Fig. 2: Performance of HF-TQS for different system sizes and couplings t/U.

a Difference between the HF-TQS ground state energy per electron and HF as a function of t/U at various system sizes N_e. The solid lines show the difference between ED and the HF ground state energy. The inset shows the absolute value of the relative error \(\delta {E}_{{{{\rm{ED}}}}}=\left\vert {E}_{{{{\rm{HF}}}}-{{{\rm{TQS}}}}}-{E}_{{{{\rm{ED}}}}}\right\vert\). The corresponding converged α values [according to Equation (14)] are shown in panel c. The gray regions indicate the vicinity of the metal-insulator transition. We fix n_U = 4 × 10³ in this region to highlight how this parameter controls the accessible corrections. Therefore, the break in trend for the corrections at N_e≥14 highlights that a larger n_U is necessary to correctly capture them. To illustrate this point, the white circles in panels (a, d) for N_e = 14 were computed using n_U = 17,000. We refer the reader to the main text for more details. b Convergence of the ground state energy per electron and of α (panel c) during training for t/U = 0.16 and N_e = 30. The total number of unique states \({n}_{U}^{f}\) indicates how many states are retained by the Transformer from the initial value n_U determined in Fig. 1c. Training was performed on one NVIDIA H100 GPU with the displayed network hyperparameters as defined in Fig. 1b (see also Supplementary Note B6). The total number of network parameters is denoted by #θ.