Fig. 5: Left plot: phase diagram of \({{\mathcal{H}}}_{{\rm{XXZ}}}\) and \({{\mathcal{H}}}_{{\rm{XX}}}\) models.

a, e are the ground energy levels (red lines). The energy levels of the first-excited state are also included (blue lines). b–d, f–h show the entanglement \({\mathcal{E}}\) of the ground states of \({{\mathcal{H}}}_{{\rm{XXZ}}}\) and \({{\mathcal{H}}}_{{\rm{XX}}}\) models, respectively. Theoretical calculation (solid lines), quantum FST (red points), and our neural network results (blue points) are distinguished. The measured subsystems are represented by the gray rounds schematic. Right plot: The dynamical evolution \({{\mathcal{S}}}^{(2)}(t)\) (the subsystem A = 12), \({{\mathcal{P}}}_{3}(t)\) (the subsystems A = 1, B = 2), and \({\mathcal{C}}(t)\) (the subsystem A = 12) for two sets of parameters (i) J = −0.5, g = −0.3 and (ii) J = −0.5, g = −0.75. The input layer only contains the measured single-qubit time traces in [0, π]. The trained model allows us to predict \({\mathcal{E}}(t)\) in the training window [0, π] and in the unseen future time window [π, 2π]. The theoretical results (solid lines), quantum FST results (square dots), and predicted results (diamond dots), for the two setups, are each represented by a different color of identification line (or dot).