Free-energy machine for combinatorial optimization

5 min read Original article ↗

References

  1. Du, D. & Pardalos, P. M. Handbook of Combinatorial Optimization vol. 4 (Springer Science & Business Media, 1998).

  2. Arora, S. & Barak, B. Computational Complexity: A Modern Approach (Cambridge Univ. Press, 2009).

  3. Kirkpatrick, S., Gelatt Jr, C. D. & Vecchi, M. P. Optimization by simulated annealing. Science 220, 671–680 (1983).

  4. Selman, B. et al. Noise strategies for improving local search. AAAI 94, 337–343 (1994).

    Google Scholar 

  5. Glover, F. & Laguna, M. Tabu Search (Springer, 1998).

  6. Boettcher, S. & Percus, A. G. Optimization with extremal dynamics. Phys. Rev. Lett. 86, 5211-5214 (2001).

  7. Barahona, F. On the computational complexity of Ising spin glass models. J. Phys. A 15, 3241 (1982).

    Article  MathSciNet  Google Scholar 

  8. Tiunov, E. S., Ulanov, A. E. & Lvovsky, A. Annealing by simulating the coherent Ising machine. Optics Express 27, 10288–10295 (2019).

  9. King, A. D., Bernoudy, W., King, J., Berkley, A. J. & Lanting, T. Emulating the coherent Ising machine with a mean-field algorithm. Preprint at https://arxiv.org/abs/1806.08422 (2018).

  10. Goto, H., Tatsumura, K. & Dixon, A. R. Combinatorial optimization by simulating adiabatic bifurcations in nonlinear Hamiltonian systems. Sci. Adv. 5, eaav2372 (2019).

    Article  Google Scholar 

  11. Goto, H. et al. High-performance combinatorial optimization based on classical mechanics. Sci. Adv. 7, eabe7953 (2021).

    Article  Google Scholar 

  12. Johnson, M. W. et al. Quantum annealing with manufactured spins. Nature 473, 194–198 (2011).

    Article  Google Scholar 

  13. Inagaki, T. et al. Large-scale Ising spin network based on degenerate optical parametric oscillators. Nat. Photonics 10, 415–419 (2016).

    Article  Google Scholar 

  14. Honjo, T. et al. 100,000-spin coherent Ising machine. Sci. Adv. 7, eabh0952 (2021).

    Article  Google Scholar 

  15. Pierangeli, D., Marcucci, G. & Conti, C. Large-scale photonic Ising machine by spatial light modulation. Phys. Rev. Lett. 122, 213902 (2019).

    Article  Google Scholar 

  16. Cai, F. et al. Power-efficient combinatorial optimization using intrinsic noise in memristor Hopfield neural networks. Nat. Electron. 3, 409–418 (2020).

    Article  Google Scholar 

  17. Mohseni, N., McMahon, P. L. & Byrnes, T. Ising machines as hardware solvers of combinatorial optimization problems. Nat. Rev. Phys. 4, 363–379 (2022).

    Article  Google Scholar 

  18. Kochenberger, G. et al. The unconstrained binary quadratic programming problem: a survey. J. Combin. Optim. 28, 58–81 (2014).

    Article  MathSciNet  Google Scholar 

  19. Lucas, A. Ising formulations of many NP problems. Front. Phys. 2, 5 (2014).

    Article  Google Scholar 

  20. Gardner, E. Spin glasses with p-spin interactions. Nucl. Phys. B 257, 747–765 (1985).

    Article  MathSciNet  Google Scholar 

  21. Karp, R. M. Reducibility among Combinatorial Problems (Springer, 2010).

  22. Wu, F.-Y. The Potts model. Rev. Mod. Phys. 54, 235–268 (1982).

    Article  MathSciNet  Google Scholar 

  23. Jensen, T. R. & Toft, B. Graph Coloring Problems (John Wiley & Sons, 2011).

  24. Newman, M. E. Modularity and community structure in networks. Proc. Natl Acad. Sci. USA 103, 8577–8582 (2006).

    Article  Google Scholar 

  25. Papadimitriou, C. & Steiglitz, K. Combinatorial Optimization: Algorithms and Complexity (Dover Publications, 1998).

  26. Mézard, M., Parisi, G. & Zecchina, R. Analytic and algorithmic solution of random satisfiability problems. Science 297, 812–815 (2002).

