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  • Open Access

Variational matrix product states for combinatorial optimization

Guillermo Preisser*, Conor Mc Keever, and Michael Lubasch†

  • Quantinuum, Partnership House, Carlisle Place, London SW1P 1BX, United Kingdom

  • *Contact author: guillermo.preisser@quantinuum.com
  • †Contact author: michael.lubasch@quantinuum.com
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Phys. Rev. Research 8, 033027 – Published 6 July, 2026

DOI: https://doi.org/10.1103/x7s3-ykb8

Abstract

To compute approximate solutions for combinatorial optimization problems, we describe variational methods based on the product state (PS) and matrix product state (MPS) ansätze. We perform variational energy minimization with respect to a quantum annealing Hamiltonian and utilize randomness by embedding the approaches in the metaheuristic iterated local search (ILS). The resulting quantum-inspired ILS algorithms are benchmarked on maximum cut problems of up to 50 000 variables. We show that they can outperform traditional (M)PS methods, classical ILS, the quantum approximate optimization algorithm, and other variational quantum-inspired solvers.

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References (69)

  1. C. Zhang, Y. Wu, Y. Ma, W. Song, Z. Le, Z. Cao, and J. Zhang, A review on learning to solve combinatorial optimisation problems in manufacturing, IET Collab. Intel. Manuf. 5, e12072 (2023).
  2. V. Guihaire and J.-K. Hao, Transit network design and scheduling: A global review, Transp. Res. A 42, 1251 (2008).
  3. R. Z. Farahani, E. Miandoabchi, W. Y. Szeto, and H. Rashidi, A review of urban transportation network design problems, Eur. J. Oper. Res. 229, 281 (2013).
  4. Y.-F. Liu, T.-H. Chang, M. Hong, Z. Wu, A. Man-Cho So, E. A. Jorswieck, and W. Yu, A survey of recent advances in optimization methods for wireless communications, IEEE J. Sel. Areas Commun. 42, 2992 (2024).
  5. H. R. Lourenço, O. C. Martin, and T. Stützle, Iterated local search: Framework and applications, in Handbook of Metaheuristics, edited by M. Gendreau and J.-Y. Potvin (Springer International Publishing, Cham, 2019), pp. 129–168.
  6. T. Kadowaki and H. Nishimori, Quantum annealing in the transverse Ising model, Phys. Rev. E 58, 5355 (1998).
  7. E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, Quantum computation by adiabatic evolution, arXiv:quant-ph/0001106.
  8. A. Das and B. K. Chakrabarti, Colloquium: Quantum annealing and analog quantum computation, Rev. Mod. Phys. 80, 1061 (2008).
  9. T. Albash and D. A. Lidar, Adiabatic quantum computation, Rev. Mod. Phys. 90, 015002 (2018).
  10. P. Hauke, H. G. Katzgraber, W. Lechner, H. Nishimori, and W. D. Oliver, Perspectives of quantum annealing: Methods and implementations, Rep. Prog. Phys. 83, 054401 (2020).
  11. 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).
  12. K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik, Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys. 94, 015004 (2022).
  13. J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y. Li, E. Grant, L. Wossnig, I. Rungger, G. H. Booth, and J. Tennyson, The variational quantum eigensolver: A review of methods and best practices, Phys. Rep. 986, 1 (2022).
  14. E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm, arXiv:1411.4028.
  15. L. Zhou, S.-T. Wang, S. Choi, H. Pichler, and M. D. Lukin, Quantum approximate optimization algorithm: Performance, mechanism, and implementation on near-term devices, Phys. Rev. X 10, 021067 (2020).
  16. D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, Matrix product state representations, Quantum Inf. Comput. 7, 401 (2007).
  17. F. Verstraete, V. Murg, and J. I. Cirac, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Adv. Phys. 57, 143 (2008).
  18. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. (NY) 326, 96 (2011).
  19. R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. (NY) 349, 117 (2014).
  20. M. C. Bañuls, Tensor network algorithms: A route map, Annu. Rev. Condens. Matter Phys. 14, 173 (2023).
