tools for the simulation of quantum many-body systems. Rather than viewing them as competing approaches, here we consider how these two methods can work in tandem. We introduce a novel algorithm that combines tensor networks and quantum computation to produce results that are more accurate than what could be achieved by either method used in isolation. Our algorithm is based on multiproduct formulas (MPFs)—a technique that linearly combines Trotter product formulas to reduce algorithmic error. It uses a quantum computer to calculate the expectation values and tensor networks to calculate the coefficients used in the linear combination. We present a detailed error analysis of the algorithm and demonstrate the full workflow on a one-dimensional quantum simulation problem on 50 qubits using two IBM quantum computers, ibm_torino and ibm_kyiv.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Simulating the dynamics of quantum-mechanical systems is a fundamental challenge in physics, chemistry, and material science. Decades of research have led to efficient algorithms, with modern tensor-network-based methods considered state of the art. However, these classical algorithms face a critical limitation: their performance deteriorates in the presence of high entanglement, requiring prohibitively large computational resources to maintain accuracy. Quantum computers, by contrast, are not inherently constrained by this “entanglement barrier.” However, they introduce a different challenge: noise in quantum gates restricts the number of reliable operations, limiting the expressiveness of quantum simulation algorithms. Consequently, even when a quantum computer can represent highly entangled states, algorithmic errors from limited gate depth may cause significant deviations from the true physical state.

In this work, we develop a new algorithm for quantum simulation that combines the best of classical and quantum computing such that both frameworks can simultaneously excel at their respective unique strengths. We prove that the results from this collaborative approach are more accurate than either method could achieve in isolation. The algorithm is based on multiproduct formulas, which enhance precision by linearly combining Trotter product formulas to suppress algorithmic error. Expectation values are computed on a quantum processor, while tensor networks are used to determine the optimal coefficients for the linear combination.

Article Text

References (38)

