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Limit shapes and large deviations in classical and quantum neural networks

Author

Listed:
  • Rahmati, Mohammad Reza
  • Arce, Fernando
  • Gómez-Flores, Wilfrido

Abstract

This work develops a unified probabilistic framework in which the training dynamics of both classical and quantum neural networks are governed by limit shapes: deterministic macroscopic profiles that arise from the collective averaging of microscopic stochastic fluctuations. Using the Law of Large Numbers, the Central Limit Theorem, and the Large Deviation Principle, we show that the parameter trajectories generated by stochastic gradient descent concentrate exponentially around the minimizer of an explicit rate functional. This minimizer is the classical limit shape, and fluctuations around it are Gaussian with covariance determined by the Hessian of the loss.

Suggested Citation

  • Rahmati, Mohammad Reza & Arce, Fernando & Gómez-Flores, Wilfrido, 2026. "Limit shapes and large deviations in classical and quantum neural networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 683(C).
    • Handle: RePEc:eee:phsmap:v:683:y:2026:i:c:s0378437125008337
    DOI: 10.1016/j.physa.2025.131181
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    References listed on IDEAS

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    1. Samson Wang & Enrico Fontana & M. Cerezo & Kunal Sharma & Akira Sone & Lukasz Cincio & Patrick J. Coles, 2021. "Noise-induced barren plateaus in variational quantum algorithms," Nature Communications, Nature, vol. 12(1), pages 1-11, December.
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