- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

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
  • Letter
  • Open Access

Continuous matrix-product operators for quantum fields

Erickson Tjoa* and J. Ignacio Cirac

  • *Contact author: erickson.tjoa@mpq.mpg.de
PDF

Phys. Rev. Research 8, L022013 – Published 14 April, 2026

DOI: https://doi.org/10.1103/d61y-cxx3

Abstract

In this work, we introduce an ansatz for continuous matrix-product operators for quantum field theory. We show that (1) they admit a closed-form expression in terms of finite number of matrix-valued functions without reference to any lattice parameter; (2) they are obtained as a suitable continuum limit of matrix-product operators; and (3) they preserve the entanglement area law directly in the continuum, and in particular they map a continuous matrix-product state (cMPS) to another cMPS. As an application, we use this ansatz to construct several families of continuous matrix-product unitaries beyond quantum cellular automata.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (49)

  1. J. I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys. 93, 045003 (2021).
  2. M. B. Hastings, An area law for one-dimensional quantum systems, J. Stat. Mech. (2007) P08024.
  3. D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, Matrix product state representations, Quantum Inf. Comput. 7, 401 (2006).
  4. F. Verstraete, J. J. García-Ripoll, and J. I. Cirac, Matrix product density operators: Simulation of finite-temperature and dissipative systems, Phys. Rev. Lett. 93, 207204 (2004).
  5. M. Zwolak and G. Vidal, Mixed-state dynamics in one-dimensional quantum lattice systems: A time-dependent superoperator renormalization algorithm, Phys. Rev. Lett. 93, 207205 (2004).
  6. G. De las Cuevas, N. Schuch, D. Pérez-García, and J. Ignacio Cirac, Purifications of multipartite states: limitations and constructive methods, New J. Phys. 15, 123021 (2013).
  7. M. Berta, F. G. S. L. Brandão, J. Haegeman, V. B. Scholz, and F. Verstraete, Thermal states as convex combinations of matrix product states, Phys. Rev. B 98, 235154 (2018).
  8. Y. Liu, A. Ruiz-de Alarcón, G. Styliaris, X.-Q. Sun, D. Pérez-García, and J. I. Cirac, Parent lindbladians for matrix product density operators, Phys. Rev. Res. 8, 013210 (2026).
  9. K. Kato, Exact renormalization group flow for matrix product density operators, arXiv:2410.22696.
  10. J. Garre-Rubio, L. Lootens, and A. Molnár, Classifying phases protected by matrix product operator symmetries using matrix product states, Quantum 7, 927 (2023).
  11. L. Lootens, J. Fuchs, J. Haegeman, C. Schweigert, and F. Verstraete, Matrix product operator symmetries and intertwiners in string-nets with domain walls, SciPost Phys. 10, 053 (2021).
  12. N. Bultinck, M. Mariën, D. J. Williamson, M. B. Şahinoğlu, J. Haegeman, and F. Verstraete, Anyons and matrix product operator algebras, Ann. Phys. (NY) 378, 183 (2017).
  13. A. Molnar, A. R. de Alarcón, J. Garre-Rubio, N. Schuch, J. I. Cirac, and D. Pérez-García, Matrix product operator algebras I: representations of weak Hopf algebras and projected entangled pair states, arXiv:2204.05940.
  14. B. Pirvu, V. Murg, J. I. Cirac, and F. Verstraete, Matrix product operator representations, New J. Phys. 12, 025012 (2010).
  15. 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).
  16. M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, Area laws in quantum systems: Mutual information and correlations, Phys. Rev. Lett. 100, 070502 (2008).
  17. L. Piroli and J. I. Cirac, Quantum cellular automata, tensor networks, and area laws, Phys. Rev. Lett. 125, 190402 (2020).
  18. R. Orus, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. (NY) 349, 117 (2014).
  19. J. Eisert, M. Cramer, and M. B. Plenio, Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys. 82, 277 (2010).
  20. F. G. Brandao and M. Horodecki, Exponential decay of correlations implies area law, Commun. Math. Phys. 333, 761 (2015).
  21. F. Verstraete and J. I. Cirac, Matrix product states represent ground states faithfully, Phys. Rev. B 73, 094423 (2006).
  22. 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).
  23. B. Schumacher and R. F. Werner, Reversible quantum cellular automata, arXiv:quant-ph/0405174.
