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Area of Science:

  • Quantum Information Science
  • Quantum Machine Learning
  • Computational Physics

Background:

  • Understanding quantum neural network (QNN) training dynamics is crucial for advancing quantum information science.
  • Applications span physics, chemistry, and machine learning, necessitating efficient training methodologies.

Purpose of the Study:

  • To elucidate the late-time training dynamics of QNNs with a quadratic loss function.
  • To identify critical transitions and their impact on convergence speed.

Main Methods:

  • Describing QNN dynamics using generalized Lotka-Volterra equations.
  • Developing a non-perturbative analytical theory via a restricted Haar ensemble.
  • Mapping the Hessian to an effective Hamiltonian to analyze spectral properties.

Main Results:

  • A transcritical bifurcation transition was identified in QNN training dynamics.
  • Dynamics shift from frozen-kernel to frozen-error, exhibiting a duality between the quantum neural tangent kernel and total error.
  • Polynomial convergence at the critical point contrasts with exponential convergence in other regions.
  • A linearly vanishing gap at the transition point was detected.

Conclusions:

  • Quadratic loss functions offer a training speedup compared to linear loss functions within frozen-error dynamics.
  • The theoretical findings are experimentally validated on IBM quantum devices, confirming the model's predictive power.