- 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
  • Open Access

Gauge-fixing quantum density operators at scale

Amit Jamadagni* and Eugene Dumitrescu†

  • *Contact author: gangapurama@ornl.gov
  • †Contact author: dumitrescuef@ornl.gov
PDF

Phys. Rev. Research 7, 023311 – Published 26 June, 2025

DOI: https://doi.org/10.1103/jxq6-k9ky

Abstract

We provide a theory, algorithms, and simulations of nonequilibrium quantum systems using a one-dimensional (1D) completely positive (CP), matrix-product (MP) density-operator (ρ) representation. By generalizing the matrix product state's orthogonality center, to additionally store positive classical mixture correlations, the MPρ factorization naturally emerges. In this setting, we analytically and numerically examine the virtual gauge freedoms associated with the representation of quantum density operators. Based on this perspective, we simplify algorithms in certain limits to speed up the integration of the canonical-form master-equation dynamics. This enables us to quickly evolve under the dynamics of two-body quantum channels without resorting to optimization-based methods. In addition to this technical advance, we also scale up numerical examples and discuss implications for accurately modeling hardware architectures and predicting their performance in the near term. This includes an example of the quantum to classical transition of informationally leaky, i.e., decohering, qubits. In this setting, because of loss from environmental interactions, nonlocal complex coherence correlations are converted into global incoherent classical statistical mixture correlations. Lastly, the representation of both global and local correlations is discussed. We expect this work to have applications in additional nonequilibrium settings, beyond qubit engineering.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (34)

  1. Long ranged interactions originate from localized sources and the resulting forces propagate at, or below, the speed of light.
  2. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  3. M. P. Zaletel and F. Pollmann, Isometric tensor network states in two dimensions, Phys. Rev. Lett. 124, 037201 (2020).
  4. D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, Matrix product state representations, Quantum Inf. Comput. 7, 401 (2007).
  5. Pure states have dim[p⃗]=p1=1; thermal states ρβ∝e−βH, at finite temperature T=(βkB)−1, have dim[p⃗]=dim[H].
  6. F. Verstraete, V. Murg, and J. Cirac, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Adv. Phys. 57, 143 (2008).
  7. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. (NY) 326, 96 (2011).
  8. 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).
  9. 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).
  10. A. H. Werner, D. Jaschke, P. Silvi, M. Kliesch, T. Calarco, J. Eisert, and S. Montangero, Positive tensor network approach for simulating open quantum many-body systems, Phys. Rev. Lett. 116, 237201 (2016).
  11. M. Ambroise, A. Thomas, and B. Corentin, Enabling large-depth simulation of noisy quantum circuits with positive tensor networks, arXiv:2403.00152.
  12. E. Godinez-Ramirez, R. Milbradt, and C. B. Mendl, A Riemannian approach to the Lindbladian dynamics of a locally purified tensor network, arXiv:2409.08127.
  13. S. Cheng, C. Cao, C. Zhang, Y. Liu, S.-Y. Hou, P. Xu, and B. Zeng, Simulating noisy quantum circuits with matrix product density operators, Phys. Rev. Res. 3, 023005 (2021).
  14. Y.-C. Liang, Y.-H. Yeh, P. E. M. F. Mendonça, R. Y. Teh, M. D. Reid, and P. D. Drummond, Quantum fidelity measures for mixed states, Rep. Prog. Phys. 82, 076001 (2019).
  15. M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor software library for tensor network calculations, SciPost Phys. Codebases 4 (2022).
  16. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, 2011).
  17. For example, imagine an atomic trap where one atom is ejected from the trap. In this case one may not even know which atom was lost, nullifying the notion of a third, flagged erasure state.
  18. K. Kato, Exact renormalization group flow for matrix product density operators, arXiv:2410.22696.
  19. S. Srinivasan, S. Adhikary, J. Miller, B. Pokharel, G. Rabusseau, and B. Boots, Towards a trace-preserving tensor network representation of quantum channels, in First Workshop on Quantum Tensor Networks in Machine Learning in Conjunction with NeurIPS (2021).
  20. J. Shah, C. Fechisin, Y.-X. Wang, J. T. Iosue, J. D. Watson, Y.-Q. Wang, B. Ware, A. V. Gorshkov, and C.-J. Lin, Instability of steady-state mixed-state symmetry-protected topological order to strong-to-weak spontaneous symmetry breaking, arXiv:2410.12900.
  21. S. Sun, J.-H. Zhang, Z. Bi, and Y. You, Holographic view of mixed-state symmetry-protected topological phases in open quantum systems, PRX Quantum 6, 020333 (2025).
  22. Y.-H. Chen and T. Grover, Separability transitions in topological states induced by local decoherence, Phys. Rev. Lett. 132, 170602 (2024).
  23. D. Wanisch, N. Reinić, D. Jaschke, S. Montangero, and P. Silvi, Entanglement transitions in a boundary-driven open quantum many-body system, arXiv:2502.18372.
  24. Z. Chen, Y. Bao, and S. Choi, Optimized trajectory unraveling for classical simulation of noisy quantum dynamics, Phys. Rev. Lett. 133, 230403 (2024).
  25. E. Dumitrescu, Tree tensor network approach to simulating Shor's algorithm, Phys. Rev. A 96, 062322 (2017).
  26. L. Arceci, P. Silvi, and S. Montangero, Entanglement of formation of mixed many-body quantum states via tree tensor operators, Phys. Rev. Lett. 128, 040501 (2022).
  27. M. Ballarin, Efficient tensor network emulation of quantum computers: Large scale algorithm performance analysis, Ph.D. thesis, Universita degli Studi di Padova, Padova PD, Italy, 2025.
  28. A. Dubey, Z. Zeybek, and P. Schmelcher, Simulating quantum circuits with tree tensor networks using density-matrix renormalization group algorithm, arXiv:2504.16718.
  29. D. Sulz, C. Lubich, G. Ceruti, I. Lesanovsky, and F. Carollo, Numerical simulations of long-range open quantum many-body dynamics with tree tensor networks, arXiv:2304.06075.
  30. W. Li, J. Ren, H. Yang, H. Wang, and Z. Shuai, Optimal tree tensor network operators for tensor network simulations: Applications to open quantum systems, J. Chem. Phys. 161, 054116 (2024).
  31. S. Mangini, M. Cattaneo, D. Cavalcanti, S. Filippov, M. A. C. Rossi, and G. García-Pérez, Tensor network noise characterization for near-term quantum computers, Phys. Rev. Res. 6, 033217 (2024).
  32. N. Lambert, E. Giguère, P. Menczel, B. Li, P. Hopf, G. Suárez, M. Gali, J. Lishman, R. Gadhvi, R. Agarwal et al., QuTip 5: The quantum toolbox in Python, arXiv:2412.04705.
  33. F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F, G. S. L. Brandao, D. A. Buell et al., Quantum supremacy using a programmable superconducting processor, Nature (London) 574, 505 (2019).
  34. C. Huang, F. Zhang, M. Newman, J. Cai, X. Gao, Z. Tian, J. Wu, H. Xu, H. Yu, B. Yuan et al., Classical simulation of quantum supremacy circuits, arXiv:2005.06787.

Outline

Information

Phys. Rev. Research 7, 023311– Published 26 June, 2025
  • Received 9 December 2024
  • Accepted 12 June 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