  24. D. Gross, V. Nesme, H. Vogts, and R. F. Werner, Index theory of one dimensional quantum walks and cellular automata, Commun. Math. Phys. 310, 419 (2012).
  25. T. Farrelly, A review of quantum cellular automata, Quantum 4, 368 (2020).
  26. J. I. Cirac, D. Perez-Garcia, N. Schuch, and F. Verstraete, Matrix product unitaries: structure, symmetries, and topological invariants, J. Stat. Mech. (2017) 083105.
  27. M. B. Şahinoğlu, S. K. Shukla, F. Bi, and X. Chen, Matrix product representation of locality preserving unitaries, Phys. Rev. B 98, 245122 (2018).
  28. F. Verstraete and J. I. Cirac, Continuous matrix product states for quantum fields, Phys. Rev. Lett. 104, 190405 (2010).
  29. J. Haegeman, J. I. Cirac, T. J. Osborne, and F. Verstraete, Calculus of continuous matrix product states, Phys. Rev. B 88, 085118 (2013).
  30. T. J. Osborne, J. Eisert, and F. Verstraete, Holographic quantum states, Phys. Rev. Lett. 105, 260401 (2010).
  31. J. Rincón, M. Ganahl, and G. Vidal, Lieb-liniger model with exponentially decaying interactions: A continuous matrix product state study, Phys. Rev. B 92, 115107 (2015).
  32. M. Ganahl, J. Rincón, and G. Vidal, Continuous matrix product states for quantum fields: An energy minimization algorithm, Phys. Rev. Lett. 118, 220402 (2017).
  33. A. Tilloy, Relativistic continuous matrix product states for quantum fields without cutoff, Phys. Rev. D 104, 096007 (2021).
  34. A. Tilloy, A study of the quantum Sinh-Gordon model with relativistic continuous matrix product states, arXiv:2209.05341.
  35. B. Tuybens, J. De Nardis, J. Haegeman, and F. Verstraete, Variational optimization of continuous matrix product states, Phys. Rev. Lett. 128, 020501 (2022).
  36. G. De las Cuevas, N. Schuch, D. Perez-Garcia, and J. I. Cirac, Continuum limits of matrix product states, Phys. Rev. B 98, 174303 (2018).
  37. M. Balanzó-Juandó and G. De las Cuevas, Generalized ansatz for continuous matrix product states, Phys. Rev. A 101, 052312 (2020).
  38. We note that in Ref. [48] a notion of cMPO was introduced to capture the continuous (imaginary) time evolution of a lattice system in terms of tensor networks. Our ansatz is consistent with their definition if one exchanges time by space. Furthermore, by expressing the MPOs in the Fock-like representation, it allows us to take the continuum limit directly and explicitly express it in terms of bosonic operators, in much the same way cMPS are [28].
  39. G. Styliaris, R. Trivedi, D. Perez-Garcia, and J. I. Cirac, Matrix-product unitaries: Beyond quantum cellular automata, Quantum 9, 1645 (2025).
  40. Here, we use L(x) rather than R(x) as is common in the cMPS literature [28, 29], as we will associate L(x),R(x) with left- and right-multiplication maps, respectively, when defining cMPO.
  41. See Supplemental Material at http://link.aps.org/supplemental/10.1103/d61y-cxx3 for details, which includes Refs. [44, 49].
  42. In practice, one may also need to impose additional smoothness conditions on g (such as k-differentiability) to ensure that the cMPO action on a cMPS does not worsen the well-behavedness of the initial cMPS [29].
  43. Here, we pick the convention that Θ(0)=0 (left-continuous) so that [Π(x),ψ†(x)]=0 for all x∈[−ℓ/2,ℓ/2].
  44. C. Kruszynska and B. Kraus, Local entanglability and multipartite entanglement, Phys. Rev. A 79, 052304 (2009).
  45. L. Piroli, A. Turzillo, S. K. Shukla, and J. I. Cirac, Fermionic quantum cellular automata and generalized matrix-product unitaries, J. Stat. Mech. (2021) 013107.
  46. Q. Mortier, L. Devos, L. Burgelman, B. Vanhecke, N. Bultinck, F. Verstraete, J. Haegeman, and L. Vanderstraeten, Fermionic tensor network methods, SciPost Phys. 18, 012 (2025).
  47. A. Tilloy and J. I. Cirac, Continuous tensor network states for quantum fields, Phys. Rev. X 9, 021040 (2019).
  48. W. Tang, H.-H. Tu, and L. Wang, Continuous matrix product operator approach to finite temperature quantum states, Phys. Rev. Lett. 125, 170604 (2020).
  49. R. Gilmore, Lie Groups, Physics, and Geometry: An Introduction for Physicists, Engineers and Chemists (Cambridge University Press, New York, 2008).

Outline

Information

Phys. Rev. Research 8, L022013– Published 14 April, 2026
  • Received 14 November 2025
  • Accepted 21 March 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. Open access publication funded by Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation