  Understanding the training dynamics of quantum neural networks is a
fundamental task in quantum information science with wide impact in physics,
chemistry and machine learning. In this work, we show that the late-time
training dynamics of quantum neural networks can be described by the
generalized Lotka-Volterra equations, which lead to a dynamical phase
transition. When the targeted value of cost function crosses the minimum
achievable value from above to below, the dynamics evolve from a frozen-kernel
phase to a frozen-error phase, showing a duality between the quantum neural
tangent kernel and the total error. In both phases, the convergence towards the
fixed point is exponential, while at the critical point becomes polynomial. Via
mapping the Hessian of the training dynamics to a Hamiltonian in the imaginary
time, we reveal the nature of the phase transition to be second-order with the
exponent $\nu=1$, where scale invariance and closing gap are observed at
critical point. We also provide a non-perturbative analytical theory to explain
the phase transition via a restricted Haar ensemble at late time, when the
output state approaches the steady state. The theory findings are verified
experimentally on IBM quantum devices.