  27. Chermoshentsev, D. A. et al. Polynomial unconstrained binary optimisation inspired by optical simulation. Preprint at https://arxiv.org/abs/2106.13167 (2021).

  28. Bybee, C. et al. Efficient optimization with higher-order Ising machines. Nat. Commun. 14, 6033 (2023).

    Article  Google Scholar 

  29. Kanao, T. & Goto, H. Simulated bifurcation for higher-order cost functions. Appl. Phys. Express 16, 014501 (2022).

    Article  Google Scholar 

  30. Reifenstein, S. et al. Coherent SAT solvers: a tutorial. Adv. Opt. Photonics 15, 385–441 (2023).

    Article  Google Scholar 

  31. Böhm, F., Vaerenbergh, T. V., Verschaffelt, G. & Van der Sande, G. Order-of-magnitude differences in computational performance of analog Ising machines induced by the choice of nonlinearity. Commun. Phys. 4, 149 (2021).

    Article  Google Scholar 

  32. Inagaki, T. et al. A coherent Ising machine for 2000-node optimization problems. Science 354, 603606 (2016).

    Article  Google Scholar 

  33. Schuetz, M. J., Brubaker, J. K. & Katzgraber, H. G. Combinatorial optimization with physics-inspired graph neural networks. Nat. Mach. Intell. 4, 367–377 (2022).

    Article  Google Scholar 

  34. Ushijima-Mwesigwa, H., Negre, C. F. & Mniszewski, S. M. Graph partitioning using quantum annealing on the D-wave system. In Proc. Second International Workshop on Post Moores Era Supercomputing 22–29 (Association for Computing Machinery, 2017).

  35. Walshaw, C. The graph partitioning archive. https://chriswalshaw.co.uk/partition/ (2023).

  36. Karypis, G. & Kumar, V. A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20, 359–392 (1998).

    Article  MathSciNet  Google Scholar 

  37. Sanders, P. & Schulz, C. Think locally, act globally: highly balanced graph partitioning. In Experimental Algorithms. SEA 2013. Lecture Notes in Computer Science (eds Bonifaci, V., Demetrescu, C. & Marchetti-Spaccamela, A.) 164–175 (Springer, 2013).

  38. Schulz, C. KaHIP. GitHub https://github.com/KaHIP/KaHIP (2025).

  39. Argelich, J., Li, C. M., Manya, F. & Planes, J. Max-SAT 2016. http://www.maxsat.udl.cat/16/benchmarks/index.html (2016).

  40. Molnár, B., Molnár, F., Varga, M., Toroczkai, Z. & Ercsey-Ravasz, M. A continuous-time MaxSAT solver with high analog performance. Nat. Commun. 9, 4864 (2018).

    Article  Google Scholar 

  41. Wu, D., Wang, L. & Zhang, P. Solving statistical mechanics using variational autoregressive networks. Phys. Rev. Lett. 122, 080602 (2019).

    Article  Google Scholar 

  42. Hibat-Allah, M., Inack, E. M., Wiersema, R., Melko, R. G. & Carrasquilla, J. Variational neural annealing. Nat. Mach. Intell. 3, 952–961 (2021).

    Article  Google Scholar 

  43. Holland, P. W., Laskey, K. B. & Leinhardt, S. Stochastic blockmodels: first steps. Soc. Netw. 5, 109–137 (1983).

    Article  MathSciNet  Google Scholar 

  44. Karrer, B. & Newman, M. E. Stochastic blockmodels and community structure in networks. Phys. Rev. E 83, 016107 (2011).

    Article  MathSciNet  Google Scholar 

  45. Cook, S. A. The complexity of theorem-proving procedures. In Logic, Automata, and Computational Complexity: The Works of Stephen A. Cook (ed Kapron, B. M.) 143-152 (Association for Computing Machinery, 2023).

  46. Mézard, M. et al. Replica symmetry breaking and the nature of the spin glass phase. J. Phys. 45, 843–854 (1984).

    Article  MathSciNet  Google Scholar 

  47. Mézard, M., Parisi, G. & Virasoro, M. A. Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications Vol. 9 (World Scientific, 1987).

  48. Bilbro, G. et al. Optimization by mean field annealing. In Advances in Neural Information Processing Systems (ed Touretzky, D.) (Morgan Kaufmann, 1988).

  49. Shen, Z. ZisongShen/Free_Energy_Machine. Zenodo https://doi.org/10.5281/zenodo.14874189 (2025).

Download references