Fig. 7: Dynamics in QNN in the example of XXZ model with different loss functions.
From: Dynamical transition in controllable quantum neural networks with large depth
In a and b, we show the dynamics of residual error ε(t) (equals to total error ϵ(t)) and QNTK K(t) optimized with linear loss function (black solid) and quadratic loss functions with different O0. O0 = − 22 (green) corresponds to \({O}_{0}={O}_{\min }\) at critical point and O0 = −26, −30 (red and blue) correspond to \({O}_{0} < {O}_{\min }\) in frozen error dynamics. Black dashed line indicates the exponential decay rate of the theoretical result in Eq. (25). Thin lines with light colors represent dynamics with different initializations in each case, while the thick lines represent the ensemble average. Here random Pauli ansatz (RPA) consists of L = 192 variational parameters (D = L layers) on n = 6 qubits, and the parameter of XXZ model is J = 2.
