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Time evolution of controlled many-body quantum systems with matrix product operators

Llorenç Balada Gaggioli1,2 and Jakub Mareček1

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Phys. Rev. A 112, 062612 – Published 9 December, 2025

DOI: https://doi.org/10.1103/9mfk-gg3x

Abstract

We present a method for describing the time evolution of many-body controlled quantum systems using matrix product operators (MPOs). Existing techniques for solving the time-dependent Schrödinger equation with an MPO Hamiltonian often rely on time discretization. In contrast, our approach uses the Magnus expansion and Chebyshev polynomials to model the time evolution and the MPO representation to efficiently encode the system's dynamics. This results in a scalable method that can be used efficiently for many-body controlled quantum systems. We apply this technique to quantum optimal control, specifically for a gate synthesis problem, demonstrating that it can be used for large-scale optimization problems that are otherwise impractical to formulate in a dense matrix representation.

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References (46)

  1. P. J. Ollitrault, A. Miessen, and I. Tavernelli, Molecular quantum dynamics: A quantum computing perspective, Acc. Chem. Res. 54, 4229 (2021).
  2. D. B. Williams-Young, S. H. Yuwono, A. E. DePrince, III, and C. Yang, Approximate exponential integrators for time-dependent equation-of-motion coupled cluster theory, J. Chem. Theory Comput. 19, 9177 (2023).
  3. 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, 19 (2022).
  4. J. Nys, G. Pescia, A. Sinibaldi, and G. Carleo, Ab-initio variational wave functions for the time-dependent many-electron Schrödinger equation, Nat. Commun. 15, 9404 (2024).
  5. R. Hamazaki, Exceptional dynamical quantum phase transitions in periodically driven systems, Nat. Commun. 12, 5108 (2021).
  6. G. K.-L. Chan, A. Keselman, N. Nakatani, Z. Li, and S. R. White, Matrix product operators, matrix product states, and ab initio density matrix renormalization group algorithms, J. Chem. Phys. 145, 014102 (2016).
  7. D. E. Parker, X. Cao, and M. P. Zaletel, Local matrix product operators: Canonical form, compression, and control theory, Phys. Rev. B 102, 035147 (2020).
  8. C. Hubig, I. P. McCulloch, and U. Schollwöck, Generic construction of efficient matrix product operators, Phys. Rev. B 95, 035129 (2017).
  9. M. C. Bañuls, Tensor network algorithms: A route map, Annu. Rev. Condens. Matter Phys. 14, 173 (2023).
  10. R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. (NY) 349, 117 (2014).
  11. B. Pirvu, V. Murg, J. I. Cirac, and F. Verstraete, Matrix product operator representations, New J. Phys. 12, 025012 (2010).
  12. F. Fröwis, V. Nebendahl, and W. Dür, Tensor operators: Constructions and applications for long-range interaction systems, Phys. Rev. A 81, 062337 (2010).
  13. G. M. Crosswhite, A. C. Doherty, and G. Vidal, Applying matrix product operators to model systems with long-range interactions, Phys. Rev. B 78, 035116 (2008).
  14. S. Keller, M. Dolfi, M. Troyer, and M. Reiher, An efficient matrix product operator representation of the quantum chemical Hamiltonian, J. Chem. Phys. 143, 244118 (2015).
  15. J. Haegeman, J. I. Cirac, T. J. Osborne, and F. Verstraete, Calculus of continuous matrix product states, Phys. Rev. B 88, 085118 (2013).
  16. M. Ganahl, P. Thunström, F. Verstraete, K. Held, and H. G. Evertz, Chebyshev expansion for impurity models using matrix product states, Phys. Rev. B 90, 045144 (2014).
  17. F. A. Wolf, I. P. McCulloch, O. Parcollet, and U. Schollwöck, Chebyshev matrix product state impurity solver for dynamical mean-field theory, Phys. Rev. B 90, 115124 (2014).
  18. R. Hong, Y.-X. Xiao, J. Hu, A.-C. Ji, and S.-J. Ran, Functional tensor network solving many-body Schrödinger equation, Phys. Rev. B 105, 165116 (2022).
  19. S. R. Manmana, A. Muramatsu, and R. M. Noack, Time evolution of one-dimensional quantum many body systems, AIP Conf. Proc. 789, 269 (2005).
