Fig. 1: Schematic of the bipartite reweight-annealing process.

Direct evaluation of the ratio \(\overline{Z}({J}_{0})/Z({J}_{0})\) is generally infeasible, since the corresponding weight distributions at J₀ exhibit a lack of overlap, as sketched in (a). Instead, two independent reweight-annealing (RA) processes are performed: the red path corresponds to \(\overline{Z}(J)/\overline{Z}({J}_{0})\), and the blue path corresponds to Z(J)/Z(J₀), as indicated by the colored RA process labels. Along each path, the system is gradually evolved in the annealing direction through small parameter shifts \({J}^{{\prime} }=J+\delta J\), ensuring sufficient overlap between adjacent distributions, as illustrated in (b) and (c). Once the annealing paths reach an easily solvable reference point, the target ratio \(\overline{Z}({J}_{0})/Z({J}_{0})\) can be reconstructed.