Fig. 4: The results of entanglement predicting of our neural network and comparisons with other protocols.

a The applications of our machine learning method in the model \({{\mathcal{H}}}_{{\rm{Long}}}\) and \({{\mathcal{H}}}_{2D}\). Here, \({S}_{A}^{th}\) is the real entropy of the half-length subsystem and \({S}_{A}^{ML}\) is the prediction result of our machine learning. b Comparison of our machine learning method with the randomized measurement toolbox proposed by Brydges et al.²⁴ and the classical shadows proposed by Huang et al.²² for predicting second-order entanglement entropy in 8-qubit ground states. Here, we consider the entanglement entropy of the subsystem A = {1, 2, 3, 4} and A = {3, 4, 5, 6, 7, 8}, respectively. It is well-known that machine learning methods fail to achieve 100%-accurate predictions and their accuracy is limited by the machine learning model. Thus, machine learning methods lose their advantage when the number of experiments is infinite, but in practice a finite number of experiments are scheduled.