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.

Parameterized quantum circuits can be used as quantum neural networks and have the potential to outperform their classical counterparts when trained for addressing learning problems. To date, much of the results on their performance on practical problems are heuristic in nature. In particular, the convergence rate for the training of quantum neural networks is not fully understood. Here, we analyze the dynamics of gradient descent for the training error of a class of variational quantum machine learning models. We define wide quantum neural networks as parameterized quantum circuits in the limit of a large number of qubits and variational parameters. We then find a simple analytic formula that captures the average behavior of their loss function and discuss the consequences of our findings. For example, for random quantum circuits, we predict and characterize an exponential decay of the residual training error as a function of the parameters of the system. We finally validate our analytic results with numerical experiments.

The idea of using quantum computers for machine learning has recently received attention both in academia and industry [1-13]. While proof of principle study have shown that some problems of mathematical interest quantum computers are useful [13], quantum advantage in machine learning algorithms for practical applications is still unclear [14]. On classical architectures, a first-principle theory of machine learning, especially the so-called deep learning that uses a large number of layers, is still in development. Early developments of the statistical learning theory provide rigorous guarantees on the learning capability in generic learning algorithms, but theoretical bounds obtained from information theory are sometimes weak in practical settings. The theory of neural tangent kernel (NTK) has been deemed an important tool to understand deep neural networks [15-21]. In the large-width limit, a generic neural network becomes nearly Gaussian when averaging over the initial weights and biases, and the learning capabilities become predictable. The NTK theory allows to derive analytical understanding of the neural networks dynamics, improving on statistical learning theory and shedding light on the underlying principle of deep learning [22-26]. In the quantum machine learning community, a similar first principle theory would help in understanding the training dynamics and selecting appropri-junyuliu@uchicago.edu