  21. Y. Zhou, E. M. Stoudenmire, and X. Waintal, What limits the simulation of quantum computers? Phys. Rev. X 10, 041038 (2020).
  22. T. Ayral, T. Louvet, Y. Zhou, C. Lambert, E. M. Stoudenmire, and X. Waintal, Density-matrix renormalization group algorithm for simulating quantum circuits with a finite fidelity, PRX Quantum 4, 020304 (2023).
  23. R. M. Karp, Reducibility among combinatorial problems, in Complexity of Computer Computations, edited by R. E. Miller, J. W. Thatcher, and J. D. Bohlinger, The IBM Research Symposia Series, (Springer US, Boston, MA, 1972), pp. 85–103.
  24. M. X. Goemans and D. P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM 42, 1115 (1995).
  25. J. Bowles, A. Dauphin, P. Huembeli, J. Martinez, and A. Acín, Quadratic unconstrained binary optimization via quantum-inspired annealing, Phys. Rev. Appl. 18, 034016 (2022).
  26. T. Guaita, L. Hackl, T. Shi, E. Demler, and J. I. Cirac, Generalization of group-theoretic coherent states for variational calculations, Phys. Rev. Res. 3, 023090 (2021).
  27. P. M. Schindler, T. Guaita, T. Shi, E. Demler, and J. I. Cirac, Variational ansatz for the ground state of the quantum Sherrington-Kirkpatrick model, Phys. Rev. Lett. 129, 220401 (2022).
  28. L. Fioroni and V. Savona, Entanglement-assisted variational algorithm for discrete optimization problems, Commun. Phys. 8, 438 (2025).
  29. Y. Ye, Gset dataset, 2003, https://web.stanford.edu/%7Eyyye/yyye/Gset/.
  30. M. C. Bañuls, R. Orús, J. I. Latorre, A. Pérez, and P. Ruiz-Femenía, Simulation of many-qubit quantum computation with matrix product states, Phys. Rev. A 73, 022344 (2006).
  31. J. A. Smolin and G. Smith, Classical signature of quantum annealing, Front. Phys. 2, 52 (2014).
  32. B. Bauer, L. Wang, I. Pižorn, and M. Troyer, Entanglement as a resource in adiabatic quantum optimization, arXiv:1501.06914.
  33. T. Hatomura and T. Mori, Shortcuts to adiabatic classical spin dynamics mimicking quantum annealing, Phys. Rev. E 98, 032136 (2018).
  34. S. Mugel, C. Kuchkovsky, E. Sánchez, S. Fernández-Lorenzo, J. Luis-Hita, E. Lizaso, and R. Orús, Dynamic portfolio optimization with real datasets using quantum processors and quantum-inspired tensor networks, Phys. Rev. Res. 4, 013006 (2022).
  35. R. Sreedhar, P. Vikstål, M. Svensson, A. Ask, G. Johansson, and L. García-Álvarez, The quantum approximate optimization algorithm performance with low entanglement and high circuit depth, arXiv:2207.03404.
  36. M. Dupont, N. Didier, M. J. Hodson, J. E. Moore, and M. J. Reagor, Entanglement perspective on the quantum approximate optimization algorithm, Phys. Rev. A 106, 022423 (2022).
  37. M. T. Veszeli and G. Vattay, Mean field approximation for solving QUBO problems, PLoS One 17, e0273709 (2022).
  38. M. Dupont, N. Didier, M. J. Hodson, J. E. Moore, and M. J. Reagor, Calibrating the classical hardness of the quantum approximate optimization algorithm, PRX Quantum 3, 040339 (2022).
  39. G. Lami, P. Torta, G. E. Santoro, and M. Collura, Quantum annealing for neural network optimization problems: A new approach via tensor network simulations, SciPost Phys. 14, 117 (2023).
  40. J. Lopez-Piqueres, J. Chen, and A. Perdomo-Ortiz, Symmetric tensor networks for generative modeling and constrained combinatorial optimization, Mach. Learn.: Sci. Technol. 4, 035009 (2023).
  41. A. Batsheva, A. Chertkov, G. Ryzhakov, and I. Oseledets, PROTES: Probabilistic optimization with tensor sampling, in 37th Conference on Neural Information Processing Systems (NeurIPS) (Curran Associates, Inc., Red Hook, New York, 2023).
  42. J. Alcazar, M. Ghazi Vakili, C. B. Kalayci, and A. Perdomo-Ortiz, Enhancing combinatorial optimization with classical and quantum generative models, Nat. Commun. 15, 2761 (2024).
  43. J. Gardiner and J. Lopez-Piqueres, Tensor network estimation of distribution algorithms, Neurocomputing 680, 133255 (2026).
  44. J. Lopez Piqueres and J. Chen, Cons-training tensor networks: Embedding and optimization over discrete linear constraints, SciPost Phys. 18, 192 (2025).
