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  • Open Access

Preparation circuits for matrix product states by classical variational disentanglement

Refik Mansuroglu1,* and Norbert Schuch1,2

  • *Contact author: Refik.Mansuroglu@univie.ac.at
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Phys. Rev. A 113, 042430 – Published 13 April, 2026

DOI: https://doi.org/10.1103/7x2g-twkl

Abstract

We study the classical compilation of quantum circuits for the preparation of matrix product states (MPS), which are quantum states of low entanglement with an efficient classical description. Our algorithm represents a near-term alternative to previous sequential approaches by reverse application of a disentangler, which can be found by minimizing bipartite entanglement measures after the application of a layer of parametrized disentangling gates. Since a successful disentangler is expected to decrease the bond dimension on average, such a layer-by-layer optimization remains classically efficient even for deep circuits. Additionally, as the Schmidt coefficients of all bonds are locally accessible through the canonical Γ−Λ form of an MPS, the optimization algorithm can be heavily parallelized. We discuss guarantees and limitations to trainability and show numerical results for ground states of one-dimensional, local Hamiltonians as well as artificially spread out entanglement among multiple qubits using error-correcting codes.

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References (48)

  1. L. Marti, R. Mansuroglu, and M. J. Hartmann, Efficient quantum cooling algorithm for fermionic systems, Quantum 9, 1635 (2025).
  2. J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nat. Commun. 9, 4812 (2018).
  3. F. Verstraete and J. I. Cirac, Matrix product states represent ground states faithfully, Phys. Rev. B 73, 094423 (2006).
  4. M. B. Hastings, An area law for one-dimensional quantum systems, J. Stat. Mech. (2007) P08024.
  5. N. Schuch, M. M. Wolf, K. G. H. Vollbrecht, and J. I. Cirac, On entropy growth and the hardness of simulating time evolution, New J. Phys. 10, 033032 (2008).
  6. M. Will, T. A. Cochran, E. Rosenberg, B. Jobst, N. M. Eassa, P. Roushan, M. Knap, A. Gammon-Smith, and F. Pollmann, Probing non-equilibrium topological order on a quantum processor, Nature (London) 645, 348 (2025).
  7. S. J. Evered, M. Kalinowski, A. A. Geim, T. Manovitz, D. Bluvstein, S. H. Li, N. Maskara, H. Zhou, S. Ebadi, M. Xu, J. Campo, M. Cain, S. Ostermann, S. F. Yelin, S. Sachdev, M. Greiner, V. Vuletić, and M. D. Lukin, Probing topological matter and fermion dynamics on a neutral-atom quantum computer, Nature (London) 645, 341 (2025).
  8. C. Schön, E. Solano, F. Verstraete, J. I. Cirac, and M. M. Wolf, Sequential generation of entangled multiqubit states, Phys. Rev. Lett. 95, 110503 (2005).
  9. D. Malz, G. Styliaris, Z.-Y. Wei, and J. I. Cirac, Preparation of matrix product states with log-depth quantum circuits, Phys. Rev. Lett. 132, 040404 (2024).
  10. Z.-Y. Wei and D. Malz, State preparation with parallel-sequential circuits, arXiv:2503.14645.
  11. K. C. Smith, A. Khan, B. K. Clark, S. M. Girvin, and T.-C. Wei, Constant-depth preparation of matrix product states with adaptive quantum circuits, PRX Quantum 5, 030344 (2024).
  12. S.-J. Ran, Encoding of matrix product states into quantum circuits of one- and two-qubit gates, Phys. Rev. A 101, 032310 (2020).
  13. M. S. Rudolph, J. Chen, J. Miller, A. Acharya, and A. Perdomo-Ortiz, Decomposition of matrix product states into shallow quantum circuits, Quantum Sci. Technol. 9, 015012 (2024).
  14. B. Jobst, K. Shen, C. A. Riofrío, E. Shishenina, and F. Pollmann, Efficient MPS representations and quantum circuits from the Fourier modes of classical image data, Quantum 8, 1544 (2024).
  15. K. Murota, Adiabatic encoding of pre-trained MPS classifiers into quantum circuits, arXiv:2504.09250.
  16. B. Jaderberg, G. Pennington, K. V. Marshall, L. W. Anderson, A. Agarwal, L. P. Lindoy, I. Rungger, S. Mensa, and J. Crain, Variational preparation of normal matrix product states on quantum computers, arXiv:2503.09683.
  17. N. F. Robertson, A. Akhriev, J. Vala, and S. Zhuk, Approximate quantum compiling for quantum simulation: A tensor network based approach, ACM Trans. Quantum Comput. 6, 1 (2025).
  18. M. Lubasch, J. Joo, P. Moinier, M. Kiffner, and D. Jaksch, Variational quantum algorithms for nonlinear problems, Phys. Rev. A 101, 010301(R) (2020).
  19. C. Mc Keever and M. Lubasch, Towards adiabatic quantum computing using compressed quantum circuits, PRX Quantum 5, 020362 (2024).
