Spiking neural network dynamic system modeling for computation of quantum annealing and its convergence analysis


Quantum annealing algorithm is a classical natural computing method for skeuomorphs, and its algorithm design and application research have achieved fruitful results, so it is widely integrated into the research of modern intelligent optimization algorithm. This paper attempts to use the spiking neural network (SNN) dynamic system model to simulate the operation mechanism and convergence of the quantum annealing algorithm, and compares the process of searching the optimal solution to the elastic motion in the quantum tunneling field, and the change of function value during the operation of the algorithm is the simple harmonic vibration or damped vibration of quantum. Spiking neural network dynamic system model simulates the human brain by incorporating synaptic state and time components into their operational models, which represents the process of quantum fluctuations. The local convergence in the early stage and the global convergence in the late stage of the algorithm are proved by using the qualitative theory of ordinary differential equations to solve and analyze the dynamic system model, and a reasonable theoretical explanation is given for its operation mechanism. Several typical test problems are selected for experimental verification. The experimental results show that the numerical convergence curve is consistent with the convergence conclusion of theoretical analysis. Finally, both theoretical and experimental analyses show that the SNN dynamic system model established in this paper is suitable to describe the quantum annealing algorithm for optimization.

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  1. 1.

    Stella, L., Santoro, G.E.: Quantum annealing of an Ising spin-glass by Green’s function Monte Carlo. Phys. Rev. E 75(3), 1–6 (2007)

    Article  Google Scholar 

  2. 2.

    Chatterjee, O., Chakrabartty, S.: Decentralized global optimization based on a growth transform dynamical system model. IEEE Trans. Neural Netw. Learn. Syst. 29(12), 6052–6061 (2018)

    Article  Google Scholar 

  3. 3.

    Kumar, V., Bass, G., Tomlin, C., et al.: Quantum annealing for combinatorial clustering. Quantum Inf. Process. 17(2), 1–14 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Brady, L.T., van Dam, W.: Necessary adiabatic run times in quantum optimization. Phys. Rev. A 95(3), 1–5 (2017)

    Article  Google Scholar 

  5. 5.

    Inack, E.M., Pilati, S.: Simulated quantum annealing of double-well and multiwell potentials. Phys. Rev. E 92(5), 1–10 (2015)

    Article  Google Scholar 

  6. 6.

    Somma, R.D., Boixo, S., Barnum, H., et al.: Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101(13), 1–4 (2008)

    Article  Google Scholar 

  7. 7.

    Yu, C., Heidari, A.A., Chen, H.: A quantum-behaved simulated annealing enhanced moth-flame optimization method. Appl. Math. Model. 87, 1–19 (2020)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Bodha, K.D., Yadav, V.K., Mukherjee, V.: Formulation and application of quantum-inspired tidal firefly technique for multiple-objective mixed cost-effective emission dispatch. Neural Comput. Appl. 32, 1–16 (2019)

    Google Scholar 

  9. 9.

    Raj, K.H., Setia, R.: Quantum seeded evolutionary computational technique for constrained optimization in engineering design and manufacturing. Struct. Multidiscip. Optim. 55(3), 751–766 (2017)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Athalye, V.R., Carmena, J.M., Costa, R.M.: Neural reinforcement: re-entering and refining neural dynamics leading to desirable outcomes. Curr. Opin. Neurobiol. 60, 145–154 (2020)

    Article  Google Scholar 

  11. 11.

    Woodward, A., Froese, T., Ikegami, T.: Neural coordination can be enhanced by occasional interruption of normal firing patterns: a self-optimizing spiking neural network model. Neural Netw. 62, 39–46 (2015)

    MATH  Article  Google Scholar 

  12. 12.

    Lee, W.W., Kukreja, S.L., Thakor, N.V.: Cone: convex-optimized-synaptic efficacies for temporally precise spike mapping. IEEE Trans. Neural Netw. Learn. Syst. 28(4), 849–861 (2016)

    Article  Google Scholar 

  13. 13.

    Zhao, J., Zurada, J.M., Yang, J., et al.: The convergence analysis of SpikeProp algorithm with smoothing L1/2 regularization. Neural Netw. 103, 19–28 (2018)

    MATH  Article  Google Scholar 

  14. 14.

    Chancellor, N.: Modernizing quantum annealing using local searches. New J. Phys. 19(2), 1–19 (2017)

    Article  Google Scholar 

  15. 15.

    Wang, P., Ye, X., Li, B., et al.: Multi-scale quantum harmonic oscillator algorithm for global numerical optimization. Appl. Soft Comput. 69, 655–670 (2018)

    Article  Google Scholar 

  16. 16.

    Miyahara, H., Tsumura, K., Sughiyama, Y.: Deterministic quantum annealing expectation-maximization algorithm. J. Stat. Mech. Theory Exp. 2017(11), 1–23 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Sato, I., Tanaka, S., Kurihara, K., et al.: Quantum annealing for Dirichlet process mixture models with applications to network clustering. Neurocomputing 121, 523–531 (2013)

    Article  Google Scholar 

  18. 18.

    Wang, Y., Wu, S., Zou, J.: Quantum annealing with Markov chain Monte Carlo simulations and D-wave quantum computers. Stat. Sci. 31, 362–398 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Franzke, B., Kosko, B.: Noise can speed Markov chain Monte Carlo estimation and quantum annealing. Phys. Rev. E 100(5), 1–18 (2019)

    Article  Google Scholar 

  20. 20.

