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Exploring the performance of superposition of product states: From one-dimensional to three-dimensional quantum spin systems

Apimuk Sornsaeng1,2, Itai Arad2, and Dario Poletti1,2,3,4,*

  • *Contact author: dario_poletti@sutd.edu.sg
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Phys. Rev. E 113, 035309 – Published 24 March, 2026

DOI: https://doi.org/10.1103/2p2k-3j2s

Abstract

Tensor networks are one of the best available tools to study many-body quantum systems. Tensor networks are particularly suitable for one-dimensional local Hamiltonians, while their performance for generic geometries is mainly limited by two aspects: the limitation in expressive power and the approximate extraction of information. Here we investigate the performance of the superposition-of-product-states (SPS) ansatz, a variational framework structurally related to canonical polyadic tensor decomposition. While the ansatz is generally not as expressive as tensor networks, it has the properties that it (i) is structurally independent of the geometry of the system, (ii) allows accurate extraction of information, (iii) is readily parallelizable, and (iv) allows analytical shortcuts. We first study the typical properties of the SPS ansatz for spin-12 systems, including its entanglement entropy and its trainability. We then use this ansatz for a ground-state search in tilted Ising models, including one-dimensional and three-dimensional models with short- and long-range interaction, and a random network, demonstrating that the SPS ansatz can attain high accuracy.

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References (38)

  1. L. D. Landau, E. M. Lifshitz, J. B. Sykes, J. S. Bell, and M. E. Rose, Quantum Mechanics, Non-Relativistic Theory, Vol. 3 of Course of Theoretical Physics (American Institute of Physics, 1958).
  2. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2010).
  3. W. M. Foulkes, L. Mitas, R. Needs, and G. Rajagopal, Quantum Monte Carlo simulations of solids, Rev. Mod. Phys. 73, 33 (2001).
  4. G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017).
  5. D. Wu, R. Rossi, F. Vicentini, N. Astrakhantsev, et al., Variational benchmarks for quantum many-body problems, Science 386, 296 (2024).
  6. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. NY 326, 96 (2011).
  7. R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. NY 349, 117 (2014).
  8. F. Verstraete and J. I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, arXiv:cond-mat/0407066.
  9. T. Nishino and K. Okunishi, Corner transfer matrix renormalization group method, J. Phys. Soc. Jpn. 65, 891 (1996).
  10. R. Orús and G. Vidal, Simulation of two-dimensional quantum systems on an infinite lattice revisited: Corner transfer matrix for tensor contraction, Phys. Rev. B 80, 094403 (2009).
  11. J. Jordan, R. Orús, G. Vidal, F. Verstraete, and J. I. Cirac, Classical simulation of infinite-size quantum lattice systems in two spatial dimensions, Phys. Rev. Lett. 101, 250602 (2008).
  12. M. Weyrauch and M. V. Rakov, Efficient MPS algorithm for periodic boundary conditions and applications, Ukr. J. Phys. 58, 657 (2013).
  13. E. Robeva and A. Seigal, Duality of graphical models and tensor networks, Inf. Inference J. IMA 8, 273 (2019).
  14. R. Alkabetz and I. Arad, Tensor networks contraction and the belief propagation algorithm, Phys. Rev. Res. 3, 023073 (2021).
  15. S. Sahu and B. Swingle, Efficient tensor network simulation of quantum many-body physics on sparse graphs, arXiv:2206.04701.
  16. C. Guo, D. Poletti, and I. Arad, Block belief propagation algorithm for two-dimensional tensor networks, Phys. Rev. B 108, 125111 (2023).
  17. A. W. Sandvik and G. Vidal, Variational quantum Monte Carlo simulations with tensor-network states, Phys. Rev. Lett. 99, 220602 (2007).
  18. L. Wang, I. Pižorn, and F. Verstraete, Monte Carlo simulation with tensor network states, Phys. Rev. B 83, 134421 (2011).
  19. A. J. Ferris and G. Vidal, Perfect sampling with unitary tensor networks, Phys. Rev. B 85, 165146 (2012).
  20. D. S. Rokhsar and B. G. Kotliar, Gutzwiller projection for bosons, Phys. Rev. B 44, 10328 (1991).
  21. W. Krauth, M. Caffarel, and J.-P. Bouchaud, Gutzwiller wave function for a model of strongly interacting bosons, Phys. Rev. B 45, 3137 (1992).
  22. D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Cold bosonic atoms in optical lattices, Phys. Rev. Lett. 81, 3108 (1998).
  23. F. L. Hitchcock, The expression of a tensor or a polyadic as a sum of products, J. Math. Phys. 6, 164 (1927).
  24. J. D. Carroll and J.-J. Chang, Analysis of individual differences in multidimensional scaling via an N-way generalization of “Eckart-Young” decomposition, Psychometrika 35, 283 (1970).
  25. R. A. Harshman, Foundations of the PARAFAC procedure: Models and conditions for an “explanatory” multi-modal factor analysis, UCLA Working Papers Phonetics 16, 84 (1970).
  26. P. Facchi, S. Pascazio, and F. V. Pepe, Quantum typicality and initial conditions, Phys. Scr. 90, 074057 (2015).
  27. B. Collins, C. E. González-Guillén, and D. Pérez-García, Matrix product states, random matrix theory and the principle of maximum entropy, Commun. Math. Phys. 320, 663 (2013).
  28. J. Haferkamp, C. Bertoni, I. Roth, and J. Eisert, Emergent statistical mechanics from properties of disordered random matrix product states, PRX Quantum 2, 040308 (2021).
  29. 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).
  30. M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational quantum algorithms, Nat. Rev. Phys. 3, 625 (2021).
  31. M. Larocca, S. Thanasilp, S. Wang, K. Sharma, J. Biamonte, P. J. Coles, L. Cincio, J. R. McClean, Z. Holmes, and M. Cerezo, Barren plateaus in variational quantum computing, Nat. Rev. Phys. 7, 174 (2025).
  32. K. Srimahajariyapong, S. Thanasilp, and T. Chotibut, Connecting phases of matter to the flatness of the loss landscape in analog variational quantum algorithms, Communications Physics (2026), doi:10.1038/s42005-026-02528-4.
  33. T. Barthel and Q. Miao, Absence of barren plateaus and scaling of gradients in the energy optimization of isometric tensor network states, Commun. Math. Phys. 406, 86 (2025).
  34. I. Loshchilov and F. Hutter, Decoupled weight decay regularization, arXiv:1711.05101.
  35. http://nscc.sg.
  36. J. Hauschild, J. Unfried, S. Anand, B. Andrews, M. Bintz, U. Borla, S. Divic, M. Drescher, J. Geiger, M. Hefel, K. Hémery, W. Kadow, J. Kemp, N. Kirchner, V. S. Liu, G. Möller, D. Parker, M. Rader, A. Romen, S. Scalet, et al., Tensor network Python (TeNPy) version 1, SciPost Phys. Codebases 41 (2024).
  37. M. Florido-Llinàs, Á. M. Alhambra, R. Trivedi, N. Schuch, D. Pérez-García, and J. I. Cirac, The product structure of matrix product states under permutations, PRX Quantum 6, 040338 (2025).
  38. https://github.com/SUTD-MDQS/SPS.

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Phys. Rev. E 113, 035309– Published 24 March, 2026
  • Received 17 November 2025
  • Accepted 6 February 2026
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