  20. M. Ben-Dov, D. Shnaiderov, A. Makmal, and E. G. Dalla Torre, Approximate encoding of quantum states using shallow circuits, npj Quantum Inf. 10, 65 (2024).
  21. A. Khan, B. K. Clark, and N. M. Tubman, Pre-optimizing variational quantum eigensolvers with tensor networks, arXiv:2310.12965.
  22. A. A. Melnikov, A. A. Termanova, S. V. Dolgov, F. Neukart, and M. R. Perelshtein, Quantum state preparation using tensor networks, Quantum Sci. Technol. 8, 035027 (2023).
  23. G. Vidal, Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. Lett. 91, 147902 (2003).
  24. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. (NY) 326, 96 (2011).
  25. B. Kraus and J. I. Cirac, Optimal creation of entanglement using a two-qubit gate, Phys. Rev. A 63, 062309 (2001).
  26. M. Cerezo, M. Larocca, D. García-Martín, N. L. Diaz, P. Braccia, E. Fontana, M. S. Rudolph, P. Bermejo, A. Ijaz, S. Thanasilp, E. R. Anschuetz, and Z. Holmes, Does provable absence of barren plateaus imply classical simulability? Nat. Commun. 16, 7907 (2025).
  27. L. Cigna and C. Schilling (private communication).
  28. M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor software library for tensor network calculations, SciPost Phys. Codebases 4 (2022).
  29. R. Mansuroglu and N. Schuch, Classical variational disentanglement, Software on Zenodo (2025), https://doi.org/10.5281/zenodo.15058443.
  30. N. Schuch and F. Verstraete, Computational complexity of interacting electrons and fundamental limitations of density functional theory, Nat. Phys. 5, 732 (2009).
  31. S. R. White and E. M. Stoudenmire, Multisliced gausslet basis sets for electronic structure, Phys. Rev. B 99, 081110(R) (2019).
  32. G. González-García, R. Trivedi, and J. I. Cirac, Error propagation in NISQ devices for solving classical optimization problems, PRX Quantum 3, 040326 (2022).
  33. S. Gharibian and F. Le Gall, Dequantizing the quantum singular value transformation: Hardness and applications to quantum chemistry and the quantum PCP conjecture, SIAM J. Comput. 52, 1009 (2023).
  34. C. Cade, L. Mineh, A. Montanaro, and S. Stanisic, Strategies for solving the Fermi-Hubbard model on near-term quantum computers, Phys. Rev. B 102, 235122 (2020).
  35. R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, Perfect quantum error correction code, Phys. Rev. Lett. 77, 198 (1996).
  36. A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Quantum error correction via codes over GF(4), in Proceedings of IEEE International Symposium on Information Theory (IEEE, Ulm, Germany, 2002).
  37. T. Eckstein, R. Mansuroglu, S. Wolf, L. Nützel, S. Tasler, M. Kliesch, and M. J. Hartmann, Shot-noise reduction for lattice Hamiltonians, PRX Quantum 7, 020303 (2026).
  38. H.-Y. Huang, Y. Liu, M. Broughton, I. Kim, A. Anshu, Z. Landau, and J. R. McClean, in Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC '24 (ACM, New York, 2024), pp. 1343–1351.
  39. R. Mansuroglu, A. Adil, M. J. Hartmann, Z. Holmes, and A. T. Sornborger, Quantum tensor-product decomposition from choi-state tomography, PRX Quantum 5, 030306 (2024).
  40. C. M. Keever and M. Lubasch, Classically optimized Hamiltonian simulation, Phys. Rev. Res. 5, 023146 (2023).
  41. R. Mansuroglu, T. Eckstein, L. Nützel, S. A. Wilkinson, and M. J. Hartmann, Variational Hamiltonian simulation for translational invariant systems via classical pre-processing, Quantum Sci. Technol. 8, 025006 (2023).
  42. R. Mansuroglu, F. Fischer, and M. J. Hartmann, Problem-specific classical optimization of Hamiltonian simulation, Phys. Rev. Res. 5, 043035 (2023).
  43. M. Nibbi and C. B. Mendl, Block encoding of matrix product operators, Phys. Rev. A 110, 042427 (2024).
  44. J. I. Latorre, Image compression and entanglement, arXiv:quant-ph/0510031.
  45. E. M. Stoudenmire and D. J. Schwab, Supervised learning with quantum-inspired tensor networks, Adv. Neural Inf. Proces. Syst. 29, 4799 (2016).
  46. N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Entropy scaling and simulability by matrix product states, Phys. Rev. Lett. 100, 030504 (2008).
  47. T. Chen, R. Shen, C. H. Lee, and B. Yang, SciPost Phys. 15, 170 (2023).
  48. T. Chen and T. Byrnes, Efficient preparation of the AKLT state with measurement-based imaginary time evolution, Quantum 8, 1557 (2024).

Outline

Information

Phys. Rev. A 113, 042430– Published 13 April, 2026
  • Received 12 May 2025
  • Accepted 26 March 2026
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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.

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