    Kadowaki, T.: Dynamics of open quantum systems by interpolation of von Neumann and classical master equations, and its application to quantum annealing. Phys. Rev. A 97(2), 1–9 (2018)

    Article  Google Scholar 

  21. 21.

    Hatomura, T., Mori, T.: Shortcuts to adiabatic classical spin dynamics mimicking quantum annealing. Phys. Rev. E 98(3), 1–6 (2018)

    Article  Google Scholar 

  22. 22.

    Zhou, L., Wang, S.T., Choi, S., et al.: Quantum approximate optimization algorithm: performance, mechanism, and implementation on near-term devices. Phys. Rev. X 10(2), 1–23 (2020)

    Google Scholar 

  23. 23.

    Ye, X., Wang, P., Xin, G., et al.: Multi-scale quantum harmonic oscillator algorithm with truncated mean stabilization strategy for global numerical optimization problems. IEEE Access 7, 18926–18939 (2019)

    Article  Google Scholar 

  24. 24.

    Jonke, Z., Habenschuss, S., Maass, W.: Solving constraint satisfaction problems with networks of spiking neurons. Front. Neurosci. 10, 1–16 (2016)

    Article  Google Scholar 

  25. 25.

    Sangiovanni-Vincentelli, A., Chen, L.K., Chua, L.: An efficient heuristic cluster algorithm for tearing large-scale networks. IEEE Trans. Circuits Syst. 24(12), 709–717 (1977)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Wang, N., Guo, G., Wang, B., et al.: Traffic clustering algorithm of urban data brain based on a hybrid-augmented architecture of quantum annealing and brain-inspired cognitive computing. Tsinghua Sci. Technol. 25(6), 813–825 (2020)

    Article  Google Scholar 

  27. 27.

    Guo, J., Yin, Y., Hu, X., et al.: Self-similar network model for fractional-order neuronal spiking: implications of dendritic spine functions. Nonlinear Dyn. 100, 1–15 (2020)

    Article  Google Scholar 

  28. 28.

    Venegas-Andraca, S.E., Cruz-Santos, W., McGeoch, C., et al.: A cross-disciplinary introduction to quantum annealing-based algorithms. Contemp. Phys. 59(2), 174–197 (2018)

    ADS  Article  Google Scholar 

  29. 29.

    Waidyasooriya, H.M., Hariyama, M.: A GPU-based quantum annealing simulator for fully-connected ising models utilizing spatial and temporal parallelism. IEEE Access 8, 67929–67939 (2020)

    Article  Google Scholar 

  30. 30.

    La Cour, B.R., Troupe, J.E., Mark, H.M.: Classical simulated annealing using quantum analogues. J. Stat. Phys. 164(4), 772–784 (2016)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Chang, C.C., Gambhir, A., Humble, T.S., et al.: Quantum annealing for systems of polynomial equations. Sci. Rep. 9(1), 1–9 (2019)

    Article  Google Scholar 

  32. 32.

    Huse, D.A., Fisher, D.S.: Residual energies after slow cooling of disordered systems. Phys. Rev. Lett. 57(17), 2203–2206 (1986)

    ADS  Article  Google Scholar 

  33. 33.

    Shahriari, B., Swersky, K., Wang, Z., et al.: Taking the human out of the loop: a review of Bayesian optimization. Proc. IEEE 104(1), 148–175 (2015)

    Article  Google Scholar 

  34. 34.

    Anwani, N., Rajendran, B.: Training multi-layer spiking neural networks using NormAD based spatio-temporal error backpropagation[J]. Neurocomputing 380, 67–77 (2020)

    Article  Google Scholar 

  35. 35.

    Morita, S., Nishimori, H.: Mathematical foundation of quantum annealing. J. Math. Phys. 49(12), 1–47 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Chua, L.O.: Global optimization: a Naive approach. IEEE Trans. Circuits Syst. 37(7), 966–969 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Morley, J.G., Chancellor, N., Bose, S., et al.: Quantum search with hybrid adiabatic-quantum-walk algorithms and realistic noise. Phys. Rev. A 99(2), 1–22 (2019)

    Article  Google Scholar 

  38. 38.

    Graß, T., Lewenstein, M.: Hybrid annealing: coupling a quantum simulator to a classical computer. Phys. Rev. A 95(5), 1–6 (2017)

    Article  Google Scholar 

  39. 39.

    Pastorello, D., Blanzieri, E.: Quantum annealing learning search for solving QUBO problems. Quantum Inf. Process. 18(10), 1–17 (2019)

    Article  Google Scholar 

  40. 40.

    Yang, K., Duan, Q., Wang, Y., et al.: Transiently chaotic simulated annealing based on intrinsic nonlinearity of memristors for efficient solution of optimization problems. Sci. Adv. 6(33), 1–9 (2020)

    ADS  Google Scholar 

  41. 41.

    King, J., Yarkoni, S., Raymond, J., et al.: Quantum annealing amid local ruggedness and global frustration. J. Phys. Soc. Jpn. 88(6), 1–12 (2019)

    Article  Google Scholar 

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This work is supported by the National Natural Science Foundation of China (Key Program) under Grant 61836010. The authors would like to thank their laboratory team member’s assistance.

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Correspondence to Chenhui Zhao or Donghui Guo.

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Zhao, C., Huang, Z. & Guo, D. Spiking neural network dynamic system modeling for computation of quantum annealing and its convergence analysis. Quantum Inf Process 20, 70 (2021). https://doi.org/10.1007/s11128-021-03003-5

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  • Convergence stability
  • Spiking neural network (SNN) model
  • Quantum annealing (QA)
  • Local convergence
  • Global convergence