  45. H. Nakada, K. Tanahashi, and S. Tanaka, Quick design of feasible tensor networks for constrained combinatorial optimization, Quantum 9, 1799 (2025).
  46. A. Morais, E. Osaba, I. Pastor, and I. Oregi, Comparative analysis of classical and quantum-inspired solvers: A preliminary study on the weighted max-cut problem, in Proceedings of the Genetic and Evolutionary Computation Conference Companion, GECCO’25 Companion (Association for Computing Machinery, New York, NY, 2025), pp. 2449–2457.
  47. D. Rattacaso, D. Jaschke, M. Ballarin, I. Siloi, and S. Montangero, Quantum algorithms for equational reasoning, Sci. Adv. 12, eaec2736 (2026).
  48. A. Garcia-Saez and J. I. Latorre, An exact tensor network for the 3SAT problem, Quantum Inf. Comput. 12, 283 (2012).
  49. J. D. Biamonte, J. Morton, and J. Turner, Tensor network contractions for #SAT, J. Stat. Phys. 160, 1389 (2015).
  50. Z. Zhu and H. G. Katzgraber, Do tensor renormalization group methods work for frustrated spin systems? arXiv:1903.07721.
  51. S. Kourtis, C. Chamon, E. R. Mucciolo, and A. E. Ruckenstein, Fast counting with tensor networks, SciPost Phys. 7, 060 (2019).
  52. J.-G. Liu, L. Wang, and P. Zhang, Tropical tensor network for ground states of spin glasses, Phys. Rev. Lett. 126, 090506 (2021).
  53. M. M. Rams, M. Mohseni, D. Eppens, K. Jałowiecki, and B. Gardas, Approximate optimization, sampling, and spin-glass droplet discovery with tensor networks, Phys. Rev. E 104, 025308 (2021).
  54. T. Hao, X. Huang, C. Jia, and C. Peng, A quantum-inspired tensor network algorithm for constrained combinatorial optimization problems, Front. Phys. 10, 906590 (2022).
  55. J.-G. Liu, X. Gao, M. Cain, M. D. Lukin, and S.-T. Wang, Computing solution space properties of combinatorial optimization problems via generic tensor networks, SIAM J. Sci. Comput. 45, A1239 (2023).
  56. N. Pancotti and J. Gray, One-step replica symmetry breaking in the language of tensor networks, arXiv:2306.15004.
  57. S. Yasuda, S. Sotobayashi, and Y. Minato, HOBOTAN: Efficient higher order binary optimization solver with tensor networks and PyTorch, arXiv:2407.19987.
  58. A. A. Gangat and J. Gray, Hyperoptimized approximate contraction of tensor networks for rugged-energy-landscape spin glasses on periodic square and cubic lattices, Phys. Rev. E 110, 065306 (2024).
  59. S. Patra, S. Singh, and R. Orús, Projected entangled pair states with flexible geometry, Phys. Rev. Res. 7, L012002 (2025).
  60. A. M. Ali, Explicit solution equation for every combinatorial problem via tensor networks: MeLoCoToN, arXiv:2502.05981.
  61. A. M. Dziubyna, T. Śmierzchalski, B. Gardas, M. M. Rams, and M. Mohseni, Limitations of tensor-network approaches for optimization and sampling: A comparison to quantum and classical Ising machines, Phys. Rev. Appl. 23, 054049 (2025).
  62. M. Tesoro, I. Siloi, D. Jaschke, G. Magnifico, and S. Montangero, Integer factorization via tensor-network Schnorr's sieving, Phys. Rev. A 113, 032418 (2026).
  63. M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor software library for tensor network calculations, SciPost Phys. Codebases 4 (2022).
  64. M. Fishman, S. R. White, and E. M. Stoudenmire, Codebase release 0.3 for ITensor, SciPost Phys. Codebases 4-r0.3 (2022).
  65. F. Fröwis, V. Nebendahl, and W. Dür, Tensor operators: Constructions and applications for long-range interaction systems, Phys. Rev. A 81, 062337 (2010).
  66. F. Barahona, M. Grötschel, M. Jünger, and G. Reinelt, An application of combinatorial optimization to statistical physics and circuit layout design, Oper. Res. 36, 493 (1988).
  67. A. Lucas, Ising formulations of many NP problems, Front. Phys. 2, 5 (2014).
  68. A. Chertkov, G. Ryzhakov, G. Novikov, and I. Oseledets, Optimization of functions given in the tensor train format, arXiv:2209.14808.
  69. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University Press, Cambridge, UK, 2007).

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Phys. Rev. Research 8, 033027– Published 6 July, 2026
  • Received 24 December 2025
  • Revised 3 April 2026
  • Accepted 9 June 2026
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