Quantum Graph-Based Modelling of Multi-Step Phase Dynamics in Coupled Oscillator Networks

SURBHI SHARMA; RASHMI Bhardwaj; Debabrata  Datta

doi:10.5269/bspm.81615

SURBHI SHARMA GURU GOBIND SINGH INDRAPRASTHA UNIVERSITY, DELHI,INDIA
RASHMI Bhardwaj GURU GOBIND SINGH INDRARASTHA UNIVERSITY, DELHI, INDIA
Debabrata Datta Heritage Institute of Technology, Kolkata

Abstract

Complex networked dynamical systems modeling and prediction are paramount in many fields, including neuroscience, power grids, social and ecological network modeling and prediction. Conventional graph neural networks (GNNs) are not optimal at the high dimensional and oscillatory dynamics of such systems. Quantum computing offers an alternative conceptualization of the study of such complex dynamics because quantum circuits are highly expressive and phases are encoded naturally. This paper introduces a Quantum Graph Neural Oscillator (QGNO) model that is used to simulate the multi-step phase dynamics of Kuramoto oscillator networks modeled as graphs. The framework combines parametrized quantum circuits (PQCs), phase regression (using circular mean squared error) with a decaying hybrid teacher forcing rollout process to ensure stability in long-term predictions. Experiments on a 3-node Kuramoto network demonstrate that QGNO achieves low training loss and accurately tracks phase trajectories over multiple steps, highlighting its potential for scalable quantum neural modelling of complex network dynamics.

Downloads

Download data is not yet available.

References

1. Strogatz, S. H. Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering, 2nd edn. (CRC Press, 2015).
2. Kuramoto, Y. Chemical oscillations, waves, and turbulence (Springer, 1984).
3. Acebrón, J. A., Bonilla, L. L., Pérez Vicente, C. J., Ritort, F. & Spigler, R. The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137–185 (2005).
4. Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. & Hwang, D.-U. Complex networks: Structure and dynamics. Phys. Rep. 424, 175–308 (2006).
5. Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y. & Zhou, C. Synchronization in complex networks. Phys. Rep. 469, 93–153 (2008).
6. Newman, M. Networks: An introduction (Oxford Univ. Press, 2010).
7. Nishikawa, T. & Motter, A. E. Comparative analysis of existing models for power-grid synchronization. New J. Phys. 17, 015012 (2015).
8. Chung, F. & Lu, L. Complex graphs and networks (American Mathematical Society, 2006).
9. Kipf, T. N. & Welling, M. Semi-supervised classification with graph convolutional networks. In Proc. Int. Conf. Learn. Represent. (2017).
10. Hamilton, W. L., Ying, R. & Leskovec, J. Inductive representation learning on large graphs. In Adv. Neural Inf. Process. Syst. 30 (2017).
11. Veličković, P. et al. Graph attention networks. In Proc. Int. Conf. Learn. Represent. (2018).
12. Wu, Z. et al. A comprehensive survey on graph neural networks. IEEE Trans. Neural Netw. Learn. Syst. 32, 4–24 (2021).
13. Gilmer, J., Schoenholz, S. S., Riley, P. F., Vinyals, O. & Dahl, G. E. Neural message passing for quantum chemistry. In Proc. Int. Conf. Mach. Learn. 70, 1263–1272 (2017).
14. Bronstein, M. M. et al. Geometric deep learning: Grids, groups, graphs, geodesics, and gauges. arXiv 2104.13478 (2021).
15. Battaglia, P. W. et al. Relational inductive biases, deep learning, and graph networks. arXiv 1806.01261 (2018).
16. Sanchez-Gonzalez, A. et al. Graph networks as learnable physics engines for inference and control. In Proc. Int. Conf. Mach. Learn. 80, 4470–4479 (2018).
17. Greydanus, S., Dzamba, M. & Yosinski, J. Hamiltonian neural networks. In Adv. Neural Inf. Process. Syst. 32 (2019).
18. Nielsen, M. A. & Chuang, I. L. Quantum computation and quantum information (Cambridge Univ. Press, 2010).
19. Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).
20. Schuld, M. & Petruccione, F. Supervised learning with quantum computers (Springer, 2018).
21. Farhi, E., Goldstone, J. & Gutmann, S. A quantum approximate optimization algorithm. arXiv 1411.4028 (2014).
22. Havlíček, V. et al. Supervised learning with quantum-enhanced feature spaces. Nature 567, 209–212 (2019).
23. Benedetti, M., Lloyd, E., Sack, S. & Fiorentini, M. Parameterized quantum circuits as machine learning models. Quantum Sci. Technol. 4, 043001 (2019).
24. Abbas, A. et al. The power of quantum neural networks. Nat. Comput. Sci. 1, 403–409 (2021).
25. Tacchino, F. et al. Quantum implementations of graph neural networks. Phys. Rev. A 102, 062414 (2020).
26. Wang, Y., Li, Y. & Peng, X. Variational quantum graph neural networks. arXiv 2106.06119 (2021).
27. Biamonte, J. et al. Quantum machine learning. Nature 549, 195–202 (2017).
28. Cerezo, M. et al. Variational quantum algorithms. Nat. Rev. Phys. 3, 625–644 (2021).
29. Skolik, A. et al. Quantum machine learning with graph-structured data. Quantum 6, 799 (2022).
30. Mardia, K. V. & Jupp, P. E. Directional statistics (Wiley, 2000).
31. Lathauwer, J.-P. Manifold-based learning for circular and periodic data. Signal Process. 160, 113–123 (2019).
32. Bishop, C. M. Pattern recognition and machine learning (Springer, 2006).
33. Kingma, D. P. & Ba, J. Adam: A method for stochastic optimization. In Proc. Int. Conf. Learn. Represent. (2015).
34. Pascanu, R., Mikolov, T. & Bengio, Y. On the difficulty of training recurrent neural networks. In Proc. Int. Conf. Mach. Learn. 28, 1310–1318 (2013).
35. Bengio, S. et al. Scheduled sampling for sequence prediction with recurrent neural networks. In Adv. Neural Inf. Process. Syst. 28 (2015).
36. Gal, Y. & Ghahramani, Z. Dropout as a Bayesian approximation: Representing model uncertainty in deep learning. In Proc. Int. Conf. Mach. Learn. 48, 1050–1059 (2016).
37. Dörfler, F. & Bullo, F. Synchronization in complex oscillator networks. Automatica 50, 1539–1564 (2014).
38. Kundur, P. Power system stability and control (McGraw–Hill, 1994).
39. Izhikevich, E. M. Dynamical systems in neuroscience (MIT Press, 2007).