Fig. 2: Performance of the CPD and neural network backflow on the Fermi-Hubbard model.

Percentage relative energy error for the ground state of the 4 × 4 square Fermi-Hubbard model at U/t = 8 compared to exact diagonalization results⁶⁹ at a a hole-doped filling of n = 0.875 and b half-filling. CPD backflow results (Ψ^CPD) are shown as a function of the configurational sample size in the optimization of the parameters, for two different model complexities of M = 1 (blue circles) and M = 2 (green circles). Neural network backflow results (NNB, red dashed lines) are taken from ref. ⁴⁴. CPD backflow energies are obtained as averages over 50 independent evaluations using the optimized parameters and a sample size of 2¹⁶, with error bars represented by the standard error across these evaluations.