  20. S. Szalay, M. Pfeffer, V. Murg, G. Barcza, F. Verstraete, R. Schneider, and O. Legeza, Tensor product methods and entanglement optimization for ab initio quantum chemistry, Int. J. Quantum Chem. 115, 1342 (2015).
  21. C. Mc Keever and M. Lubasch, Towards adiabatic quantum computing using compressed quantum circuits, PRX Quantum 5, 020362 (2024).
  22. C. Mc Keever and M. Lubasch, Classically optimized Hamiltonian simulation, Phys. Rev. Res. 5, 023146 (2023).
  23. J. J. García-Ripoll, Time evolution of matrix product states, New J. Phys. 8, 305 (2006).
  24. T. Li, P. Venkatesh, N. Shitara, and A. Montoya-Castillo, Numerically exact quantum dynamics with tensor networks: Predicting the decoherence of interacting spin systems, arXiv:2509.20604.
  25. S. Paeckel, T. Köhler, A. Swoboda, S. R. Manmana, U. Schollwöck, and C. Hubig, Time-evolution methods for matrix-product states, Ann. Phys. (NY) 411, 167998 (2019).
  26. M. V. Damme, J. Haegeman, I. McCulloch, and L. Vanderstraeten, Efficient higher-order matrix product operators for time evolution, SciPost Phys. 17, 135 (2024).
  27. 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).
  28. N. F. Robertson, B. Pokharel, B. Fuller, E. Switzer, O. Shtanko, M. Amico, A. Byrne, A. D'Urbano, S. Hayes-Shuptar, A. Akhriev, N. Keenan, S. Bravyi, and S. Zhuk, Tensor network enhanced dynamic multiproduct formulas, PRX Quantum 6, 020360 (2025).
  29. T. Chanda, P. Sierant, and J. Zakrzewski, Time dynamics with matrix product states: Many-body localization transition of large systems revisited, Phys. Rev. B 101, 035148 (2020).
  30. M. L. Wall and L. D. Carr, Out-of-equilibrium dynamics with matrix product states, New J. Phys. 14, 125015 (2012).
  31. 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. Reson. 172, 296 (2005).
  32. P. Doria, T. Calarco, and S. Montangero, Optimal control technique for many-body quantum dynamics, Phys. Rev. Lett. 106, 190501 (2011).
  33. M. M. Müller, A. Kölle, R. Löw, T. Pfau, T. Calarco, and S. Montangero, Room-temperature Rydberg single-photon source, Phys. Rev. A 87, 053412 (2013).
  34. S. van Frank, M. Bonneau, J. Schmiedmayer, S. Hild, C. Gross, M. Cheneau, I. Bloch, T. Pichler, A. Negretti, T. Calarco, and S. Montangero, Optimal control of complex atomic quantum systems, Sci. Rep. 6, 34187 (2016).
  35. T. Caneva, T. Calarco, R. Fazio, G. E. Santoro, and S. Montangero, Speeding up critical system dynamics through optimized evolution, Phys. Rev. A 84, 012312 (2011).
  36. J. H. M. Jensen, F. S. Møller, J. J. Sørensen, and J. F. Sherson, Approximate dynamics lead to more optimal control: Efficient exact derivatives Phys. Rev. A 103, 062612 (2021).
  37. D. I. Bondar, L. B. Gaggioli, G. Korpas, J. Marecek, J. Vala, and K. Jacobs, Globally optimal control of quantum dynamics, Phys. Rev. Res. 7, 043202 (2025).
  38. L. B. Gaggioli, D. I. Bondar, J. Vala, R. Ovsiannikov, and J. Mareček, Unitary gate synthesis via polynomial optimization, arXiv:2508.01356.
  39. W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math. 7, 649 (1954).
  40. D. Fang, D. Liu, and S. Zhu, High-order magnus expansion for Hamiltonian simulation, arXiv:2509.06054.
  41. S. Blanes, F. Casas, J. Oteo, and J. Ros, The Magnus expansion and some of its applications, Phys. Rep. 470, 151 (2009).
  42. H. Tal-Ezer and R. Kosloff, An accurate and efficient scheme for propagating the time dependent Schrödinger equation, J. Chem. Phys. 81, 3967 (1984).
  43. J. Johansson, P. Nation, and F. Nori, QuTiP: An open-source Python framework for the dynamics of open quantum systems, Comput. Phys. Commun. 183, 1760 (2012).
  44. J. B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim. 11, 796 (2001).
  45. J. Wang, V. Magron, and J.-B. Lasserre, TSSOS: A moment-SOS hierarchy that exploits term sparsity, SIAM J. Optim. 31, 30 (2021).
  46. https://github.com/llorebaga/QCMPO.

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Phys. Rev. A 112, 062612– Published 9 December, 2025
  • Received 18 April 2025
  • Accepted 29 October 2025
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