  1. N. Harle, O. Shtanko, and R. Movassagh, Observing and braiding topological Majorana modes on programmable quantum simulators, Nat. Commun. 14, 2286 (2023).
  2. J. Knolle, D. L. Kovrizhin, J. T. Chalker, and Roderich Moessner, Dynamics of a two-dimensional quantum spin liquid: Signatures of emergent majorana fermions and fluxes, Phys. Rev. Lett. 112, 207203 (2014).
  3. J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, J. High Energy Phys. 2016, 1 (2016).
  4. A. I. Larkin and Yu. N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, Sov. Phys. JETP 28, 1200 (1969).
  5. S. H. Shenker and D. Stanford, Multiple shocks, J. High Energy Phys. 2014, 1 (2014).
  6. A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise, in Talk given at the Fundamental Physics Prize Symposium (2014), Vol. 10, p. 33.
  7. P. Calabrese and J. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech.: Theory Exp. 2005, P04010 (2005).
  8. S. Lloyd, Universal quantum simulators, Science 273, 1073 (1996).
  9. 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).
  10. J. Haah, M. B. Hastings, R. Kothari, and G. Hao Low, Quantum algorithm for simulating real time evolution of lattice Hamiltonians, SIAM J. Comput. 52, FOCS18 (2021).
  11. A. M. Childs, D. Maslov, Y. Nam, N. J. Ross, and Y. Su, Toward the first quantum simulation with quantum speedup, Proc. Natl. Acad. Sci. 115, 9456 (2018).
  12. G. H. Low and I. L. Chuang, Optimal Hamiltonian simulation by quantum signal processing, Phys. Rev. Lett. 118, 010501 (2017).
  13. A. M. Childs, A. Ostrander, and Y. Su, Faster quantum simulation by randomization, Quantum 3, 182 (2019).
  14. S. Bravyi, O. Dial, J. M. Gambetta, D. Gil, and Z. Nazario, The future of quantum computing with superconducting qubits, J. Appl. Phys. 132, 160902 (2022).
  15. G. Vidal, Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. Lett. 91, 147902 (2003).
  16. A. M. Childs and N. Wiebe, Hamiltonian simulation using linear combinations of unitary operations. arXiv:1202.5822.
  17. G. H. Low, V. Kliuchnikov, and N. Wiebe, Well-conditioned multiproduct Hamiltonian simulation. arXiv:1907.11679.
  18. A. Carrera Vazquez, D. J. Egger, D. Ochsner, and S. Woerner, Well-conditioned multi-product formulas for hardware-friendly Hamiltonian simulation, Quantum 7, 1067 (2023).
  19. G. Rendon, J. Watkins, and N. Wiebe, Improved error scaling for trotter simulations through extrapolation. arXiv:2212.14144.
  20. S. Zhuk, N. F. Robertson, and S. Bravyi, Trotter error bounds and dynamic multi-product formulas for Hamiltonian simulation, Phys. Rev. Res. 6, 033309 (2024).
  21. R. Mansuroglu, F. Fischer, and M. J. Hartmann, Problem-specific classical optimization of Hamiltonian simulation, Phys. Rev. Res. 5, 043035 (2023).
  22. N. Robertson, A. Akhriev, J. Vala, and S. Zhuk, Approximate quantum compiling for quantum simulation: A tensor network based approach, ACM Trans. Quantum Comput. 6, 20 (2025).
  23. G. Vidal, Efficient simulation of one-dimensional quantum many-body systems, Phys. Rev. Lett. 93, 040502 (2004).
  24. J. Haegeman, J. Ignacio Cirac, T. J. Osborne, I. Pižorn, H. Verschelde, and F. Verstraete, Time-dependent variational principle for quantum lattices, Phys. Rev. Lett. 107, 070601 (2011).
  25. J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, and F. Verstraete, Unifying time evolution and optimization with matrix product states, Phys. Rev. B 94, 165116 (2016).
  26. M. P. Zaletel, R. S. K. Mong, C. Karrasch, J. E. Moore, and F. Pollmann, Time-evolving a matrix product state with long-ranged interactions, Phys. Rev. B 91, 165112 (2015).
  27. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011).
  28. 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.
  29. J. Hauschild and F. Pollmann, Efficient numerical simulations with tensor networks: Tensor network python (TeNPy). SciPost Phys. Lect. Notes, 5 (2018). Code available from https://github.com/tenpy/tenpy.
  30. B. Pokharel, N. Anand, B. Fortman, and D. A. Lidar, Demonstration of fidelity improvement using dynamical decoupling with superconducting qubits, Phys. Rev. Lett. 121, 220502 (2018).
  31. N. Ezzell, B. Pokharel, L. Tewala, G. Quiroz, and D. A. Lidar, Dynamical decoupling for superconducting qubits: A performance survey, Phys. Rev. Appl. 20, 064027 (2023).
  32. E. Van Den Berg, Z. K. Minev, and K. Temme, Model-free readout-error mitigation for quantum expectation values, Phys. Rev. A 105, 032620 (2022).
  33. J. Tindall and M. Fishman, Gauging tensor networks with belief propagation, SciPost Phys. 15, 222 (2023).
  34. H. N. Phien, I. P. McCulloch, and G. Vidal, Fast convergence of imaginary time evolution tensor network algorithms by recycling the environment, Phys. Rev. B 91, 115137 (2015).
  35. M. P. Zaletel and F. Pollmann, Isometric tensor network states in two dimensions, Phys. Rev. Lett. 124, 037201 (2020).
  36. G. Evenbly, Gauge fixing, canonical forms, and optimal truncations in tensor networks with closed loops, Phys. Rev. B 98, 085155 (2018).
  37. Y. Kim, A. Eddins, S. Anand, K. Xuan Wei, E. Van Den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme et al., Evidence for the utility of quantum computing before fault tolerance, Nature 618, 500 (2023).
  38. P. D. Nation and M. Treinish, Suppressing quantum circuit errors due to system variability, PRX Quantum 4, 010327 (2023).

Outline

Information

PRX Quantum 6, 020360– Published 27 June, 2025
  • Received 10 October 2024
  • Accepted 29 May 2025
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